Frequently Asked Questions

In

Quantitative Finance

Frequently Asked Questions

In

Quantitative Finance

Including key models, important formulæ,

common contracts, a history of quantitative

finance, sundry lists, brainteasers and more

www.wilmott.com

Paul Wilmott

Copyright  2007 Paul Wilmott.

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To my parents

Contents

Preface

xiii

1 Quantitative Finance Timeline

1

2 FAQs

19

3 The Most Popular Probability Distributions

and Their Uses in Finance

231

4 Ten Different Ways to Derive Black–Scholes

251

5 Models and Equations

275

6 The Black–Scholes Formulæ and the Greeks

299

7 Common Contracts

305

8 Popular Quant Books

327

9 The Most Popular Search Words and Phrases

on Wilmott.com

341

10 Brainteasers

349

11 Paul & Dominic’s Guide to Getting

a Quant Job

391

Frequently Asked

Questions

1. What are the different types of Mathematics

found in Quantitative Finance?

20

2. What is arbitrage?

25

3. What is put-call parity?

28

4. What is the central limit theorem and what

are its implications for finance?

31

5. How is risk defined in mathematical terms?

36

6. What is value at risk and how is it used?

40

7. What is CrashMetrics?

44

8. What is a coherent risk measure and what

are its properties?

48

9. What is Modern Portfolio Theory?

51

10. What is the Capital Asset Pricing Model?

54

11. What is Arbitrage Pricing Theory?

58

12. What is Maximum Likelihood Estimation?

61

13. What is cointegration?

67

14. What is the Kelly criterion?

70

15. Why Hedge?

73

16. What is marketing to market and how does it

affect risk management in derivatives trading? 79

17. What is the Efficient Markets Hypothesis?

83

x FREQUENTLY ASKED QUESTIONS

18. What are the most useful performance

measures?

87

19. What is a utility function and how is it used?

90

20. What is Brownian Motion and what are its

uses in finance?

94

21. What is Jensen’s Inequality and what is its

role in finance?

97

22. What is Itˆ

o’s lemma?

100

23. Why does risk-neutral valuation work?

103

24. What is Girsanov’s theorem and why is it

important in finance?

107

25. What are the ‘greeks’?

110

26. Why do quants like closed-form solutions?

116

27. What are the forward and backward

equations?

119

28. Which numerical method should I use and

when?

123

29. What is Monte Carlo Simulation?

132

30. What is the finite-difference method?

136

31. What is a jump-diffusion model and how does

it affect option values?

142

32. What is meant by ‘complete’ and ‘incomplete’

markets?

146

33. What is volatility?

151

34. What is the volatility smile?

157

35. What is GARCH?

164

36. How do I dynamically hedge?

170

37. What is dispersion trading?

176

FREQUENTLY ASKED QUESTIONS

xi

38. What is bootstrapping using discount factors? 179

39. What is the LIBOR Market Model and its

principle applications in finance?

183

40. What is meant by the ‘value’ of a contract?

188

41. What is calibration?

191

42. What is the market price of risk?

194

43. What is the difference between the

equilibrium approach and the no-arbitrage

approach to modelling?

198

44. How good is the assumption of normal

distributions for financial returns?

201

45. How robust is the Black–Scholes model?

206

46. Why is the lognormal distribution important? 209

47. What are copulas and how are they used in

quantitative finance?

212

48. What is the asymptotic analysis and how is

it used in financial modelling?

216

49. What is a free-boundary problem and what is

the optimal-stopping time for an American

option?

220

50. What are low discrepancy numbers?

225

Preface

xiv PREFACE

This book grew out of a suggestion by wilmott.com Mem-

ber ‘bayes’ for a Forum (as in ‘internet discussion

group’) dedicated to gathering together answers to

the most common quanty questions. We responded

positively, as is our wont, and the Wilmott Quantita-

tive Finance FAQs Project was born. This Forum may

be found at www.wilmott.com/faq. (There anyone may

read the FAQ answers, but to post a message you must

be a member. Fortunately, this is entirely free!) The

FAQs project is one of the many collaborations between

Members of wilmott.com.

As well as being an ongoing online project, the FAQs

have inspired the book you are holding. It includes

FAQs and their answers and also sections on common

models and formulæ, many different ways to derive the

Black-Scholes model, the history of quantitative finance,

a selection of brainteasers and a couple of sections for

those who like lists (there are lists of the most popular

quant books and search items on wilmott.com). Right at

the end is an excerpt from Paul and Dominic’s Guide to

Getting a Quant Job, this will be of interest to those of

you seeking their first quant role.

FAQs in QF is not a shortcut to an in-depth knowledge

of quantitative finance. There is no such shortcut. How-

ever, it will give you tips and tricks of the trade, and

insight, to help you to do your job or to get you through

initial job interviews. It will serve as an aide memoire

to fundamental concepts (including why theory and

practice diverge) and some of the basic Black–Scholes

formulæ and greeks. The subject is forever evolving,

and although the foundations are fairly robust and

static there are always going to be new products and

models. So, if there are questions you would like to see

answered in future editions please drop me an email at

paul@wilmott.com.

PREFACE

xv

I would like to thank all Members of the forum for their

participation and in particular the following, more pro-

lific, Members for their contributions to the online FAQs

and Brainteasers: Aaron, adas, Alan, bayes, Cuchulainn,

exotiq, HA, kr, mj, mrbadguy, N, Omar, reza, Waagh-

Bakri and zerdna. Thanks also to DCFC for his advice

concerning the book.

I am grateful to Caitlin Cornish, Emily Pears, Graham

Russel, Jenny McCall, Sarah Stevens, Steve Smith, Tom

Clark and Viv Wickham at John Wiley & Sons Ltd for

their continued support, and to Dave Thompson for his

entertaining cartoons.

I am also especially indebted to James Fahy for making

the Forum happen and run smoothly.

Mahalo and aloha to my ever-encouraging wife, Andrea.

About the author

Paul Wilmott is one of the most well-known names in

derivatives and risk management. His academic and

practitioner credentials are impeccable, having writ-

ten over 100 research papers on mathematics and

finance, and having been a partner in a highly prof-

itable volatility arbitrage hedge fund. Dr Wilmott is a

consultant, publisher, author and trainer, the propri-

etor of wilmott.com and the founder of the Certificate in

Quantitative Finance (7city.com/cqf). He is the Editor in

Chief of the bimonthly quant magazine Wilmott and the

author of the student text Paul Wilmott Introduces Quan-

titative Finance, which covers classical quant finance

from the ground up, and Paul Wilmott on Quantitative

Finance, the three-volume research-level epic. Both are

also published by John Wiley & Sons.

Chapter 1

The Quantitative

Finance Timeline

2 Frequently Asked Questions In Quantitative Finance

T here follows a speedy, roller-coaster of a ride

through the history of quantitative finance, passing

through both the highs and lows. Where possible I give

dates, name names and refer to the original sources.1

1827 Brown

The Scottish botanist, Robert Brown, gave

his name to the random motion of small particles in a

liquid. This idea of the random walk has permeated

many scientific fields and is commonly used as the

model mechanism behind a variety of unpredictable

continuous-time processes. The lognormal random walk

based on Brownian motion is the classical paradigm for

the stock market. See Brown (1827).

1900 Bachelier

Louis Bachelier was the first to quantify

the concept of Brownian motion. He developed a mathe-

matical theory for random walks, a theory rediscovered

later by Einstein. He proposed a model for equity prices,

a simple normal distribution, and built on it a model

for pricing the almost unheard of options. His model

contained many of the seeds for later work, but lay

‘dormant’ for many, many years. It is told that his thesis

was not a great success and, naturally, Bachelier’s work

was not appreciated in his lifetime. See Bachelier (1995).

1905 Einstein

Albert Einstein proposed a scientific foun-

dation for Brownian motion in 1905. He did some other

clever stuff as well. See Stachel (1990).

1911 Richardson

Most option models result in diffusion-

type equations. And often these have to be solved

numerically. The two main ways of doing this are Monte

1A version of this chapter was first published in New Direc-

tions in Mathematical Finance, edited by Paul Wilmott and Hen-

rik Rasmussen, John Wiley & Sons, 2002.

Chapter 1: Quantitative Finance Timeline

3

Carlo and finite differences (a sophisticated version of

the binomial model). The very first use of the finite-

difference method, in which a differential equation is

discretized into a difference equation, was by Lewis

Fry Richardson in 1911, and used to solve the dif-

fusion equation associated with weather forecasting.

See Richardson (1922). Richardson later worked on the

mathematics for the causes of war.

1923 Wiener

Norbert Wiener developed a rigorous the-

ory for Brownian motion, the mathematics of which was

to become a necessary modelling device for quantita-

tive finance decades later. The starting point for almost

all financial models, the first equation written down in

most technical papers, includes the Wiener process as

the representation for randomness in asset prices. See

Wiener (1923).

1950s Samuelson

The 1970 Nobel Laureate in Economics,

Paul Samuelson, was responsible for setting the tone

for subsequent generations of economists. Samuelson

‘mathematized’ both macro and micro economics. He

rediscovered Bachelier’s thesis and laid the foundations

for later option pricing theories. His approach to deriva-

tive pricing was via expectations, real as opposed to the

much later risk-neutral ones. See Samuelson (1995).

1951 Itô

Where would we be without stochastic or Itô

calculus? (Some people even think finance is only about

Itô calculus.) Kiyosi Itô showed the relationship between

a stochastic differential equation for some independent

variable and the stochastic differential equation for a

function of that variable. One of the starting points for

classical derivatives theory is the lognormal stochastic

differential equation for the evolution of an asset. Itô’s

lemma tells us the stochastic differential equation for

the value of an option on that asset.

4 Frequently Asked Questions In Quantitative Finance

In mathematical terms, if we have a Wiener process

X with increments dX that are normally distributed

with mean zero and variance dt then the increment of a

function F ( X) is given by

dF

d 2 F

dF =

dX + 1

dt

dX

2 dX 2

This is a very loose definition of Itô’s lemma but will

suffice. See Itô (1951).

1952 Markowitz

Harry Markowitz was the first to pro-

pose a modern quantitative methodology for portfolio

selection. This required knowledge of assets’ volatili-

ties and the correlation between assets. The idea was

extremely elegant, resulting in novel ideas such as

‘efficiency’ and ‘market portfolios.’ In this Modern Port-

folio Theory, Markowitz showed that combinations of

assets could have better properties than any individual

assets. What did ‘better’ mean? Markowitz quantified a

portfolio’s possible future performance in terms of its

expected return and its standard deviation. The latter

was to be interpreted as its risk. He showed how to opti-

mize a portfolio to give the maximum expected return

for a given level of risk. Such a portfolio was said to be

‘efficient.’ The work later won Markowitz a Nobel Prize

for Economics but is rarely used in practice because of

the difficulty in measuring the parameters volatility, and

especially correlation, and their instability.

1963 Sharpe, Lintner and Mossin

William Sharpe of Stanford,

John Lintner of Harvard and Norwegian economist Jan

Mossin independently developed a simple model for

pricing risky assets. This Capital Asset Pricing Model

(CAPM) also reduced the number of parameters needed

for portfolio selection from those needed by Markowitz’s

Modern Portfolio Theory, making asset allocation theory

more practical. See Sharpe (1963), Lintner (1963) and

Mossin (1963).

Chapter 1: Quantitative Finance Timeline

5

1966 Fama

Eugene Fama concluded that stock prices

were unpredictable and coined the phrase ‘‘market effi-

ciency.’’ Although there are various forms of market

efficiency, in a nutshell the idea is that stock market

prices reflect all publicly available information, that no

person can gain an edge over another by fair means.

See Fama (1966).

1960s Sobol’, Faure, Hammersley, Haselgrove, Halton. . .

Many

people were associated with the definition and devel-

opment of quasi random number theory or low-

discrepancy sequence theory. The subject concerns the

distribution of points in an arbitrary number of dimen-

sions so as to cover the space as efficiently as possible,

with as few points as possible. The methodology is

used in the evaluation of multiple integrals among other

things. These ideas would find a use in finance almost

three decades later. See Sobol’ (1967), Faure (1969),

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1-1: They may not look like it, but these dots are distributed

deterministically so as to have very useful properties.

6 Frequently Asked Questions In Quantitative Finance

Hammersley and Handscomb (1964), Haselgrove (1961)

and Halton (1960).

1968 Thorp

Ed Thorp’s first claim to fame was that he

figured out how to win at casino Blackjack, ideas that

were put into practice by Thorp himself and written

about in his best-selling Beat the Dealer, the ‘‘book that

made Las Vegas change its rules.’’ His second claim to

fame is that he invented and built, with Claude Shannon,

the information theorist, the world’s first wearable com-

puter. His third claim to fame is that he was the first to

use the ‘correct’ formulæ for pricing options, formulæ

that were rediscovered and originally published several

years later by the next three people on our list. Thorp

used these formulæ to make a fortune for himself and

his clients in the first ever quantitative finance-based

hedge fund. See Thorp (2002) for the story behind the

discovery of the Black–Scholes formulæ.

1973 Black, Scholes and Merton

Fischer Black, Myron

Scholes and Robert Merton derived the Black–Scholes

equation for options in the early seventies, publish-

ing it in two separate papers in 1973 (Black & Scholes,

1973, and Merton, 1973). The date corresponded almost

exactly with the trading of call options on the Chicago

Board Options Exchange. Scholes and Merton won the

Nobel Prize for Economics in 1997. Black had died

in 1995.

The Black–Scholes model is based on geometric Brown-

ian motion for the asset price S

dS = µS dt + σ S dX.

The Black–Scholes partial differential equation for the

value V of an option is then

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ rS

− rV = 0 .

∂t

2

∂S 2

∂S

Chapter 1: Quantitative Finance Timeline

7

1974 Merton, again

In 1974 Robert Merton (Merton, 1974)

introduced the idea of modelling the value of a company

as a call option on its assets, with the company’s debt

being related to the strike price and the maturity of

the debt being the option’s expiration. Thus was born

the structural approach to modelling risk of default,

for if the option expired out of the money (i.e. assets

had less value than the debt at maturity) then the firm

would have to go bankrupt.

Credit risk became big, huge, in the 1990s. Theory and

practice progressed at rapid speed during this period,

urged on by some significant credit-led events, such as

the Long Term Capital Management mess. One of the

principals of LTCM was Merton who had worked on

credit risk two decades earlier. Now the subject really

took off, not just along the lines proposed by Merton

but also using the Poisson process as the model for

the random arrival of an event, such as bankruptcy

or default. For a list of key research in this area see

Schönbucher (2003).

1977 Boyle

Phelim Boyle related the pricing of options

to the simulation of random asset paths. He showed

how to find the fair value of an option by generating lots

of possible future paths for an asset and then looking

at the average that the option had paid off. The future

important role of Monte Carlo simulations in finance

was assured. See Boyle (1977).

1977 Vasicek

So far quantitative finance hadn’t had much

to say about pricing interest rate products. Some people

were using equity option formulæ for pricing interest

rate options, but a consistent framework for interest

rates had not been developed. This was addressed by

Vasicek. He started by modelling a short-term interest

rate as a random walk and concluded that interest rate

8 Frequently Asked Questions In Quantitative Finance

160

140

120

100

80

Asset

60

40

20

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time

Figure 1-2: Simulations like this can be easily used to value

derivatives.

derivatives could be valued using equations similar to

the Black–Scholes partial differential equation.

Oldrich Vasicek represented the short-term interest rate

by a stochastic differential equation of the form

dr = µ( r, t) dt + σ ( r, t) dX.

The bond pricing equation is a parabolic partial differ-

ential equation, similar to the Black–Scholes equation.

See Vasicek (1977).

1979 Cox, Ross, Rubinstein

Boyle had shown how to price

options via simulations, an important and intuitively rea-

sonable idea, but it was these three, John Cox, Stephen

Ross and Mark Rubinstein, who gave option pricing

capability to the masses.

The Black–Scholes equation was derived using stochas-

tic calculus and resulted in a partial differential

equation. This was not likely to endear it to the thou-

sands of students interested in a career in finance. At

Chapter 1: Quantitative Finance Timeline

9

uS

S

δ t

vS

Figure 1-3: The branching structure of the binomial model.

that time these were typically MBA students, not the

mathematicians and physicists that are nowadays found

on Wall Street. How could MBAs cope? An MBA was

a necessary requirement for a prestigious career in

finance, but an ability to count beans is not the same as

an ability to understand mathematics. Fortunately Cox,

Ross and Rubinstein were able to distil the fundamen-

tal concepts of option pricing into a simple algorithm

requiring only addition, subtraction, multiplication and

(twice) division. Even MBAs could now join in the fun.

See Cox, Ross and Rubinstein (1979).

1979–81 Harrison, Kreps, Pliska

Until these three came

onto the scene quantitative finance was the domain of

either economists or applied mathematicians. Mike Har-

rison and David Kreps, in 1979, showed the relationship

between option prices and advanced probability theory,

originally in discrete time. Harrison and Stan Pliska in

1981 used the same ideas but in continuous time. From

that moment until the mid 1990s applied mathemati-

cians hardly got a look in. Theorem, proof everywhere

10 Frequently Asked Questions In Quantitative Finance

you looked. See Harrison and Kreps (1979) and Harrison

and Pliska (1981).

1986 Ho and Lee

One of the problems with the Vasicek

framework for interest rate derivative products was that

it didn’t give very good prices for bonds, the simplest

of fixed income products. If the model couldn’t even

get bond prices right, how could it hope to correctly

value bond options? Thomas Ho and Sang-Bin Lee found

a way around this, introducing the idea of yield curve

fitting or calibration. See Ho and Lee (1986).

1992 Heath, Jarrow and Morton

Although Ho and Lee

showed how to match theoretical and market prices for

simple bonds, the methodology was rather cumbersome

and not easily generalized. David Heath, Robert Jarrow

and Andrew Morton took a different approach. Instead

of modelling just a short rate and deducing the whole

yield curve, they modelled the random evolution of the

whole yield curve. The initial yield curve, and hence the

value of simple interest rate instruments, was an input

to the model. The model cannot easily be expressed

in differential equation terms and so relies on either

Monte Carlo simulation or tree building. The work was

well known via a working paper, but was finally pub-

lished, and therefore made respectable in Heath, Jarrow

and Morton (1992).

1990s Cheyette, Barrett, Moore, Wilmott

When there are

many underlyings, all following lognormal random walks

you can write down the value of any European non

path-dependent option as a multiple integral, one dimen-

sion for each asset. Valuing such options then becomes

equivalent to calculating an integral. The usual methods

for quadrature are very inefficient in high dimensions,

but simulations can prove quite effective. Monte Carlo

evaluation of integrals is based on the idea that an inte-

gral is just an average multiplied by a ‘volume.’ And

Chapter 1: Quantitative Finance Timeline

11

since one way of estimating an average is by picking

numbers at random we can value a multiple integral

by picking integrand values at random and summing.

With N function evaluations, taking a time of O( N) you

can expect an accuracy of O(1 /N 1 / 2), independent of

the number of dimensions. As mentioned above, break-

throughs in the 1960s on low-discrepancy sequences

showed how clever, non-random, distributions could

be used for an accuracy of O(1 /N), to leading order.

(There is a weak dependence on the dimension.) In

the early 1990s several groups of people were simul-

taneously working on valuation of multi-asset options.

Their work was less of a breakthrough than a transfer

of technology.

They used ideas from the field of number theory

and applied them to finance. Nowadays, these low-

discrepancy sequences are commonly used for option

valuation whenever random numbers are needed. A few

years after these researchers made their work public,

a completely unrelated group at Columbia University

successfully patented the work. See Oren Cheyette

(1990) and John Barrett, Gerald Moore and Paul Wilmott

(1992).

1994 Dupire, Rubinstein, Derman and Kani

Another discovery

was made independently and simultaneously by three

groups of researchers in the subject of option pricing

with deterministic volatility. One of the perceived prob-

lems with classical option pricing is that the assumption

of constant volatility is inconsistent with market prices

of exchange-traded instruments. A model is needed that

can correctly price vanilla contracts, and then price

exotic contracts consistently. The new methodology,

which quickly became standard market practice, was

to find the volatility as a function of underlying and

time that when put into the Black–Scholes equation and

solved, usually numerically, gave resulting option prices

12 Frequently Asked Questions In Quantitative Finance

which matched market prices. This is what is known as

an inverse problem, use the ‘answer’ to find the coeffi-

cients in the governing equation. On the plus side, this

is not too difficult to do in theory. On the minus side the

practice is much harder, the sought volatility function

depending very sensitively on the initial data. From a

scientific point of view there is much to be said against

the methodology. The resulting volatility structure never

matches actual volatility, and even if exotics are priced

consistently it is not clear how to best hedge exotics

with vanillas so as to minimize any model error. Such

concerns seem to carry little weight, since the method

is so ubiquitous. As so often happens in finance, once a

technique becomes popular it is hard to go against the

majority. There is job safety in numbers. See Emanuel

Derman and Iraj Kani (1994), Bruno Dupire (1994) and

Mark Rubinstein (1994).

1996 Avellaneda and Parás

Marco Avellaneda and Anto-

nio Parás were, together with Arnon Levy and Terry

Lyons, the creators of the uncertain volatility model

for option pricing. It was a great breakthrough for the

rigorous, scientific side of finance theory, but the best

was yet to come. This model, and many that succeeded

it, was non linear. Nonlinearity in an option pricing

model means that the value of a portfolio of contracts

is not necessarily the same as the sum of the values

of its constituent parts. An option will have a different

value depending on what else is in the portfolio with it,

and an exotic will have a different value depending on

what it is statically hedged with. Avellaneda and Parás

defined an exotic option’s value as the highest possible

marginal value for that contract when hedged with any

or all available exchange-traded contracts. The result

was that the method of option pricing also came with

its own technique for static hedging with other options.

Chapter 1: Quantitative Finance Timeline

13

Prior to their work the only result of an option pricing

model was its value and its delta, only dynamic hedging

was theoretically necessary. With this new concept,

theory became a major step closer to practice. Another

result of this technique was that the theoretical price

of an exchange-traded option exactly matched its mar-

ket price. The convoluted calibration of volatility surface

models was redundant. See Avellaneda and Parás (1996).

1997 Brace, Gatarek and Musiela

Although the HJM inter-

est rate model had addressed the main problem with

stochastic spot rate models, and others of that ilk, it

still had two major drawbacks. It required the existence

of a spot rate and it assumed a continuous distribution

of forward rates. Alan Brace, Dariusz Gatarek and Marek

Musiela (1997) got around both of those difficulties by

introducing a model which only relied on a discrete set

of rates, ones that actually are traded. As with the HJM

model the initial data are the forward rates so that bond

prices are calibrated automatically. One specifies a num-

ber of random factors, their volatilities and correlations

between them, and the requirement of no arbitrage then

determines the risk-neutral drifts. Although B, G and M

have their names associated with this idea many others

worked on it simultaneously.

2000 Li

As already mentioned, the 1990s saw an explo-

sion in the number of credit instruments available,

and also in the growth of derivatives with multiple

underlyings. It’s not a great step to imagine contracts

depending of the default of many underlyings. Examples

of these are the ubiquitous Collateralized Debt Obliga-

tions (CDOs). But to price such complicated instruments

requires a model for the interaction of many com-

panies during the process of default. A probabilistic

approach based on copulas was proposed by David Li

14 Frequently Asked Questions In Quantitative Finance

(2000). The copula approach allows one to join together

(hence the word ‘copula’) default models for individual

companies in isolation to make a model for the proba-

bilities of their joint default. The idea has been adopted

universally as a practical solution to a complicated

problem.

2002 Hagan, Kumar, Lesniewski, Woodward

There has always

been a need for models that are both fast and match

traded prices well. The interest-rate model of Pat Hagan,

Deep Kumar, Andrew Lesniewski & Diana Woodward

(2002) which has come to be called the SABR (stochas-

tic, α, β, ρ) model is a model for a forward rate and its volatility, both of which are stochastic. This model is

made tractable by exploiting an asymptotic approxima-

tion to the governing equation that is highly accurate in

practice. The asymptotic analysis simplifies a problem

that would otherwise have to be solved numerically.

Although asymptotic analysis has been used in financial

problems before, for example in modelling transaction

costs, this was the first time it really entered main-

stream quantitative finance.

References and Further Reading

Avellaneda, M, Levy, A & Parás, A 1995 Pricing and hedging

derivative securities in markets with uncertain volatilities.

Applied Mathematical Finance 2 73–88

Avellaneda, M & Parás, A 1994 Dynamic hedging portfolios

for derivative securities in the presence of large transaction

costs. Applied Mathematical Finance 1 165–194

Avellaneda, M & Parás, A 1996 Managing the volatility risk of

derivative securities: the Lagrangian volatility model. Applied

Mathematical Finance 3 21–53

Chapter 1: Quantitative Finance Timeline

15

Avellaneda, M & Buff, R 1997 Combinatorial implications of

nonlinear uncertain volatility models: the case of barrier

options. Courant Institute, NYU

Bachelier, L 1995 Théorie de la Spéculation. Jacques Gabay

Barrett, JW, Moore, G & Wilmott, P 1992 Inelegant efficiency.

Risk magazine 5 (9) 82–84

Black, F & Scholes, M 1973 The pricing of options and corpo-

rate liabilities. Journal of Political Economy 81 637–59

Boyle, P 1977 Options: a Monte Carlo approach. Journal of

Financial Economics 4 323–338

Brace, A, Gatarek, D & Musiela, M 1997 The market model of

interest rate dynamics. Mathematical Finance 7 127–154

Brown, R 1827 A Brief Account of Microscopical Observations.

London

Cheyette, O 1990 Pricing options on multiple assets. Adv. Fut.

Opt. Res. 4 68–91

Cox, JC, Ross, S & Rubinstein M 1979 Option pricing: a simpli-

fied approach. Journal of Financial Economics 7 229–263

Derman, E, Ergener, D & Kani, I 1997 Static options replication.

In Frontiers in Derivatives. (Ed. Konishi, A & Dattatreya, RE)

Irwin

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7

(2) 32–39 (February)

Dupire, B 1993 Pricing and hedging with smiles. Proc AFFI

Conf, La Baule June 1993

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18–20

(January)

Fama, E 1965 The behaviour of stock prices. Journal of Business

38 34–105

Faure, H 1969 Résultat voisin d’un théreme de Landau sur le

nombre de points d’un réseau dans une hypersphere. C. R.

Acad. Sci. Paris S´

er. A 269 383–386

16 Frequently Asked Questions In Quantitative Finance

Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Man-

aging smile risk. Wilmott magazine, September

Halton, JH 1960 On the efficiency of certain quasi-random

sequences of points in evaluating multi-dimensional integrals.

Num. Maths. 2 84–90

Hammersley, JM & Handscomb, DC 1964 Monte Carlo Methods.

Methuen, London

Harrison, JM & Kreps, D 1979 Martingales and arbitrage in

multiperiod securities markets. Journal of Economic Theory

20 381–408

Harrison, JM & Pliska, SR 1981 Martingales and stochastic

integrals in the theory of continuous trading. Stochastic

Processes and their Applications 11 215–260

Haselgrove, CB 1961 A method for numerical integration. Math-

ematics of Computation 15 323–337

Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the

term structure of interest rates: a new methodology. Econo-

metrica 60 77–105

Ho, T & Lee, S 1986 Term structure movements and pric-

ing interest rate contingent claims. Journal of Finance 42

1129–1142

Itô, K 1951 On stochastic differential equations. Memoirs of the

Am. Math. Soc. 4 1–51

Li, DX 2000 On default correlation: a copula function approach.

RiskMetrics Group

Lintner, J 1965 Security prices, risk, and maximal gains from

diversification. Journal of Finance 20 587–615

Markowitz, H 1959 Portfolio Selection: efficient diversification of

investment. John Wiley www.wiley.com

Merton, RC 1973 Theory of rational option pricing. Bell Journal

of Economics and Management Science 4 141–83

Merton, RC 1974 On the pricing of corporate debt: the risk

structure of interest rates. Journal of Finance 29 449–70

Chapter 1: Quantitative Finance Timeline

17

Merton, RC 1992 Continuous-time Finance. Blackwell

Mossin, J 1966 Equilibrium in a capital asset market.

Econometrica 34 768–83

Niederreiter, H 1992 Random Number Generation and Quasi-

Monte Carlo Methods. SIAM

Ninomiya, S & Tezuka, S 1996 Toward real-time pricing of

complex financial derivatives. Applied Mathematical Finance

3 1–20

Paskov, SH 1996 New methodologies for valuing derivatives.

In Mathematics of Derivative Securities (Eds Pliska, SR and

Dempster, M)

Paskov, SH & Traub, JF 1995 Faster valuation of financial

derivatives. Journal of Portfolio Management Fall 113–120

Richardson, LF 1922 Weather Prediction by Numerical Process.

Cambridge University Press

Rubinstein, M 1994 Implied binomial trees. Journal of Finance

69 771–818

Samuelson, P 1955 Brownian motion in the stock market.

Unpublished

Schönbucher, PJ 2003 Credit Derivatives Pricing Models. John

Wiley & Sons

Sharpe, WF 1985 Investments. Prentice–Hall

Sloan, IH & Walsh, L 1990 A computer search of rank two

lattice rules for multidimensional quadrature. Mathematics of

Computation 54 281–302

Sobol’, IM 1967 On the distribution of points in cube and the

approximate evaluation of integrals. USSR Comp. Maths and

Math. Phys. 7 86–112

Stachel, J (ed.) 1990 The Collected Papers of Albert Einstein.

Princeton University Press

Thorp, EO 1962 Beat the Dealer. Vintage

Thorp, EO & Kassouf, S 1967 Beat the Market. Random House

18 Frequently Asked Questions In Quantitative Finance

Thorp, EO 2002 Wilmott magazine, various papers

Traub, JF & Wozniakowski, H 1994 Breaking intractability.

Scientific American Jan 102–107

Vasicek, OA 1977 An equilibrium characterization of the term

structure. Journal of Financial Economics 5 177–188

Wiener, N 1923 Differential space. J. Math. and Phys. 58 131–74

Chapter 2

FAQs

20 Frequently Asked Questions In Quantitative Finance

What are the Different Types of

Mathematics Found in Quantitative

Finance?

Short Answer

The fields of mathematics most used in quantitative

finance are those of probability theory and differen-

tial equations. And, of course, numerical methods are

usually needed for producing numbers.

Example

The classical model for option pricing can be writ-

ten as a partial differential equation. But the same

model also has a probabilistic interpretation in terms

of expectations.

Long Answer

The real-world subject of quantitative finance uses tools

from many branches of mathematics. And financial

modelling can be approached in a variety of different

ways. For some strange reason the advocates of differ-

ent branches of mathematics get quite emotional when

discussing the merits and demerits of their method-

ologies and those of their ‘opponents.’ Is this a terri-

torial thing, what are the pros and cons of martingales

and differential equations, what is all this fuss and will

it end in tears before bedtime?

Here’s a list of the various approaches to modelling

and a selection of useful tools. The distinction between a

‘modelling approach’ and a ‘tool’ will start to become clear.

Modelling approaches:

• Probabilistic

Chapter 2: FAQs

21

• Deterministic

• Discrete: difference equations

• Continuous: differential equations

Useful tools:

• Simulations

• Approximations

• Asymptotic analysis

• Series solutions

• Discretization methods

• Green’s functions

While these are not exactly arbitrary lists, they are

certainly open to some criticism or addition. Let’s first

take a look at the modelling approaches.

Probabilistic:

One of the main assumptions about the

financial markets, at least as far as quantitative finance

goes, is that asset prices are random. We tend to think

of describing financial variables as following some ran-

dom path, with parameters describing the growth of

the asset and its degree of randomness. We effectively

model the asset path via a specified rate of growth,

on average, and its deviation from that average. This

approach to modelling has had the greatest impact over

the last 30 years, leading to the explosive growth of the

derivatives markets.

Deterministic:

The idea behind this approach is that our

model will tell us everything about the future. Given

enough data, and a big enough brain, we can write

down some equations or an algorithm for predicting the

future. Interestingly, the subjects of dynamical systems

and chaos fall into this category. And, as you know,

chaotic systems show such sensitivity to initial condi-

tions that predictability is in practice impossible. This

22 Frequently Asked Questions In Quantitative Finance

is the ‘butterfly effect,’ that a butterfly flapping its wings

in Brazil will ‘cause’ rainfall over Manchester. (And what

doesn’t!) A topic popular in the early 1990s, this has not

lived up to its promises in the financial world.

Discrete/Continuous:

Whether probabilistic or determinis-

tic the eventual model you write down can be discrete

or continuous. Discrete means that asset prices and/or

time can only be incremented in finite chunks, whether

a dollar or a cent, a year or a day. Continuous means

that no such lower increment exists. The mathemat-

ics of continuous processes is often easier than that

of discrete ones. But then when it comes to number

crunching you have to anyway turn a continuous model

into a discrete one.

In discrete models we end up with difference equations.

An example of this is the binomial model for option

pricing. Time progresses in finite amounts, the time

step. In continuous models we end up with differential

equations. The equivalent of the binomial model in dis-

crete space is the Black–Scholes model, which has con-

tinuous asset price and continuous time. Whether bino-

mial or Black–Scholes, both of these models come from

the probabilistic assumptions about the financial world.

Now let’s take a look at some of the tools available.

Simulations:

If the financial world is random then we

can experiment with the future by running simulations.

For example, an asset price may be represented by

its average growth and its risk, so let’s simulate what

could happen in the future to this random asset. If

we were to take such an approach we would want to

run many, many simulations. There’d be little point in

running just the one, we’d like to see a range of possible

future scenarios.

Chapter 2: FAQs

23

Simulations can also be used for non-probabilistic prob-

lems. Just because of the similarities between mathe-

matical equations a model derived in a deterministic

framework may have a probabilistic interpretation.

Discretization methods:

The complement to simulation

methods, there are many types of these. The best known

of these are the finite-difference methods which are

discretizations of continuous models such as Black–

Scholes.

Depending on the problem you are solving, and unless

it’s very simple, you will probably go down the sim-

ulation or finite-difference routes for your number

crunching.

Approximations:

In modelling we aim to come up with a

solution representing something meaningful and use-

ful, such as an option price. Unless the model is really

simple, we may not be able to solve it easily. This is

where approximations come in. A complicated model

may have approximate solutions. And these approxi-

mate solutions might be good enough for our purposes.

Asymptotic analysis:

This is an incredibly useful technique,

used in most branches of applicable mathematics, but

until recently almost unknown in finance. The idea is

simple, find approximate solutions to a complicated

problem by exploiting parameters or variables that

are either large or small, or special in some way. For

example, there are simple approximations for vanilla

option values close to expiry.

Green’s functions:

This is a very special technique that

only works in certain situations. The idea is that solu-

tions to some difficult problems can be built up from

solutions to special solutions of a similar problem.

24 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Wilmott, P 2001 Paul Wilmott Introduces Quantitative Finance,

second edition. John Wiley & Sons

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

25

What is Arbitrage?

Short Answer

Arbitrage is making a sure profit in excess of the risk-

free rate of return. In the language of quantitative

finance we can say an arbitrage opportunity is a port-

folio of zero value today which is of positive value in the

future with positive probability and of negative value in

the future with zero probability.

The assumption that there are no arbitrage opportun-

ities in the market is fundamental to classical finance

theory. This idea is popularly known as ‘there’s no such

thing as a free lunch.’

Example

An at-the-money European call option with a strike of

$100 and an expiration of six months is worth $8. A

European put with the same strike and expiration is

worth $6. There are no dividends on the stock and a

six-month zero-coupon bond with a principal of $100 is

worth $97.

Buy the call and a bond, sell the put and the stock,

which will bring in $ − 8 − 97 + 6 + 100 = $1. At expira-

tion this portfolio will be worthless regardless of the

final price of the stock. You will make a profit of $1

with no risk. This is arbitrage. It is an example of the

violation of put-call parity.

Long Answer

The principle of no arbitrage is one of the founda-

tions of classical finance theory. In derivatives theory

it is assumed during the derivation of the binomial

model option pricing algorithm and in the Black–Scholes

model. In these cases it is rather more complicated than

the simple example given above. In the above example

26 Frequently Asked Questions In Quantitative Finance

we set up a portfolio that gave us an immediate profit,

and that portfolio did not have to be touched until

expiration. This is a case of a static arbitrage. Another

special feature of the above example is that it does not

rely on any assumptions about how the stock price

behaves. So the example is that of model-independent

arbitrage. However, when deriving the famous option-

pricing models we rely on a dynamic strategy, called

delta hedging, in which a portfolio consisting of an

option and stock is constantly adjusted by purchase

or sale of stock in a very specific manner.

Now we can see that there are several types of arbitrage

that we can think of. Here is a list and description of the

most important.

• A static arbitrage is an arbitrage that does not

require rebalancing of positions

• A dynamic arbitrage is an arbitrage that requires

trading instruments in the future, generally contingent

on market states

• A statistical arbitrage is not an arbitrage but simply a

likely profit in excess of the risk-free return (perhaps

even suitably adjusted for risk taken) as predicted by

past statistics

• Model-independent arbitrage is an arbitrage which

does not depend on any mathematical model of

financial instruments to work. For example, an

exploitable violation of put-call parity or a violation of

the relationship between spot and forward prices, or

between bonds and swaps

• Model-dependent arbitrage does require a model. For

example, options mispriced because of incorrect

volatility estimate. To profit from the arbitrage you

need to delta hedge and that requires a model

Not all apparent arbitrage opportunities can be exploited

in practice. If you see such an opportunity in quoted

Chapter 2: FAQs

27

prices on a screen in front of you then you are likely to

find that when you try to take advantage of them they

just evaporate. Here are several reasons for this.

• Quoted prices are wrong or not tradeable

• Option and stock prices were not quoted

synchronously

• There is a bid-offer spread you have not accounted

for

• Your model is wrong, or there is a risk factor you

have not accounted for

References and Further Reading

Merton, RC 1973 Theory of rational option pricing. Bell Journal

of Economics and Management Science 4 141–83

Wilmott, P 2001 Paul Wilmott Introduces Quantitative Finance.

John Wiley & Sons

28 Frequently Asked Questions In Quantitative Finance

What is Put-Call Parity?

Short Answer

Put-call parity is a relationship between the prices of

a European-style call option and a European-style put

option, as long as they have the same strike and expira-

tion:

Call price − Put price = Stock price

− Strike price (present valued from expiration) .

Example:

Stock price is $98, a European call option

struck at $100 with an expiration of nine months has

a value of $9.07. The nine-month, continuously com-

pounded, interest rate is 4.5%. What is the value of the

put option with the same strike and expiration?

By rearranging the above expression we find

Put price = 9 . 07 − 98 + 100 e−0 . 045×0 . 75 = 7 . 75 .

The put must therefore be worth $7.75.

Long Answer

This relationship,

C − P = S − K e− r( T− t),

between European calls (value C) and puts (value P)

with the same strike ( K) and expiration ( T ) valued at

time t is a result of a simple arbitrage argument. If you

buy a call option, at the same time write a put, and sell

stock short, what will your payoff be at expiration? If

the stock is above the strike at expiration you will have

S − K from the call, 0 from the put and − S from the

stock. A total of − K. If the stock is below the strike at

expiration you will have 0 from the call, − S again from

the stock, and −( K − S) from the short put. Again a total

of − K. So, whatever the stock price is at expiration this

Chapter 2: FAQs

29

portfolio will always be worth − K, a guaranteed amount.

Since this amount is guaranteed we can discount it back

to the present. We must have

C − P − S = − K e− r( T− t) .

This is put-call parity.

Another way of interpreting put-call parity is in terms

of implied volatility. Calls and puts with the same strike

and expiration must have the same implied volatility.

The beauty of put-call parity is that it is a model-

independent relationship. To value a call on its own

we need a model for the stock price, in particular its

volatility. The same is true for valuing a put. But to

value a portfolio consisting of a long call and a short

put (or vice versa), no model is needed. Such model-

independent relationships are few and far between in

finance. The relationship between forward and spot

prices is one, and the relationships between bonds and

swaps is another.

In practice options don’t have a single price, they have

two prices, a bid and an offer (or ask). This means

that when looking for violations of put-call parity you

must use bid (offer) if you are going short (long) the

options. This makes the calculations a little bit messier.

If you think in terms of implied volatility then it’s much

easier to spot violations of put-call parity. You must

look for non-overlapping implied volatility ranges. For

example, suppose that the bid/offer on a call is 22%/25%

in implied volatility terms and that on a put (same strike

and expiration) is 21%/23%. There is an overlap between

these two ranges (22–23%) and so no arbitrage opportu-

nity. However, if the put prices were 19%/21% then there

would be a violation of put-call parity and hence an easy

arbitrage opportunity. Don’t expect to find many (or,

indeed, any) of such simple free-money opportunities in

30 Frequently Asked Questions In Quantitative Finance

practice though. If you do find such an arbitrage then

it usually disappears by the time you put the trade on.

See Kamara & Miller (1995) for details of the statistics

of no-arbitrage violations.

When there are dividends on the underlying stock dur-

ing the life of the options then we must adjust the

equation to allow for this. We now find that

C − P = S − Present value of all dividends − E e− r( T− t) .

This, of course, assumes that we know what the divi-

dends will be.

If interest rates are not constant then just discount the

strike back to the present using the value of a zero-

coupon bond with maturity the same as expiration of

the option. Dividends should similarly be discounted.

When the options are American, put-call parity does

not hold. This is because the short position could be

exercised against you, leaving you with some exposure

to the stock price. Therefore you don’t know what you

will be worth at expiration. In the absence of dividends

it is theoretically never optimal to exercise an American

call before expiration, whereas an American put should

be exercised if the stock falls low enough.

References and Further Reading

Kamara, A & Millet, T 1995 Daily and Intradaily Tests of Euro-

pean Put-Call Parity. Journal of Financial and Quantitative

Analysis, December 519–539

Chapter 2: FAQs

31

What is the Central Limit Theorem

and What are its Implications for

Finance?

Short Answer

The distribution of the average of a lot of random num-

bers will be normal (also known as Gaussian) even

when the individual numbers are not normally dis-

tributed.

Example

Play a dice game where you win $10 if you throw a six,

but lose $1 if you throw anything else. The distribution

of your profit after one coin toss is clearly not normal,

it’s bimodal and skewed, but if you play the game thou-

sands of times your total profit will be approximately

normal.

Long Answer

Let X 1, X 2, . . . , Xn be a sequence of random variables which are independent and identically distributed (i.i.d.),

with finite mean, m and standard deviation s. The sum

n

Sn =

Xi

i=1

√

has mean mn and standard deviation s n. The Central

Limit Theorem says that as n gets larger the distribution

of Sn tends to the normal distribution. More accurately,

if we work with the scaled quantity

Sn − mn

Sn =

√

s n

then the distribution of Sn converges to the normal

distribution with zero mean and unit standard devia-

tion as n tends to infinity. The cumulative distribution

32 Frequently Asked Questions In Quantitative Finance

for Sn approaches that for the standardized normal

distribution.

In the next figure is the distribution for the above coin-

tossing experiment.

In the figure after is what your total profit will be like

after one thousand tosses.

Your expected profit after one toss is

1 × 10 + 5 × (−1) = 5 ≈ 0 . 833 .

6

6

6

Your variance is therefore

1

2

2

× 10 − 5

+ 5 × −1 − 5

= 605,

6

6

6

6

54

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-1

0

1

2

3

4

5

6

7

8

9

10

Figure 2-1: Probabilities in a simple coin-tossing experiment: one

toss.

Chapter 2: FAQs

33

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

0

-1000

-450

100

650

1200

1750

2300

2850

3400

3950

4500

5050

5600

6150

6700

7250

7800

8350

8900

9450

10000

Figure 2-2: Probabilities in a simple coin-tossing experiment: one

thousand tosses.

so a standard deviation of

605 / 54 ≈ 1 . 097. After one

thousand tosses your expected profit is

1000 × 5 ≈ 833 . 3

6

and your standard deviation is

1000 × 605 ≈ 34 . 7

54

See how the standard deviation has grown much less

than the expectation. That’s because of the square-

root rule.

In finance we often assume that equity returns are nor-

mally distributed. We could argue that this ought to be

34 Frequently Asked Questions In Quantitative Finance

the case by saying that returns over any finite period,

one day, say, are made up of many, many trades over

smaller time periods, with the result that the returns

over the finite timescale are normal thanks to the

Central Limit Theorem. The same argument could be

applied to the daily changes in exchange rate rates, or

interest rates, or risk of default, etc. We find ourselves

using the normal distribution quite naturally for many

financial processes.

As often with mathematical ‘laws’ there is the ‘legal’

small print, in this case the conditions under which the

Central Limit Theorem applies. These are as follows.

• The random numbers must all be drawn from the

same distribution

• The draws must all be independent

• The distribution must have finite mean and standard

deviation

Of course, financial data may not satisfy all of these,

or indeed, any. In particular, it turns out that if you

try to fit equity returns data with non-normal distribu-

tions you often find that the best distribution is one

that has infinite variance. Not only does it complicate

the nice mathematics of normal distributions and the

Central Limit Theorem, it also results in infinite volatil-

ity. This is appealing to those who want to produce the

best models of financial reality but does rather spoil

many decades of financial theory and practice based on

volatility as a measure of risk for example.

However, you can get around these three restrictions to

some extent and still get the Central Limit Theorem, or

something very much like it. For example, you don’t

need to have completely identical distributions. As

long as none of the random variables has too much

more impact on the average than the others then it

Chapter 2: FAQs

35

still works. You are even allowed to have some weak

dependence between the variables.

A generalization that is important in finance applies to

distributions with infinite variance. If the tails of the

individual distributions have a power-law decay, | x|−1− α

with 0 < α < 2 then the average will tend to a stable

Lévy distribution.

If you add random numbers and get normal, what hap-

pens when you multiply them? To answer this question

we must think in terms of logarithms of the random

numbers.

Logarithms of random numbers are themselves random

(let’s stay with logarithms of strictly positive numbers).

So if you add up lots of logarithms of random numbers

you will get a normal distribution. But, of course, a

sum of logarithms is just the logarithm of a product,

therefore the logarithm of the product must be normal,

and this is the definition of lognormal: the product of

positive random numbers converges to lognormal.

This is important in finance because a stock price after

a long period can be thought of as its value on some

starting day multiplied by lots of random numbers, each

representing a random return. So whatever the distribu-

tion of returns is, the logarithm of the stock price will

be normally distributed. We tend to assume that equity

returns are normally distributed, and equivalently, equi-

ties themselves are lognormally distributed.

References

Feller, W 1968 An Introduction to Probability Theory and Its

Applications. 3rd ed. New York, Wiley

36 Frequently Asked Questions In Quantitative Finance

How is Risk Defined in Mathematical

Terms?

Short Answer

In layman’s terms, risk is the possibility of harm or loss.

In finance it refers to the possibility of a monetary loss

associated with investments.

Example

The most common measure of risk is simply standard

deviation of portfolio returns. The higher this is, the

more randomness in a portfolio, and this is seen as a

bad thing.

Long Answer

Financial risk comes in many forms:

• Market risk: The possibility of loss due to movements

in the market, either as a whole or specific

investments

• Credit risk: The possibility of loss due to default on a

financial obligation

• Model risk: The possibility of loss due to errors in

mathematical models, often models of derivatives.

Since these models contain parameters, such as

volatility, we can also speak of parameter risk,

volatility risk, etc.

• Operational risk: The possibility of loss due to

people, procedures or systems. This includes human

error and fraud

• Legal risk: The possibility of loss due to legal action

or the meaning of legal contracts

Before looking at the mathematics of risk we should

understand the difference between risk, randomness

and uncertainty, all of which are important.

Chapter 2: FAQs

37

When measuring risk we often use probabilistic con-

cepts. But this requires having a distribution for the

randomness in investments, a probability density func-

tion, for example. With enough data or a decent enough

model we may have a good idea about the distribution

of returns. However, without the data, or when embark-

ing into unknown territory we may be completely in the

dark as to probabilities. This is especially true when

looking at scenarios which are incredibly rare, or have

never even happened before. For example, we may have

a good idea of the results of an alien invasion, after all,

many scenarios have been explored in the movies, but

what is the probability of this happening? When you do

not know the probabilities then you have what Knight

(1921) termed ‘uncertainty.’

We can categorize these issues, following Knight, as

follows.

1. For risk the probabilities that specified events will

occur in the future are measurable and known, i.e.,

there is randomness but with a known probability

distribution. This can be further divided.

(a) a priori risk, such as the outcome of the roll of a

fair die

(b) estimable risk, where the probabilities can be

estimated through statistical analysis of the past,

for example, the probability of a one-day fall of ten

percent in the S&P index

2. With uncertainty the probabilities of future events

cannot be estimated or calculated.

In finance we tend to concentrate on risk with prob-

abilities we estimate, we then have all the tools of

statistics and probability for quantifying various aspects

of that risk. In some financial models we do attempt

to address the uncertain. For example the uncertain

volatility work of Avellaneda et al. (1995). Here volatility

38 Frequently Asked Questions In Quantitative Finance

is uncertain, is allowed to lie within a specified range,

but the probability of volatility having any value is not

given. Instead of working with probabilities we now

work with worst-case scenarios. Uncertainty is therefore

more associated with the idea of stress testing port-

folios. CrashMetrics is another example of worst-case

scenarios and uncertainty.

A starting point for a mathematical definition of risk is

simply as standard deviation. This is sensible because of

the results of the Central Limit Theorem (CLT), that if

you add up a large number of investments what matters

as far as the statistical properties of the portfolio are

just the expected return and the standard deviation

of individual investments, and the resulting portfolio

returns are normally distributed. The normal distribu-

tion being symmetrical about the mean, the potential

downside can be measured in terms of the standard

deviation.

However, this is only meaningful if the conditions for

the CLT are satisfied. For example, if we only have a

small number of investments, or if the investments are

correlated, or if they don’t have finite variance, . . . then

standard deviation may not be relevant.

Another mathematical definition of risk is semi variance,

in which only downside deviations are used in the calcu-

lation. This definition is used in the Sortino performance

measure.

Artzner et al. (1997) proposed a set of properties that a

measure of risk should satisfy for it to be sensible. Such

risk measures are called coherent.

Chapter 2: FAQs

39

References

Artzner, P, Delbaen, F, Eber, J-M & Heath, D 1997 Thinking

coherently. Risk magazine 10 (11) 68–72.

Avellaneda, M, Levy, A & Parás, A 1995 Pricing and hedging

derivative securities in markets with uncertain volatilities.

Applied Mathematical Finance 2 73–88

Avellaneda, M & Parás, A 1996 Managing the volatility risk of

derivative securities: the Lagrangian volatility model. Applied

Mathematical Finance 3 21–53

Knight, FH 1921 Risk, Uncertainty, and Profit. Hart, Schaffner,

and Marx Prize Essays, no. 31. Boston and New York:

Houghton Mifflin

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

40 Frequently Asked Questions In Quantitative Finance

What is Value at Risk and How is it

Used?

Short Answer

Value at Risk, or VaR for short, is a measure of the

amount that could be lost from a position, portfolio,

desk, bank, etc. VaR is generally understood to mean

the maximum loss an investment could incur at a given

confidence level over a specified time horizon. There

are other risk measures used in practice but this is the

simplest and most common.

Example

An equity derivatives hedge fund estimates that its

Value at Risk over one day at the 95% confidence level

is $500,000. This is interpreted as one day out of 20 the

fund expects to lose more than half a million dollars.

Long Answer

VaR calculations often assume that returns are normally

distributed over the time horizon of interest. Inputs for

a VaR calculation will include details of the portfolio

composition, the time horizon, and parameters govern-

ing the distribution of the underlyings. The latter set of

parameters includes average growth rate, standard devi-

ations (volatilities) and correlations. (If the time horizon

is short you can ignore the growth rate, as it will only

have a small effect on the final calculation.)

With the assumption of normality, VaR is calculated by

a simple formula if you have a simple portfolio, or by

simulations if you have a more complicated portfolio.

The difference between simple and complicated is

essentially the difference between portfolios without

derivatives and those with. If your portfolio only con-

tains linear instruments then calculations involving

Chapter 2: FAQs

41

normal distributions, standard deviations, etc. can all be

done analytically. This is also the case if the time hori-

zon is short so that derivatives can be approximated by

a position of delta in the underlying.

The simulations can be quite straightforward, albeit

rather time consuming. Simulate many realizations of

all of the underlyings up to the time horizon using

traditional Monte Carlo methods. For each realization

calculate the portfolio’s value. This will give you a dis-

tribution of portfolio values at the time horizon. Now

look at where the tail of the distribution begins, the left-

hand 5% tail if you want 95% confidence, or the 1% tail

if you are working to 99% etc.

If you are working entirely with normal distributions

then going from one confidence level to another is just

a matter of looking at a table of numbers for the stan-

dardized normal distribution, see the table below. As

long as your time horizon is sufficiently short for the

growth to be unimportant you can use the square-root

rule to go from one time horizon to another. (The VaR

will scale with the square root of the time horizon,

this assumes that the portfolio return is also normally

distributed.)

An alternative to using a parameterized model for the

underlyings is to simulate straight from historical data,

bypassing the normal-distribution assumption alto-

gether.

VaR is a very useful concept in practice for the following

reasons.

• VaR is easily calculated for individual instruments,

entire portfolios, or at any level right up to an entire

bank or fund

42 Frequently Asked Questions In Quantitative Finance

Table 2.1: Degree of confidence and the

relationship with deviation from the mean.

Degree of

Number of standard

confidence

deviations from

the mean

99%

2.326342

98%

2.053748

97%

1.88079

96%

1.750686

95%

1.644853

90%

1.281551

• You can adjust the time horizon depending on your

trading style. If you hedge every day you may want a

one-day horizon, if you buy and hold for many

months, then a longer horizon would be relevant

• It can be broken down into components, so you can

examine different classes of risk, or you can look at

the marginal risk of adding new positions to

your book

• It can be used to constrain positions of individual

traders or entire hedge funds

• It is easily understood, by management, by investors,

by people who are perhaps not that technically

sophisticated

Of course, there are also valid criticisms as well.

• It does not tell you what the loss will be beyond the

VaR value

• VaR is concerned with typical market conditions, not

the extreme events

• It uses historical data, ‘‘like driving a car by looking

in the rear-view mirror only’’

Chapter 2: FAQs

43

• Within the time horizon positions could change

dramatically (due to normal trading or due to

hedging or expiration of derivatives)

A common criticism of traditional VaR has been that

it does not satisfy all of certain commonsense criteria.

Artzner et al. (1997) specify criteria that make a risk

measure coherent. And VaR as described above is

not coherent.

Prudence would suggest that other risk-measurement

methods are used in conjunction with VaR, including

but not limited to, stress testing under different real

and hypothetical scenarios, including the stressing of

volatility especially for portfolios containing derivatives.

References and Further Reading

Artzner, P, Delbaen, F, Eber, J-M & Heath, D 1997 Thinking

coherently. Risk magazine 10 (11) 68–72

44 Frequently Asked Questions In Quantitative Finance

What is CrashMetrics?

Short Answer

CrashMetrics is a stress-testing methodology for eval-

uating portfolio performance in the event of extreme

movements in financial markets. Like CAPM it relates

moves in individual stocks to the moves in one or more

indices but only during large moves. It is applicable to

portfolios of equities and equity derivatives.

Example

Your portfolio contains many individual stocks and

many derivatives of different types. It is perfectly

constructed to profit from your view on the market

and its volatility. But what if there is a dramatic fall in

the market, perhaps 5%, what will the effect be on your

P&L? And if the fall is 10%, 20% . . . ?

Long Answer

CrashMetrics is a very simple risk-management tool for

examining the effects of a large move in the market as

a whole. It is therefore of use for studying times when

diversification does not work.

If your portfolio consists of a single underlying equity

and its derivatives then the change in its value during a

crash, δ, can be written as

δ = F( δS),

where F (·) is the ‘formula’ for the portfolio, meaning

option-pricing formulæ for all of the derivatives and

equity in the portfolio, and δS is the change in the

underlying.

Chapter 2: FAQs

45

In CrashMetrics the risk in this portfolio is measured as

the worst case over some range of equity moves:

worst-case loss =

min

F ( δS) .

− δS−≤ δS≤ δS+

This is the number that would be quoted as the possible

downside during a dramatic move in the market.

This downside can be reduced by adding derivatives

to the portfolio in an optimal fashion. This is called

Platinum Hedging. For example, if you want to use

some out-of-the-money puts to make this worst case not

so bad then you could optimize by choosing λ so that

the worst case of

F ( δS) + λF ∗( δS) − | λ| C

represents an acceptable level of downside risk. Here

F ∗(·) is the ‘formula’ for the change in value of the

hedging contract, C is the ‘cost’ associated with each

hedging contract and λ is the quantity of the contract

which is to be determined. In practice there would be

many such hedging contracts, not necessarily just an

out-of-the-money put, so you would sum over all of

them and then optimize.

CrashMetrics deals with any number of underlyings

by exploiting the high degree of correlation between

equities during extreme markets. We can relate the

return on the i th stock to the return on a representative

index, x, during a crash by

δSi = κix,

Si

where κi is a constant crash coefficient. For example,

if the kappa for stock XYZ is 1.2 it means that when

46 Frequently Asked Questions In Quantitative Finance

the index falls by 10% XYZ will fall by 12%. The crash

coefficient therefore allows a portfolio with many under-

lyings to be interpreted during a crash as a portfolio

on a single underlying, the index. We therefore consider

the worst case of

δ = F( δS 1, . . . , δSN) = F( κ 1 xS 1, . . . , κNxSN) as our measure of downside risk.

Again Platinum Hedging can be applied when we have

many underlyings. We must consider the worst case of

M

δ = F( κ 1 xS 1, . . . , κNxSN) +

λkFk( κ 1 xS 1, . . . , κNxSN)

k=1

M

−

| λk| Ck,

k=1

where F is the original portfolio and the Fks are the

available hedging contracts.

CrashMetrics is very robust because

• it does not use unstable parameters such as

volatilities or correlations

• it does not rely on probabilities, instead considers

worst cases

CrashMetrics is a good risk tool because

• it is very simple and fast to implement

• it can be used to optimize portfolio insurance against

market crashes

CrashMetrics is used for

• analyzing derivatives portfolios under the threat of a

crash

• optimizing portfolio insurance

Chapter 2: FAQs

47

• reporting risk

• providing trading limits to avoid intolerable

performance during a crash

References and Further Reading

Hua, P & Wilmott, P 1997 Crash courses. Risk magazine 10 (6)

64–67

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

48 Frequently Asked Questions In Quantitative Finance

What is a Coherent Risk Measure and

What are its Properties?

Short Answer

A risk measure is coherent if it satisfies certain simple,

mathematical properties. One of these properties,

which some popular measures do not possess is sub-

additivity, that adding together two risky portfolios

cannot increase the measure of risk.

Example

Artzner et al. (1997) give a simple example of traditional

VaR which violates this, and illustrates perfectly the

problems of measures that are not coherent. Portfolio

X consists only of a far out-of-the-money put with one

day to expiry. Portfolio Y consists only of a far out-of-

the-money call with one day to expiry. Let us suppose

that each option has a probability of 4% of ending up in

the money. For each option individually, at the 95% con-

fidence level the one-day traditional VaR is effectively

zero. Now put the two portfolios together and there is

a 92% chance of not losing anything, 100% less two lots

of 4%. So at the 95% confidence level there will be a

significant VaR. Putting the two portfolios together has

in this example increased the risk. ‘‘A merger does not

create extra risk’’ (Artzner et al., 1997).

Long Answer

A common criticism of traditional VaR has been that

it does not satisfy all of certain commonsense criteria.

Artzner et al. (1997) defined the following set of sensible

criteria that a measure of risk, ρ( X ) where X is a set of outcomes, should satisfy. These are as follows.

1. Sub-additivity: ρ( X + Y ) ≤ ρ( X) + ρ( Y ). This just says that if you add two portfolios together the total risk

Chapter 2: FAQs

49

can’t get any worse than adding the two risks

separately. Indeed, there may be cancellation effects

or economies of scale that will make the risk better.

2. Monotonicity: If X ≤ Y for each scenario then

ρ( X) ≤ ρ( Y). If one portfolio has better values than another under all scenarios then its risk will be

better.

3. Positive homogeneity: For all λ > 0, ρ( λX) = λρ( X).

Double your portfolio then you double your risk.

4. Translation invariance: For all constant c,

ρ( X + c) = ρ( X) − c. Think of just adding cash to a portfolio, this would come off your risk.

A risk measure that satisfies all of these is called coher-

ent. The traditional, simple VaR measure is not coherent

since it does not satisfy the sub-additivity condition.

Sub-additivity is an obvious requirement for a risk

measure, otherwise there would be no risk benefit to

adding uncorrelated new trades into a book. If you have

two portfolios X and Y then this benefit can be defined

as

ρ( X) + ρ( Y) − ρ( X + Y) .

Sub-additivity says that this can only be non negative.

Lack of sub-additivity is a risk measure and can be

exploited in a form of regulatory arbitrage. All a bank

has to do is create subsidiary firms, in a reverse form

of the above example, to save regulatory capital.

With a coherent measure of risk, specifically because of

its sub-additivity, one can simply add together risks of

individual portfolios to get a conservative estimate of

the total risk.

Coherent measures

Straightforward, no-nonsense, standard

deviation is coherent. This is not an entirely satisfactory

measure since it does not focus on the particularly

50 Frequently Asked Questions In Quantitative Finance

damaging tail events. Another measure that is coherent

is Expected Shortfall. This is calculated as the average

of all the P&Ls making up the tail percentile of interest.

Suppose we are working with the 5% percentile, rather

than quoting this number (this would be traditional

VaR) instead calculate the average of all the P&Ls in

this 5% tail.

Attribution

Having calculated a coherent measure of

risk, one often wants to attribute this to smaller units.

For example, a desk has calculated its risk and wants to

see how much each trader is responsible for. Similarly,

one may want to break down the risk into contributions

from each of the greeks in a derivatives portfolio. How

much risk is associated with direction of the market,

and how much is associated with volatility exposure,

for example.

References and Further Reading

Acerbi, C & Tasche, D On the coherence of expected shortfall.

www-m1.mathematik.tu-muenchen.de/m4/Papers/Tasche/shortfall.pdf

Artzner, P, Delbaen, F, Eber, J-M & Heath, D 1997 Thinking

coherently. Risk magazine 10 (11) 68–72

Chapter 2: FAQs

51

What is Modern Portfolio Theory?

Short Answer

The Modern Portfolio Theory (MPT) of Harry Markowitz

(1952) introduced the analysis of portfolios of invest-

ments by considering the expected return and risk

of individual assets and, crucially, their interrelation-

ship as measured by correlation. Prior to this investors

would examine investments individually, build up port-

folios of favoured stocks, and not consider how they

related to each other. In MPT diversification plays an

important role.

Example

Should you put all your money in a stock that has low

risk but also low expected return, or one with high

expected return but which is far riskier? Or perhaps

divide your money between the two. Modern Portfolio

Theory addresses this question and provides a frame-

work for quantifying and understanding risk and return.

Long Answer

In MPT the return on individual assets are represented

by normal distributions with certain mean and standard

deviation over a specified period. So one asset might

have an annualized expected return of 5% and an annu-

alized standard deviation (volatility) of 15%. Another

might have an expected return of −2% and a volatility of

10%. Before Markowitz, one would only have invested in

the first stock, or perhaps sold the second stock short.

Markowitz showed how it might be possible to better

both of these simplistic portfolios by taking into account

the correlation between the returns on these stocks.

In the MPT world of N assets there are 2 N + N( N − 1) / 2

parameters: expected return, one per stock; standard

deviation, one per stock; correlations, between any two

52 Frequently Asked Questions In Quantitative Finance

stocks (choose two from N without replacement, order

unimportant). To Markowitz all investments and all port-

folios should be compared and contrasted via a plot

of expected return versus risk, as measured by stan-

dard deviation. If we write µA to represent the expected

return from investment or portfolio A (and similarly

for B, C, etc.) and σB for its standard deviation then

investment/portfolio A is at least as good as B if

µA ≥ µB

and

σA ≤ σB.

The mathematics of risk and return is very simple.

Consider a portfolio, , of N assets, with Wi being

the fraction of wealth invested in the i th asset. The

expected return is then

N

µ =

Wiµi

i=1

and the standard deviation of the return, the risk, is

N

N

σ

=

WiWjρijσiσj,

i=1 j=1

where ρij is the correlation between the i th and j th

investments, with ρii = 1.

Markowitz showed how to optimize a portfolio by find-

ing the W s giving the portfolio the greatest expected

return for a prescribed level of risk. The curve in the

risk-return space with the largest expected return for

each level of risk is called the efficient frontier.

According to the theory, no one should hold portfolios

that are not on the efficient frontier. Furthermore, if

you introduce a risk-free investment into the universe of

assets, the efficient frontier becomes the tangential line

shown below. This line is called the Capital Market Line

Chapter 2: FAQs

53

50%

45%

40%

35%

30%

market portfolio

n

25%

Retur

20%

15%

10%

5%

0%

0%

10%

20%

30%

40%

50%

60%

70%

80%

Risk

Figure 2-3: Reward versus risk, a selection of risky assets and the

efficient frontier (bold).

and the portfolio at the point at which it is tangential

is called the Market Portfolio. Now, again according to

the theory, no one ought to hold any portfolio of assets

other than the risk-free investment and the Market Port-

folio.

Harry Markowitz, together with Merton Miller and

William Sharpe, was awarded the Nobel Prize for Eco-

nomic Science in 1990.

References and Further Reading

Markowitz, HM 1952 Portfolio selection. Journal of Finance 7

(1) 77–91

Ingersoll, JE Jr 1987 Theory of Financial Decision Making. Row-

man & Littlefield

54 Frequently Asked Questions In Quantitative Finance

What is the Capital Asset Pricing

Model?

Short Answer

The Capital Asset Pricing Model (CAPM) relates the

returns on individual assets or entire portfolios to the

return on the market as a whole. It introduces the con-

cepts of specific risk and systematic risk. Specific risk

is unique to an individual asset, systematic risk is that

associated with the market. In CAPM investors are com-

pensated for taking systematic risk but not for taking

specific risk. This is because specific risk can be diver-

sified away by holding many different assets.

Example

A stock has an expected return of 15% and a volatility of

20%. But how much of that risk and return are related

to the market as a whole? Because the less that can

be attributed to the behaviour of the market then the

better that stock will be for diversification purposes.

Long Answer

CAPM simultaneously simplified Markowitz’s Modern

Portfolio Theory (MPT), made it more practical and

introduced the idea of specific and systematic risk.

Whereas MPT has arbitrary correlation between all

investments, CAPM, in its basic form, only links invest-

ments via the market as a whole.

CAPM is an example of an equilibrium model, as

opposed to a no-arbitrage model such as Black–Scholes.

The mathematics of CAPM is very simple. We relate the

random return on the i th investment, Ri, to the random

Chapter 2: FAQs

55

return on the market as a whole (or some representative

index), RM by

Ri = αi + βiRM + i.

The i is random with zero mean and standard deviation

ei, and uncorrelated with the market return RM and the

other j. There are three parameters associated with

each asset, αi, βi and ei. In this representation we can see that the return on an asset can be decomposed into

three parts: a constant drift; a random part common

with the index; a random part uncorrelated with the

index, i. The random part i is unique to the i th asset.

Notice how all the assets are related to the index but

are otherwise completely uncorrelated.

Let us denote the expected return on the index by µM

and its standard deviation by σM . The expected return

on the i th asset is then

µi = αi + βiµM

and the standard deviation

σi = β 2 σ 2 + e 2 .

i

M

i

If we have a portfolio of such assets then the return is

given by

δ

N

N

N

N

=

W

W

+ R

W

+

W

iRi =

iαi

M

iβi

i i.

i=1

i=1

i=1

i=1

From this it follows that

N

N

µ =

Wiαi + E[ RM ]

Wiβi .

i=1

i=1

Writing

N

N

α =

Wiαi

and

β =

Wiβi,

i=1

i=1

56 Frequently Asked Questions In Quantitative Finance

we have

µ = α + βE[ RM] = α + βµM.

Similarly the risk in is measured by

N

N

N

σ

=

WiWjβiβjσ 2 +

W 2 e 2 .

M

i

i

i=1 j=1

i=1

Note that if the weights are all about the same, N−1, then

the final terms inside the square root are also O( N−1).

Thus this expression is, to leading order as N → ∞,

N

σ

=

Wiβi σM = | β| σM.

i=1

Observe that the contribution from the uncorrelated

s to the portfolio vanishes as we increase the num-

ber of assets in the portfolio; this is the risk associated

with the diversifiable risk. The remaining risk, which is

correlated with the index, is the undiversifiable system-

atic risk.

Multi-index versions of CAPM can be constructed. Each

index being representative of some important financial

or economic variable.

The parameters alpha and beta are also commonly

referred to in the hedge-fund world. Performance reports

for trading strategies will often quote the alpha and beta

of the strategy. A good strategy will have a high, pos-

itive alpha with a beta close to zero. With beta being

small you would expect performance to be unrelated

to the market as a whole and with large, positive alpha

you would expect good returns whichever way the mar-

ket was moving. Small beta also means that a strategy

should be a valuable addition to a portfolio because of

its beneficial diversification.

Sharpe shared the 1990 Nobel Prize in Economics with

Harry Markowitz and Merton Miller.

Chapter 2: FAQs

57

References and Further Reading

Lintner, J 1965 The valuation of risk assets and the selection

of risky investments in stock portfolios and capital budgets.

Rev. of Econ. and Stats 47

Mossin, J 1966 Equilibrium in a Capital Asset Market. Econo-

metrica 34 768–783

Sharpe, WF 1964 Capital asset prices: A theory of market

equilibrium under conditions of risk. J. of Finance 19 (3)

425–442

Tobin, J 1958 Liquidity preference as behavior towards risk.

Rev. of Economic Studies 25

58 Frequently Asked Questions In Quantitative Finance

What is Arbitrage Pricing Theory?

Short Answer

The Arbitrage Pricing Theory (APT) of Stephen Ross

(1976) represents the returns on individual assets as

a linear combination of multiple random factors. These

random factors can be fundamental factors or statistical.

For there to be no arbitrage opportunities there must

be restrictions on the investment processes.

Example

Suppose that there are five dominant causes of ran-

domness across investments. These five factors might

be market as a whole, inflation, oil prices, etc. If you

are asked to invest in six different, well diversified

portfolios then either one of these portfolios will have

approximately the same risk and return as a suitable

combination of the other five, or there will be an arbi-

trage opportunity.

Long Answer

Modern Portfolio Theory represents each asset by its

own random return and then links the returns on dif-

ferent assets via a correlation matrix. In the Capital

Asset Pricing Model returns on individual assets are

related to returns on the market as a whole together

with an uncorrelated stock-specific random component.

In Arbitrage Pricing Theory returns on investments

are represented by a linear combination of multiple

random factors, with as associated factor weighting.

Portfolios of assets can also be decomposed in this

way. Provided the portfolio contains a sufficiently large

number of assets then the stock-specific component can

be ignored. Being able to ignore the stock-specific risk

is the key to the ‘‘A’’ in ‘‘APT.’’

Chapter 2: FAQs

59

We write the random return on the i th asset as

n

Ri = αi +

βjiRj + i,

j=1

where the Rj are the factors, the α s and β s are con-

stants and i is the stock-specific risk. A portfolio of

these assets has return

N

N

n

N

aiRi =

aiαi +

aiβji Rj + · · · ,

i=1

i=1

j=1

i=1

where the · · · can be ignored if the portfolio is well

diversified.

Suppose that we think that five factors are sufficient to

represent the economy. We can therefore decompose

any portfolio into a linear combination of these five

factors, plus some supposedly negligible stock-specific

risks. If we are shown six diversified portfolios we can

decompose each into the five random factors. Since

there are more portfolios than factors we can find a

relationship between (some of) these portfolios, effec-

tively relating their values, otherwise there would be

an arbitrage. Note that the arbitrage argument is an

approximate one, relating diversified portfolios, on the

assumption that the stock-specific risks are negligible

compared with the factor risks.

In practice we can choose the factors to be macro-

economic or statistical. Here are some possible macro-

economic variables.

• an index level

• GDP growth

• an interest rate (or two)

60 Frequently Asked Questions In Quantitative Finance

• a default spread on corporate bonds

• an exchange rate

Statistical variables come from an analysis of a covari-

ance of asset returns. From this one extracts the factors

by some suitable decomposition.

The main differences between CAPM and APT is that

CAPM is based on equilibrium arguments to get to the

concept of the Market Portfolio whereas APT is based

on a simple approximate arbitrage argument. Although

APT talks about arbitrage, this must be contrasted with

the arbitrage arguments we see in spot versus forward

and in option pricing. These are genuine exact arbi-

trages (albeit the latter being model dependent). In APT

the arbitrage is only approximate.

References and Further Reading

Ross, S 1976 The Arbitrage Theory of Capital Asset Pricing. J.

of Economic Theory 13 341–360

Chapter 2: FAQs

61

What is Maximum Likelihood

Estimation?

Short Answer

Maximum Likelihood Estimation (MLE) is a statisti-

cal technique for estimating parameters in a proba-

bility distribution. We choose parameters that maxi-

mize the a priori probability of the final outcome actu-

ally happening.

Example

You have three hats containing normally distributed

random numbers. One hat’s numbers have mean of zero

and standard deviation 0.1. This is hat A. Another hat’s

numbers have mean of zero and standard deviation 1.

This is hat B. The final hat’s numbers have mean of

zero and standard deviation 10. This is hat C. You don’t

know which hat is which.

You pick a number out of one hat, it is −2.6. Which hat

do you think it came from? MLE can help you answer

this question.

Long Answer

A large part of statistical modelling concerns finding

model parameters. One popular way of doing this is

Maximum Likelihood Estimation.

The method is easily explained by a very simple ex-

ample. You are attending a maths conference. You

arrive by train at the city hosting the event. You take

a taxi from the train station to the conference venue.

The taxi number is 20,922. How many taxis are there in

the city?

This is a parameter estimation problem. Getting into

a specific taxi is a probabilistic event. Estimating the

62 Frequently Asked Questions In Quantitative Finance

number of taxis in the city from that event is a question

of assumptions and statistical methodology.

For this problem the obvious assumptions to make are:

1. Taxi numbers are strictly positive integers

2. Numbering starts at one

3. No number is repeated

4. No number is skipped

We will look at the probability of getting into taxi num-

ber 20,922 when there are N taxis in the city. This

couldn’t be simpler, the probability of getting into any

specific taxi is

1 .

N

Which N maximizes the probability of getting into taxi

number 20,922? The answer is

N = 20, 922 .

This example explains the concept of MLE: Choose

parameters that maximize the probability of the outcome

actually happening.

Another example, more closely related to problems

in quantitative finance is the hat example above. You

have three hats containing normally distributed random

numbers. One hat’s numbers have mean of zero and

standard deviation 0.1. This is hat A. Another hat’s

numbers have mean of zero and standard deviation 1.

This is hat B. The final hat’s numbers have mean of zero

and standard deviation 10. This is hat C.

You pick a number out of one hat, it is −2.6. Which hat

do you think it came from?

The ‘probability’ of picking the number −2.6 from hat

A (having a mean of zero and a standard deviation of

Chapter 2: FAQs

63

0.1) is

1

√

exp − 2 . 62

= 6 10−147 .

2 π 0 . 1

2 × 0 . 12

Very, very unlikely!

(N.B. The word ‘probability’ is in inverted commas to

emphasize the fact that this is the value of the proba-

bility density function, not the actual probability. The

probability of picking exactly −2.6 is, of course, zero.)

The ‘probability’ of picking the number −2.6 from hat B

(having a mean of zero and a standard deviation of 1) is

1

√

exp − 2 . 62

= 0 . 014,

2 π 1

2 × 12

and from hat C (having a mean of zero and a standard

deviation of 10)

1

√

exp − 2 . 62

= 0 . 039 .

2 π 10

2 × 102

We would conclude that hat C is the most likely, since it

has the highest probability for picking the number −2.6.

We now pick a second number from the same hat, it is

0.37. This looks more likely to have come from hat B.

We get the following table of probabilities.

Hat

−2.6

0.37

Joint

A

6 10−147

0.004

2 10−149

B

0.014

0.372

0.005

C

0.039

0.040

0.002

The second column represents the probability of draw-

ing the number −2.6 from each of the hats, the third

64 Frequently Asked Questions In Quantitative Finance

column represents the probability of drawing 0.37 from

each of the hats, and the final column is the joint prob-

ability, that is, the probability of drawing both numbers

from each of the hats.

Using the information about both draws, we can see that

the most likely hat is now B.

Now let’s make this into precisely a quant finance

problem.

Find the volatility

You have one hat containing normally

distributed random numbers, with a mean of zero and a

standard deviation of σ which is unknown. You draw N

numbers φi from this hat. Estimate σ .

Q. What is the ‘probability’ of drawing φi from a Normal

distribution with mean zero and standard deviation σ ?

A. It is

φ 2

1

i

√

−

e 2 σ 2 .

2 πσ

Q. What is the ‘probability’ of drawing all of the numbers

φ 1, φ 2, . . . , φN from independent Normal distributions with mean zero and standard deviation σ ?

A. It is

N

φ 2

1

i

√

−

e 2 σ 2 .

2 πσ

i=1

Now choose the σ that maximizes this quantity. This is

easy. First take logarithms of this expression, and the

differentiate with respect to σ and set result equal to

zero:

N

d

− N ln( σ) − 1

φ 2 = 0 .

dσ

2 σ 2

i

i=1

Chapter 2: FAQs

65

(A multiplicative factor has been ignored here.) I.e.

N

N

− + 1

φ 2 = 0 .

σ

σ 3

i

i=1

Therefore our best guess for σ is given by

N

σ 2 = 1

φ 2 .

N

i

i=1

You should recognize this as a measure of the variance.

Quants’ salaries

In the figure are the results of a 2004

survey on www.wilmott.com concerning the salaries of

quants using the Forum (or rather, those answering the

question!).

This distribution looks vaguely lognormal, with distribu-

tion

1

(

√

ln E − ln E 0 ) 2

exp −

,

2 πσ E

2 σ 2

where E is annual earnings, σ is the standard deviation

and E 0 the mean. We can use MLE find σ and E 0.

It turns out that the mean E 0 = $133,284, with σ = 0 . 833.

Figure 2-4: Distribution of quants’ salaries.

66 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Eliason, SR 1993 Maximum Likelihood Estimation: Logic and

Practice. Sage

Chapter 2: FAQs

67

What is Cointegration?

Short Answer

Two time series are cointegrated if a linear combination

has constant mean and standard deviation. In other

words, the two series never stray too far from one

another. Cointegration is a useful technique for studying

relationships in multivariate time series, and provides

a sound methodology for modelling both long-run and

short-run dynamics in a financial system.

Example

Suppose you have two stocks S 1 and S 2 and you find

that S 1 − 3 S 2 is stationary, so that this combination

never strays too far from its mean. If one day this

‘spread’ is particularly large then you would have sound

statistical reasons for thinking the spread might shortly

reduce, giving you a possible source of statistical

arbitrage profit. This can be the basis for pairs trading.

Long Answer

The correlations between financial quantities are noto-

riously unstable. Nevertheless correlations are regularly

used in almost all multivariate financial problems. An

alternative statistical measure to correlation is cointe-

gration. This is probably a more robust measure of the

linkage between two financial quantities but as yet there

is little derivatives theory based on the concept.

Two stocks may be perfectly correlated over short

timescales yet diverge in the long run, with one grow-

ing and the other decaying. Conversely, two stocks may

follow each other, never being more than a certain dis-

tance apart, but with any correlation, positive, negative

or varying. If we are delta hedging then maybe the short

timescale correlation matters, but not if we are hold-

ing stocks for a long time in an unhedged portfolio. To

68 Frequently Asked Questions In Quantitative Finance

see whether two stocks stay close together we need a

definition of stationarity. A time series is stationary if

it has finite and constant mean, standard deviation and

autocorrelation function. Stocks, which tend to grow,

are not stationary. In a sense, stationary series do not

wander too far from their mean.

Testing for the stationarity of a time series Xt involves a

linear regression to find the coefficients a, b and c in Xt = aXt−1 + b + ct.

If it is found that | a| > 1 then the series is unstable.

If −1 ≤ a < 1 then the series is stationary. If a = 1 then

the series is non stationary. As with all things statistical,

we can only say that our value for a is accurate with a

certain degree of confidence. To decide whether we

have got a stationary or non-stationary series requires

us to look at the Dickey–Fuller statistic to estimate the

degree of confidence in our result. So far, so good, but

from this point on the subject of cointegration gets

complicated.

How is this useful in finance? Even though individual

stock prices might be non stationary it is possible for

a linear combination (i.e., a portfolio) to be stationary.

Can we find λ

N

i, with

λ

i=1 i = 1, such that

N

λiSi

i=1

is stationary? If we can, then we say that the stocks are

cointegrated.

For example, suppose we find that the S&P500 is coin-

tegrated with a portfolio of 15 stocks. We can then use

these fifteen stocks to track the index. The error in this

tracking portfolio will have constant mean and standard

deviation, so should not wander too far from its average.

Chapter 2: FAQs

69

This is clearly easier than using all 500 stocks for the

tracking (when, of course, the tracking error would be

zero).

We don’t have to track the index, we could track any-

thing we want, such as e 0 . 2 t to choose a portfolio that gets a 20% return. We could analyze the cointegration

properties of two related stocks, Nike and Reebok, for

example, to look for relationships. This would be pairs

trading. Clearly there are similarities with MPT and

CAPM in concepts such as means and standard devi-

ations. The important difference is that cointegration

assumes far fewer properties for the individual time

series. Most importantly, volatility and correlation do

not appear explicitly.

Another feature of cointegration is Granger causality

which is where one variable leads and another lags.

This is of help in explaining why there is any dynamic

relationship between several financial quantities.

References and Further Reading

Alexander, CO 2001 Market Models. John Wiley & Sons

Engle, R & Granger, C 1987 Cointegration and error correction:

representation, estimation and testing. Econometrica 55

251–276

70 Frequently Asked Questions In Quantitative Finance

What is the Kelly criterion?

Short Answer

The Kelly criterion is a technique for maximizing ex-

pected growth of assets by optimally investing a fixed

fraction of your wealth in a series of investments. The

idea has long been used in the world of gambling.

Example

You own a biased coin that will lands heads up with

probability p > 1 . You find someone willing to bet any

2

amount against you at evens. They are willing to bet any

number of times. Clearly you can make a lot of money

with this special coin. You start with $1000. How much

of this should you bet?

Long Answer

Let’s work with the above example. The first observa-

tion is that you should bet an amount proportional to

how much you have. As you win and your wealth grows

you will bet a larger amount. But you shouldn’t bet too

much. If you bet all $1000 you will eventually toss a tail

and lose everything and will be unable to continue. If

you bet too little then it will take a long time for you to

make a decent amount.

The Kelly criterion is to bet a certain fraction of your

wealth so as to maximize your expected growth of wealth.

We use φ to denote the random variable taking value

1 with probability p and −1 with probability 1 − p and

f to denote the fraction of our wealth that we bet. The

growth of wealth after each toss of the coin is then the

random amount

ln(1 + f φ) .

Chapter 2: FAQs

71

The expected growth rate is

p ln(1 + f ) + (1 − p) ln(1 − f ) .

This function is plotted below for p = 0 . 55.

This expected growth rate is maximized by the choice

f = 2 p − 1 .

This is the Kelly fraction.

A betting fraction of less than this would be a conserva-

tive strategy. Anything to the right will add volatility to

returns, and decrease the expected returns. Too far to

the right and the expected return becomes negative.

This money management principle can be applied to

any bet or investment, not just the coin toss. More

0.01

0

-0.1

0.1

0.3

0.5

0.7

0.9

1.1

-0.01

Betting fraction

rn

retu

-0.02

ected

Exp

-0.03

-0.04

-0.05

72 Frequently Asked Questions In Quantitative Finance

generally, if the investment has an expected return of

µ and a standard deviation σ µ then the expected

growth for an investment fraction of f is

E[ln(1 + f φ)]

which can be approximated by Taylor series

f φ − 1 f 2 φ 2 + · · · .

2

The Kelly fraction, which comes from maximizing this

expression, is therefore

µ

f =

.

σ 2

In practice, because the mean and standard deviation

are rarely known accurately, one would err on the side

of caution and bet a smaller fraction. A common choice

is half Kelly.

Other money management strategies are, of course,

possible, involving target wealth, probability of ruin, etc.

References and Further Reading

Kelly, JL 1956 A new interpretation of information rate. Bell

Systems Tech. J. 35 917–926

Poundstone, W 2005 Fortune’s Formula. Hill & Wang

Chapter 2: FAQs

73

Why Hedge?

Short Answer

‘Hedging’ in its broadest sense means the reduction of

risk by exploiting relationships or correlation (or lack of

correlation) between various risky investments. The pur-

pose behind hedging is that it can lead to an improved

risk/return. In the classical Modern Portfolio Theory

framework, for example, it is usually possible to con-

struct many portfolios having the same expected return

but with different variance of returns (‘risk’). Clearly, if

you have two portfolios with the same expected return

the one with the lower risk is the better investment.

Example

You buy a call option, it could go up or down in value

depending on whether the underlying go up or down. So

now sell some stock short. If you sell the right amount

short then any rises or falls in the stock position will

balance the falls or rises in the option, reducing risk.

Long Answer

To help understand why hedge it is useful to look at the

different types of hedging.

The two main classifications

Probably the most important

distinction between types of hedging is between model-

independent and model-dependent hedging strategies.

Model-independent hedging: An example of such hedg-

ing is put-call parity. There is a simple relationship

between calls and puts on an asset (when they are

both European and with the same strikes and expiries),

the underlying stock and a zero-coupon bond with the

same maturity. This relationship is completely inde-

pendent of how the underlying asset changes in value.

Another example is spot-forward parity. In neither case

74 Frequently Asked Questions In Quantitative Finance

do we have to specify the dynamics of the asset, not

even its volatility, to find a possible hedge. Such model-

independent hedges are few and far between.

Model-dependent hedging: Most sophisticated finance

hedging strategies depend on a model for the underly-

ing asset. The obvious example is the hedging used in

the Black–Scholes analysis that leads to a whole the-

ory for the value of derivatives. In pricing derivatives

we typically need to at least know the volatility of the

underlying asset. If the model is wrong then the option

value and any hedging strategy could also be wrong.

Delta hedging

One of the building blocks of derivatives

theory is delta hedging. This is the theoretically per-

fect elimination of all risk by using a very clever hedge

between the option and its underlying. Delta hedging

exploits the perfect correlation between the changes

in the option value and the changes in the stock price.

This is an example of ‘dynamic’ hedging; the hedge must

be continually monitored and frequently adjusted by the

sale or purchase of the underlying asset. Because of the

frequent rehedging, any dynamic hedging strategy is

going to result in losses due to transaction costs. In

some markets this can be very important.

The ‘underlying’ in a delta-hedged portfolio could be a

traded asset, a stock for example, or it could be another

random quantity that determines a price such as a risk

of default. If you have two instruments depending on the

same risk of default, you can calculate the sensitivities,

the deltas, of their prices to this quantity and then buy

the two instruments in amounts inversely proportional

to these deltas (one long, one short). This is also delta

hedging.

If two underlyings are very highly correlated you can

use one as a proxy for the other for hedging purposes.

Chapter 2: FAQs

75

You would then only be exposed to basis risk. Be careful

with this because there may be times when the close

relationship breaks down.

If you have many financial instruments that are uncorre-

lated with each other then you can construct a portfolio

with much less risk than any one of the instruments

individually. With a large such portfolio you can theo-

retically reduce risk to negligible levels. Although this

isn’t strictly hedging it achieves the same goal.

Gamma hedging

To reduce the size of each rehedge

and/or to increase the time between rehedges, and thus

reduce costs, the technique of gamma hedging is often

employed. A portfolio that is delta hedged is insensitive

to movements in the underlying as long as those move-

ments are quite small. There is a small error in this due

to the convexity of the portfolio with respect to the

underlying. Gamma hedging is a more accurate form of

hedging that theoretically eliminates these second-order

effects. Typically, one hedges one, exotic, say, contract

with a vanilla contract and the underlying. The quan-

tities of the vanilla and the underlying are chosen so

as to make both the portfolio delta and the portfolio

gamma instantaneously zero.

Vega hedging

The prices and hedging strategies are only

as good as the model for the underlying. The key param-

eter that determines the value of a contract is the

volatility of the underlying asset. Unfortunately, this is a

very difficult parameter to measure. Nor is it usually a

constant as assumed in the simple theories. Obviously,

the value of a contract depends on this parameter,

and so to ensure that a portfolio value is insensitive to

this parameter we can vega hedge. This means that we

hedge one option with both the underlying and another

option in such a way that both the delta and the vega,

the sensitivity of the portfolio value to volatility, are

76 Frequently Asked Questions In Quantitative Finance

zero. This is often quite satisfactory in practice but is

usually theoretically inconsistent; we should not use a

constant volatility (basic Black–Scholes) model to cal-

culate sensitivities to parameters that are assumed not

to vary. The distinction between variables (underlying

asset price and time) and parameters (volatility, divi-

dend yield, interest rate) is extremely important here.

It is justifiable to rely on sensitivities of prices to vari-

ables, but usually not sensitivity to parameters. To get

around this problem it is possible to independently

model volatility, etc., as variables themselves. In such a

way it is possible to build up a consistent theory.

Static hedging

There are quite a few problems with delta

hedging, on both the practical and the theoretical side.

In practice, hedging must be done at discrete times

and is costly. Sometimes one has to buy or sell a pro-

hibitively large number of the underlying in order to

follow the theory. This is a problem with barrier options

and options with discontinuous payoff. On the theoreti-

cal side, the model for the underlying is not perfect, at

the very least we do not know parameter values accu-

rately. Delta hedging alone leaves us very exposed to

the model, this is model risk. Many of these problems

can be reduced or eliminated if we follow a strategy

of static hedging as well as delta hedging; buy or sell

more liquid traded contracts to reduce the cashflows in

the original contract. The static hedge is put into place

now, and left until expiry. In the extreme case where

an exotic contract has all of its cashflows matched by

cashflows from traded options then its value is given by

the cost of setting up the static hedge; a model is not

needed. (But then the option wasn’t exotic in the first

place.)

Superhedging

In incomplete markets you cannot eliminate

all risk by classical dynamic delta hedging. But some-

times you can superhedge meaning that you construct

Chapter 2: FAQs

77

a portfolio that has a positive payoff whatever hap-

pens to the market. A simple example of this would be

to superhedge a short call position by buying one of

the stock, and never rebalancing. Unfortunately, as you

can probably imagine, and certainly as in this example,

superhedging might give you prices that differ vastly

from the market.

Margin hedging

Often what causes banks, and other

institutions, to suffer during volatile markets is not

the change in the paper value of their assets but the

requirement to suddenly come up with a large amount

of cash to cover an unexpected margin call. Examples

where margin has caused significant damage are Met-

allgesellschaft and Long Term Capital Management.

Writing options is very risky. The downside of buy-

ing an option is just the initial premium, the upside may

be unlimited. The upside of writing an option is limited,

but the downside could be huge. For this reason, to

cover the risk of default in the event of an unfavourable

outcome, the clearing houses that register and settle

options insist on the deposit of a margin by the writers

of options. Margin comes in two forms, the initial mar-

gin and the maintenance margin. The initial margin is

the amount deposited at the initiation of the contract.

The total amount held as margin must stay above a pre-

scribed maintenance margin. If it ever falls below this

level then more money (or equivalent in bonds, stocks,

etc.) must be deposited. The amount of margin that

must be deposited depends on the particular contract.

A dramatic market move could result in a sudden large

margin call that may be difficult to meet. To prevent this

situation it is possible to margin hedge. That is, set up a

portfolio such that margin calls on one part of the port-

folio are balanced by refunds from other parts. Usually

over-the-counter contracts have no associated margin

requirements and so won’t appear in the calculation.

78 Frequently Asked Questions In Quantitative Finance

Crash (Platinum) hedging

The final variety of hedging is

specific to extreme markets. Market crashes have at

least two obvious effects on our hedging. First of all,

the moves are so large and rapid that they cannot

be traditionally delta hedged. The convexity effect is

not small. Second, normal market correlations become

meaningless. Typically all correlations become one (or

minus one). Crash or Platinum hedging exploits the

latter effect in such a way as to minimize the worst

possible outcome for the portfolio. The method, called

CrashMetrics, does not rely on parameters such as

volatilities and so is a very robust hedge. Platinum

hedging comes in two types: hedging the paper value

of the portfolio and hedging the margin calls.

References and Further Reading

Taleb, NN 1997 Dynamic Hedging. John Wiley & Sons

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

79

What is Marking to Market and How

Does it Affect Risk Management in

Derivatives Trading?

Short Answer

Marking to market means valuing an instrument at the

price at which it is currently trading in the market. If

you buy an option because you believe it is undervalued

then you will not see any profit appear immediately, you

will have to wait until the market value moves into line

with your own estimate. With an option this may not

happen until expiration. When you hedge options you

have to choose whether to use a delta based on the

implied volatility or your own estimate of volatility. If

you want to avoid fluctuations in your mark-to-market

P&L you will hedge using the implied volatility, even

though you may believe this volatility to be incorrect.

Example

A stock is trading at $47, but you think it is seriously

undervalued. You believe that the value should be $60.

You buy the stock. How much do you tell people your

little ‘portfolio’ is worth? $47 or $60? If you say $47

then you are marking to market, if you say $60 you

are marking to (your) model. Obviously this is open to

serious abuse and so it is usual, and often a regulatory

requirement, to quote the mark-to-market value. If you

are right about the stock value then the profit will be

realized as the stock price rises. Patience, my son.

Long Answer

If instruments are liquid, exchange traded, then marking

to market is straightforward. You just need to know

the most recent market-traded price. Of course, this

doesn’t stop you also saying what you believe the value

80 Frequently Asked Questions In Quantitative Finance

to be, or the profit you expect to make. After all, you

presumably entered the trade because you thought you

would make a gain.

Hedge funds will tell their investors their Net Asset

Value based on the mark-to-market values of the liquid

instruments in their portfolio. They may estimate future

profit, although this is a bit of a hostage to fortune.

With futures and short options there are also margins to

be paid, usually daily, to a clearing house as a safeguard

against credit risk. So if prices move against you you

may have to pay a maintenance margin. This will be

based on the prevailing market values of the futures

and short options. (There is no margin on long options

positions because they are paid for up front, from which

point the only way is up.)

Marking to market of exchange-traded instruments is

clearly very straightforward. But what about exotic or

over-the-counter (OTC) contracts? These are not traded

actively, they may be unique to you and your counter-

party. These instruments have to be marked to model.

And this obviously raises the question of which model

to use. Usually in this context the ‘model’ means the

volatility, whether in equity markets, FX or fixed income.

So the question about which model to use becomes a

question about which volatility to use.

Here are some possible ways of marking OTC contracts.

• The trader uses his own volatility. Perhaps his best

forecast going forward. This is very easy to abuse, it

is very easy to rack up an imaginary profit this way.

Whatever volatility is used it cannot be too far from

the market’s implied volatilities on liquid options with

the same underlying.

Chapter 2: FAQs

81

• Use prices obtained from brokers. This has the

advantage of being real, tradeable prices, and

unprejudiced. The main drawback is that you can’t be

forever calling brokers for prices with no intention of

trading. They get very annoyed. And they won’t give

you tickets to Wimbledon anymore.

• Use a volatility model that is calibrated to vanillas.

This has the advantage of giving prices that are

consistent with the information in the market, and

are therefore arbitrage free. Although there is always

the question of which volatility model to use,

deterministic, stochastic, etc., so ‘arbitrage freeness’

is in the eye of the modeller. It can also be time

consuming to have to crunch prices frequently.

One subtlety concerns the marking method and the

hedging of derivatives. Take the simple case of a vanilla

equity option bought because it is considered cheap.

There are potentially three different volatilities here:

implied volatility; forecast volatility; hedging volatility.

In this situation the option, being exchanged traded,

would probably be marked to market using the implied

volatility, but the ultimate profit will depend on the real-

ized volatility (let’s be optimistic and assume it is as

forecast) and also how the option is hedged. Hedging

using implied volatility in the delta formula theoreti-

cally eliminates the otherwise random fluctuations in

the mark-to-market value of the hedged option port-

folio, but at the cost of making the final profit path

dependent, directly related to realized gamma along the

stock’s path.

By marking to market, or using a model-based marking

that is as close to this as possible, your losses will

be plain to see. If your theoretically profitable trade is

doing badly you will see your losses mounting up. You

may be forced to close your position if the loss gets

to be too large. Of course, you may have been right

82 Frequently Asked Questions In Quantitative Finance

in the end, just a bit out in the timing. The loss could

have reversed, but if you have closed out your position

previously then tough. Having said that, human nature is

such that people tend to hold onto losing positions too

long on the assumption that they will recover, yet close

out winning positions too early. Marking to market will

therefore put some rationality back into your trading.

References and Further Reading

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

83

What is the Efficient Markets

Hypothesis?

Short Answer

An efficient market is one where it is impossible to beat

the market because all information about securities is

already reflected in their prices.

Example

Or rather a counter-example, ‘‘I’d be a bum in the street

with a tin cup if the markets were efficient,’’ Warren

Buffett.

Long Answer

The concept of market efficiency was proposed by

Eugene Fama in the 1960s. Prior to that it had been

assumed that excess returns could be made by careful

choice of investments. Here and in the following the

references to ‘excess returns’ refers to profit above the

risk-free rate not explained by a risk premium, i.e., the

reward for taking risk. Fama argued that since there

are so many active, well-informed and intelligent mar-

ket participants securities will be priced to reflect all

available information. Thus was born the idea of the

efficient market, one where it is impossible to beat the

market.

There are three classical forms of the Efficient Markets

Hypothesis (EMH). These are weak form, semi-strong

form and strong form.

Weak-form efficiency

In weak-form efficiency excess returns

cannot be made by using investment strategies based on

historical prices or other historical financial data. If this

form of efficiency is true then it will not be possible to

84 Frequently Asked Questions In Quantitative Finance

make excess returns by using methods such as techni-

cal analysis. A trading strategy incorporating historical

data, such as price and volume information, will not

systematically outperform a buy-and-hold strategy. It is

often said that current prices accurately incorporate all

historical information, and that current prices are the

best estimate of the value of the investment. Prices will

respond to news, but if this news is random then price

changes will also be random. Technical analysis will not

be profitable.

Semi-strong form efficiency

In the semi-strong form of the

EMH a trading strategy incorporating current publicly

available fundamental information (such as financial

statements) and historical price information will not sys-

tematically outperform a buy-and-hold strategy. Share

prices adjust instantaneously to publicly available new

information, and no excess returns can be earned by

using that information. Fundamental analysis will not be

profitable.

Strong-form efficiency

In strong-form efficiency share prices

reflect all information, public and private, fundamental

and historical, and no one can earn excess returns.

Inside information will not be profitable.

Of course, tests of the EMH should always allow for

transaction costs associated with trading and the inter-

nal efficiency of trade execution.

A weaker cousin of EMH is the Adaptive Market Hypoth-

esis of Andrew Lo. This idea is related to behavioural

finance and proposes that market participants adapt to

changing markets, information, models, etc., in such a

way as to lead to market efficiency but in the mean-

time there may well be exploitable opportunities for

Chapter 2: FAQs

85

excess returns. This is commonly seen when new con-

tracts, exotic derivatives, are first created leading to

a short period of excess profit before the knowledge

diffuses and profit margins shrink. The same is true of

previously neglected sources of convexity and there-

fore value. A profitable strategy can exist for a while

but perhaps others find out about it, or because of the

exploitation of the profit opportunity, either way that

efficiency disappears.

The Grossman–Stiglitz paradox says that if a market

were efficient, reflecting all available information, then

there would be no incentive to acquire the information

on which prices are based. Essentially the job has been

done for everyone. This is seen when one calibrates

a model to market prices of derivatives, without ever

studying the statistics of the underlying process.

The validity of the EMH, whichever form, is of great

importance because it determines whether anyone can

outperform the market, or whether successful investing

is all about luck. EMH does not require investors to

behave rationally, only that in response to news or data

there will be a sufficiently large random reaction that

an excess profit cannot be made. Market bubbles, for

example, do not invalidate EMH provided they cannot

be exploited.

There have been many studies of the EMH, and the

validity of its different forms. Many early studies con-

cluded in favour of the weak form. Bond markets and

large-capitalization stocks are thought to be highly effi-

cient, smaller stocks less so. Because of different quality

of information among investors and because of an emo-

tional component, real estate is thought of as being

quite inefficient.

86 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Fama, EF 1965 Random Walks in Stock Market Prices. Financial

Analysts Journal September/October

Lo, A 2004 The Adaptive Markets Hypothesis: Market Efficiency

from an evolutionary perspective. J. of Portfolio Management

30 15–29

Chapter 2: FAQs

87

What are the Most Useful Performance

Measures?

Short Answer

Performance measures are used to quantify the results

of a trading strategy. They are usually adjusted for risk.

The most popular is the Sharpe ratio.

Example

One stock has an average growth of 10% per annum,

another is 30% per annum. You’d rather invest in the

second, right? What if I said that the first had a volatility

of only 5%, whereas the second was 20%, does that

make a difference?

Long Answer

Performance measures are used to determine how suc-

cessful an investment strategy has been. When a hedge

fund or trader is asked about past performance the

first question is usually ‘‘What was your return?’’ Later

maybe ‘‘What was your worst month?’’ These are both

very simple measures of performance. The more sen-

sible measures make allowance for the risk that has

been taken, since a high return with low risk is much

better than a high return with a lot of risk.

Sharpe Ratio

The Sharpe ratio is probably the most

important non-trivial risk-adjusted performance measure.

It is calculated as

µ − r

Sharpe ratio =

σ

where µ is the return on the strategy over some spec-

ified period, r is the risk-free rate over that period and

σ is the standard deviation of returns. The Sharpe ratio

will be quoted in annualized terms. A high Sharpe ratio

is intended to be a sign of a good strategy.

88 Frequently Asked Questions In Quantitative Finance

If returns are normally distributed then the Sharpe ratio

is related to the probability of making a return in excess

of the risk-free rate. In the expected return versus risk

diagram of Modern Portfolio Theory the Sharpe ratio is

the slope of the line joining each investment to the risk-

free investment. Choosing the portfolio that maximizes

the Sharpe ratio will give you the Market Portfolio. We

also know from the Central Limit Theorem that if you

have many different investments all that matters is the

mean and the standard deviation. So as long as the CLT

is valid the Sharpe ratio makes sense.

The Sharpe ratio has been criticized for attaching equal

weight to upside ‘risk’ as downside risk since the stan-

dard deviation incorporates both in its calculation. This

may be important if returns are very skewed.

Modigliani-Modigliani Measure

The Modigliani-Modigliani or

M2 measure is a simple linear transformation of the

Sharpe ratio:

M2 = r + v × Sharpe

where v is the standard deviation of returns of the relevant

benchmark. This is easily interpreted as the return you

would expect from your portfolio if it were (de)leveraged

to have the same volatility as the benchmark.

Sortino Ratio

The Sortino ratio is calculated in the same

way as the Sharpe ratio except that it uses the square

root of the semi-variance as the denominator measuring

risk. The semi variance is measured in the same way

as the variance except that all data points with positive

return are replaced with zero, or with some target value.

This measure ignores upside ‘risk’ completely. How-

ever, if returns are expected to be normally distributed

the semi variance will be statistically noisier than the

Chapter 2: FAQs

89

variance because fewer data points are used in its cal-

culation.

Treynor Ratio

The Treynor or Reward-to-variability Ratio

is another Sharpe-like measure, but now the denomin-

ator is the systematic risk, measured by the portfolio’s

beta, (see Capital Asset Pricing Model), instead of the

total risk:

µ − r

Treynor ratio =

.

β

In a well-diversified portfolio Sharpe and Treynor are

similar, but Treynor is more relevant for less diversified

portfolios or individual stocks.

Information Ratio

The Information ratio is a different

type of performance measure in that it uses the idea

of tracking error. The numerator is the return in excess

of a benchmark again, but the denominator is the stan-

dard deviation of the differences between the portfolio

returns and the benchmark returns, the tracking error.

µ − r

Information ratio =

.

Tracking error

This ratio gives a measure of the value added by a

manager relative to their benchmark.

References and Further Reading

Modigliani, F & Modigliani, L 1997 Risk-adjusted performance.

J. Portfolio Manag. 23 (2) 45–54

Sharpe, WF 1966 Mutual Fund Performance. Journal of Business

January 119–138

Sortino FA & van der Meer, R 1991 Downside risk. J. Portfolio

Manag. 27–31

Treynor, JL 1966 How to rate management investment funds.

Harvard Business Review 43 63–75

90 Frequently Asked Questions In Quantitative Finance

What is a Utility Function and How is

it Used?

Short Answer

A utility function represents the ‘worth,’ ‘happiness’ or

‘satisfaction’ associated with goods, services, events,

outcomes, levels of wealth, etc. It can be used to rank

outcomes, to aggregate ‘happiness’ across individuals

and to value games of chance.

Example

You own a valuable work of art, you are going to put

it up for auction. You don’t know how much you will

make but the auctioneer has estimated the chances of

achieving certain amounts. Someone then offers you a

guaranteed amount provided you withdraw the painting

from the auction. Should you take the offer or take your

chances? Utility theory can help you make that decision.

Long Answer

The idea is not often used in practice in finance but

is common in the literature, especially economics lit-

erature. The utility function allows the ranking of the

otherwise incomparable, and is used to explain people’s

actions; rational people are supposed to act so as to

increase their utility.

When a meaningful numerical value is used to represent

utility this is called cardinal utility. One can then talk

about one thing having three times the utility of another,

and one can compare utility from person to person. If

the ordering of utility is all that matters (so that one

is only concerned with ranking of preferences, not the

numerical value) then this is called ordinal utility.

If we denote a utility function by U ( W ) where W is

the ‘wealth,’ then one would expect utility functions to

Chapter 2: FAQs

91

have certain commonsense properties. In the following

denotes differentiation with respect to W .

• The function U( W) can vary among investors, each

will have a different attitude to risk for example.

• U ( W) ≥ 0: more is preferred to less. If it is a strict

inequality then satiation is not possible, the investor

will always prefer more than he has. This slope

measures the marginal improvement in utility with

changes in wealth.

• Usually U ( W) < 0: the utility function is strictly concave. Since this is the rate of change of the

marginal ‘happiness,’ it gets harder and harder to

increase happiness as wealth increases. An investor

with a concave utility function is said to be risk

averse. This property is often referred to as the law

of diminishing returns.

The final point in the above leads to definitions for mea-

surement of risk aversion. The absolute risk aversion

function is defined as

A( W ) = − U ( W ) .

U ( W )

The relative risk aversion function is defined as

WU ( W )

R( W ) = −

= WA( W) .

U ( W )

Utility functions are often used to analyze random

events. Suppose a monetary amount is associated with

the number of spots on a rolled dice. You could

calculate the expected winnings as the average of all

of the six amounts. But what if the amounts were

$1, $2, $3, $4, $5 and $6,000,000? Would the average,

$1,000,002.5, be meaningful? Would you be willing to pay

$1,000,000 to enter this as a bet? After all, you expect

to make a profit. A more sensible way of valuing this

game might be to look at the utility of each of the six

92 Frequently Asked Questions In Quantitative Finance

outcomes, and then average the utility. This leads on to

the idea of certainty equivalent wealth.

When the wealth is random, and all outcomes can be

assigned a probability, one can ask what amount of

certain wealth has the same utility as the expected

utility of the unknown outcomes. Simply solve

U ( Wc) = E[ U ( W )] .

The quantity of wealth Wc that solves this equation is

called the certainty equivalent wealth. One is therefore

indifferent between the average of the utilities of the

random outcomes and the guaranteed amount Wc. As an

example, consider the above dice-rolling game, suppos-

ing our utility function is U ( W ) = − 1 η e− ηW . With η = 1

200

180

160

t

140

ivalenu

120

eqty

100

ainrteC 80

60

40

20

0

0.001

0.01

0.1

1

10

100

ln(eta)

Figure 2-5: Certainty equivalent as a function of the risk-aversion

parameter for example in the text.

Chapter 2: FAQs

93

we find that the certainty equivalent is $2.34. So we

would pay this amount or less to play the game. Above

is a plot of the certainty equivalent for this example as a

function of the risk-aversion parameter η. Observe how

this decreases the greater the risk aversion.

References and Further Reading

Ingersoll, JE Jr 1987 Theory of Financial Decision Making. Row-

man & Littlefield

94 Frequently Asked Questions In Quantitative Finance

What is Brownian Motion and What

are its Uses in Finance?

Short Answer

Brownian Motion is a stochastic process with station-

ary independent normally distributed increments and

which also has continuous sample paths. It is the most

common stochastic building block for random walks in

finance.

Example

Pollen in water, smoke in a room, pollution in a river,

are all examples of Brownian motion. And this is the

common model for stock prices as well.

Long Answer

Brownian motion (BM) is named after the Scottish

botanist who first described the random motions of

pollen grains suspended in water. The mathematics

of this process were formalized by Bachelier, in an

option-pricing context, and by Einstein. The math-

ematics of BM is also that of heat conduction and

diffusion.

Mathematically, BM is a continuous, stationary, stochas-

tic process with independent normally distributed incre-

ments. If Wt is the BM at time t then for every t, τ ≥ 0, Wt+ τ − Wt is independent of { Wu : 0 ≤ u ≤ t}, and has a normal distribution with zero mean and variance τ .

The important properties of BM are as follows.

• Finiteness: the scaling of the variance with the time

step is crucial to BM remaining finite.

• Continuity: the paths are continuous, there are no

discontinuities. However, the path is fractal, and not

differentiable anywhere.

Chapter 2: FAQs

95

• Markov: the conditional distribution of Wt given

information up until τ < t depends only on Wτ .

• Martingale: given information up until τ < t the

conditional expectation of Wt is Wτ .

• Quadratic variation: if we divide up the time 0 to t in

a partition with n + 1 partition points ti = it/n then

n

2

Wt − W

→ t.

j

tj−1

j=1

• Normality: Over finite time increments ti−1 to ti,

Wt − W

is normally distributed with mean zero

i

ti−1

and variance ti − ti−1.

BM is a very simple yet very rich process, extremely

useful for representing many random processes es-

pecially those in finance. Its simplicity allows calcula-

tions and analysis that would not be possible with other

processes. For example, in option pricing it results in

simple closed-form formulæ for the prices of vanilla

options. It can be used as a building block for random

walks with characteristics beyond those of BM itself.

For example, it is used in the modelling of interest rates

via mean-reverting random walks. Higher-dimensional

versions of BM can be used to represent multi-factor

random walks, such as stock prices under stochas-

tic volatility.

One of the unfortunate features of BM is that it gives

returns distributions with tails that are unrealistically

shallow. In practice, asset returns have tails that are

much fatter than that given by the normal distribution

of BM. There is even some evidence that the distribu-

tion of returns have infinite second moment. Despite

this, and the existence of financial theories that do

incorporate such fat tails, BM motion is easily the

most common model used to represent random walks

in finance.

96 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Bachelier, L 1995 Théorie de la Spéculation. Jacques Gabay

Brown, R 1827 A Brief Account of Microscopical Observations.

London

Stachel, J (ed.) 1990 The Collected Papers of Albert Einstein.

Princeton University Press

Wiener, N 1923 Differential space. J. Math. and Phys. 58 131–74

Chapter 2: FAQs

97

What is Jensen’s Inequality and What

is its Role in Finance?

Short Answer

Jensen’s Inequality states1 that if f (·) is a convex func-

tion and x is a random variable then

E [ f ( x)] ≥ f (E[ x] ) .

This justifies why non-linear instruments, options, have

inherent value.

Example

You roll a die, square the number of spots you get, you

win that many dollars. For this exercise f ( x) is x 2, a

convex function. So E [ f ( x)] is 1 + 2 + 9 + 16 + 25 + 36 =

91 divided by 6, so 15 1 / 6. But E[ x] is 3 1 / 2 so f (E[ x] ) is 12 1 / 4.

Long Answer

A function f (·) is convex on an interval if for every x

and y in that interval

f ( λx + (1 − λ) y) ≥ λf ( x) + (1 − λ) f ( y) for any 0 ≤ λ ≤ 1. Graphically this means that the line

joining the points ( x, f ( x)) and ( y, f ( y)) is nowhere lower than the curve. (Concave is the opposite, simply − f is

convex.)

Jensen’s inequality and convexity can be used to explain

the relationship between randomness in stock prices

and the value inherent in options, the latter typically

having some convexity.

Suppose that a stock price S is random and we want

to consider the value of an option with payoff P( S). We

1This is the probabilistic interpretation of the inequality.

98 Frequently Asked Questions In Quantitative Finance

could calculate the expected stock price at expiration

as E[ ST ], and then the payoff at that expected price

P( E[ ST ]). Alternatively we could look at the various

option payoffs and then calculate the expected payoff

as E[ P( ST )]. The latter makes more sense, and is indeed

the correct way to value options, provided the expec-

tation is with respect to the risk-neutral stock price. If

the payoff is convex then

E[ P( ST )] ≥ P( E[ ST ]) .

We can get an idea of how much greater the left-hand

side is than the right-hand side by using a Taylor series

approximation around the mean of S. Write

S = S + ,

where S = E[ S], so E[ ] = 0. Then

E [ f ( S)] = E f ( S + ) = E f ( S) + f ( S) + 1 2 f ( S) + · · ·

2

≈ f ( S) + 1 f ( S) E 2

2

= f ( E[ S]) + 1 f ( E[ S]) E 2 .

2

Therefore the left-hand side is greater than the right by

approximately

1 f ( E[ S]) E 2 .

2

This shows the importance of two concepts

• f ( E[ S]): The convexity of an option. As a rule this adds value to an option. It also means that any

intuition we may get from linear contracts (forwards

and futures) might not be helpful with non-linear

instruments such as options.

• E 2 : Randomness in the underlying, and its

variance. Modelling randomness is the key to

modelling options.

The lesson to learn from this is that whenever a con-

tract has convexity in a variable or parameter, and that

Chapter 2: FAQs

99

variable or parameter is random, then allowance must

be made for this in the pricing. To do this correctly

requires a knowledge of the amount of convexity and

the amount of randomness.

References and Further Reading

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

100 Frequently Asked Questions In Quantitative Finance

What is Itô’s Lemma?

Short Answer

Itô’s lemma is a theorem in stochastic calculus. It tells

you that if you have a random walk, in y, say, and a

function of that randomly walking variable, call it f ( y, t), then you can easily write an expression for the random walk in f . A function of a random variable is itself

random in general.

Example

The obvious example concerns the random walk

dS = µS dt + σ S dX

commonly used to model an equity price or exchange

rate, S. What is the stochastic differential equation for

the logarithm of S, ln S?

The answer is

d(ln S) = µ − 1 σ 2 dt + σ dX.

2

Long Answer

Let’s begin by stating the theorem. Given a random

variable y satisfying the stochastic differential equation

dy = a( y, t) dt + b( y, t) dX, where dX is a Wiener process, and a function f ( y, t) that is differentiable with respect to t and twice differentiable with respect to y, then f satisfies the following

stochastic differential equation

∂f

∂f

∂ 2 f

∂f

df =

+ a( y, t)

+ 1 b( y, t)2

dt + b( y, t)

dX.

∂t

∂y

2

∂y 2

∂y

Itô’s lemma is to stochastic variables what Taylor series

is to deterministic. You can think of it as a way of

expanding functions in a series in dt, just like Taylor

series. If it helps to think of it this way then you must

remember the simple rules of thumb as follows.

Chapter 2: FAQs

101

1. Whenever you get dX 2 in a Taylor series expansion

of a stochastic variable you must replace it with dt.

2. Terms that are O( dt 3 / 2) or smaller must be ignored.

This means that dt 2, dX 3, dt dX, etc. are too small to

keep.

It is difficult to overstate the importance of Itô’s lemma

in quantitative finance. It is used in many of the deriva-

tions of the Black–Scholes option pricing model and

the equivalent models in the fixed-income and credit

worlds. If we have a random walk model for a stock

price S and an option on that stock, with value V ( S, t), then Itô’s lemma tells us how the option price changes

with changes in the stock price. From this follows the

idea of hedging, by matching random fluctuations in S

with those in V . This is important both in the theory of

derivatives pricing and in the practical management of

market risk.

Even if you don’t know how to prove Itô’s lemma you

must be able to quote it and use the result.

Sometimes we have a function of more than one

stochastic quantity. Suppose that we have a function

f ( y 1, y 2, . . . , yn, t) of n stochastic quantities and time such that

dyi = ai( y 1, y 2, . . . , yn, t) dt + bi( y 1, y 2, . . . , yn, t) dXi, where the n Wiener processes dXi have correlations ρij

then





n

∂

n

n

f

∂ 2 f

df =  ∂f +

a

+ 1

ρ

 dt

∂

i

ij bibj

t

∂y

2

i

∂yi ∂yj

i=1

i=1 j=1

n

∂

+

f

bi

dX

∂

i.

yi

i=1

102 Frequently Asked Questions In Quantitative Finance

We can understand this (if not entirely legitimately

derive it) via Taylor series by using the rules of thumb

dX 2 = dt and dX

i

idXj = ρijdt.

Another extension that is often useful in finance is to

incorporate jumps in the independent variable. These

are usually modelled by a Poisson process. This is dq

such dq = 1 with probability λ dt and is 0 with probabil-

ity 1 − λ dt. Returning to the single independent variable

case for simplicity, suppose y satisfies

dy = a( y, t) dt + b( y, t) dX + J( y, t) dq where dq is a Poisson process and J is the size of the

jump or discontinuity in y (when dq = 1) then

∂f

∂f

∂ 2 f

∂f

df =

+ a( y, t)

+ 1 b( y, t)2

dt + b( y, t)

dX

∂t

∂y

2

∂y 2

∂y

+ (f ( y + J( y, t)) − f ( y, t) ) dq.

And this is Itô in the presence of jumps.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Neftci, S 1996 An Introduction to the Mathematics of Financial

Derivatives. Academic Press

Wilmott, P 2001 Paul Wilmott Introduces Quantitative Finance.

John Wiley & Sons

Chapter 2: FAQs

103

Why Does Risk-Neutral Valuation

Work?

Short Answer

Risk-neutral valuation means that you can value options

in terms of their expected payoffs, discounted from

expiration to the present, assuming that they grow on

average at the risk-free rate.

Option value = Expected present value of payoff

(under a risk-neutral random walk).

Therefore the real rate at which the underlying grows on

average doesn’t affect the value. Of course, the volatil-

ity, related to the standard deviation of the underlying’s

return, does matter. In practice, it’s usually much, much

harder to estimate this average growth than the volatil-

ity, so we are rather spoiled in derivatives, that we only

need to estimate the relatively stable parameter, volatil-

ity.2 The reason that this is true is that by hedging an

option with the underlying we remove any exposure to

the direction of the stock, whether it goes up or down

ceases to matter. By eliminating risk in this way we also

remove any dependence on the value of risk. End result

is that we may as well imagine we are in a world in

which no one values risk at all, and all tradeable assets

grow at the risk-free rate on average.

For any derivative product, as long as we can hedge it

dynamically and perfectly (supposing we can as in the

case of known, deterministic volatility and no defaults)

the hedged portfolio loses its randomness and behaves

like a bond.

2I should emphasize the word ‘relatively.’ Volatility does vary

in reality, but probably not as much as the growth rate.

104 Frequently Asked Questions In Quantitative Finance

Example

A stock whose value is currently $44.75 is growing on

average by 15% per annum. Its volatility is 22%. The

interest rate is 4%. You want to value a call option with

a strike of $45, expiring in two months’ time. What can

you do?

First of all, the 15% average growth is totally irrelevant.

The stock’s growth and therefore its real direction does

not affect the value of derivatives. What you can do is

simulate many, many future paths of a stock with an

average growth of 4% per annum, since that is the risk-

free interest rate, and a 22% volatility, to find out where

it may be in two months’ time. Then calculate the call

payoff for each of these paths. Present value each of

these back to today, and calculate the average over all

paths. That’s your option value.

Long Answer

Risk-neutral valuation of derivatives exploits the per-

fect correlation between the changes in the value of an

option and its underlying asset. As long as the under-

lying is the only random factor then this correlation

should be perfect. So if an option goes up in value with

a rise in the stock then a long option and sufficiently

short stock position shouldn’t have any random fluc-

tuations, therefore the stock hedges the option. The

resulting portfolio is risk free.

Of course, you need to know the correct number of the

stock to sell short. That’s called the ‘delta’ and usu-

ally comes from a model. Because we usually need a

mathematical model to calculate the delta, and because

quantitative finance models are necessarily less than

perfect, the theoretical elimination of risk by delta

hedging is also less than perfect in practice. There are

several such imperfections with risk-neutral valuation.

First, it requires continuous rebalancing of the hedge.

Chapter 2: FAQs

105

Delta is constantly changing so you must always be

buying or selling stock to maintain a risk-free position.

Obviously, this is not possible in practice. Second, it

hinges on the accuracy of the model. The underlying

has to be consistent with certain assumptions, such as

being Brownian motion without any jumps, and with

known volatility.

One of the most important side effects of risk-neutral

pricing is that we can value derivatives by doing simula-

tions of the risk-neutral path of underlyings, to calculate

payoffs for the derivatives. These payoffs are then

discounted to the present, and finally averaged. This

average that we find is the contract’s fair value.

Here are some further explanations of risk-neutral pricing.

Explanation 1: If you hedge correctly in a Black–Scholes

world then all risk is eliminated. If there is no risk then

we should not expect any compensation for risk. We can

therefore work under a measure in which everything

grows at the risk-free interest rate.

Explanation 2: If the model for the asset is dS = µS dt +

σS dX then the µ s cancel in the derivation of the Black–

Scholes equation.

Explanation 3: Two measures are equivalent if they have

the same sets of zero probability. Because zero proba-

bility sets don’t change, a portfolio is an arbitrage under

one measure if and only if it is one under all equivalent

measures. Therefore a price is non-arbitrageable in the

real world if and only if it is non-arbitrageable in the

risk-neutral world. The risk-neutral price is always non-

arbitrageable. If everything has a discounted asset price

process which is a martingale then there can be no

arbitrage. So if we change to a measure in which all the

fundamental assets, for example the stock and bond,

106 Frequently Asked Questions In Quantitative Finance

are martingales after discounting, and then define the

option price to be the discounted expectation making

it into a martingale too, we have that everything is a

martingale in the risk-neutral world. Therefore there is

no arbitrage in the real world.

Explanation 4: If we have calls with a continuous dis-

tribution of strikes from zero to infinity then we can

synthesize arbitrarily well any payoff with the same

expiration. But these calls define the risk-neutral proba-

bility density function for that expiration, and so we can

interpret the synthesized option in terms of risk-neutral

random walks. When such a static replication is possi-

ble then it is model independent, we can price complex

derivatives in terms of vanillas. (Of course, the contin-

uous distribution requirement does spoil this argument

to some extent.)

It should be noted that risk-neutral pricing only works

under assumptions of continuous hedging, zero transac-

tion costs, continuous asset paths, etc. Once we move

away from this simplifying world we may find that it

doesn’t work.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Neftci, S 1996 An Introduction to the Mathematics of Financial

Derivatives. Academic Press

Chapter 2: FAQs

107

What is Girsanov’s Theorem, and Why

is it Important in Finance?

Short Answer

Girsanov’s theorem is the formal concept underlying

the change of measure from the real world to the risk-

neutral world. We can change from a Brownian motion

with one drift to a Brownian motion with another.

Example

The classical example is to start with

dS = µS dt + σ S dWt

with W being Brownian motion under one measure (the

real-world measure) and converting it to

dS = rS dt + σ S d ˜

Wt

under a different, the risk-neutral, measure.

Long Answer

First a statement of the theorem. Let Wt be a Brownian

motion with measure P and sample space . If

γt is a previsible process satisfying the constraint

E

1

T

P exp

γ 2

< ∞ then there exists an equivalent

2

0

t

measure Q on such that

t

˜

Wt = Wt +

γsds

0

is a Brownian motion.

It will be helpful if we explain some of the more techni-

cal terms in this theorem.

Sample space:

All possible future states or outcomes.

108 Frequently Asked Questions In Quantitative Finance

(Probability) Measure:

In layman’s terms, the measure

gives the probabilities of each of the outcomes in the

sample space.

Previsible:

A previsible process is one that only depends

on the previous history.

Equivalent:

Two measures are equivalent if they have

the same sample space and the same set of ‘possibil-

ities.’ Note the use of the word possibilities instead

of probabilities. The two measures can have different

probabilities for each outcome but must agree on what

is possible.

Another way of writing the above is in differential form

d ˜

Wt = dWt + γt dt.

One important point about Girsanov’s theorem is its

converse, that every equivalent measure is given by

a drift change. This implies that in the Black–Scholes

world there is only the one equivalent risk-neutral mea-

sure. If this were not the case then there would be

multiple arbitrage-free prices.

For many problems in finance Girsanov theorem is not

necessarily useful. This is often the case in the world

of equity derivatives. Straightforward Black–Scholes

does not require any understanding of Girsanov. Once

you go beyond basic Black–Scholes it becomes more

useful. For example, suppose you want to derive the

valuation partial differential equations for options under

stochastic volatility. The stock price follows the real-

world processes, P,

dS = µS dt + σ S dX 1

and

dσ = a( S, σ , t) dt + b( S, σ , t) dWX 2,

Chapter 2: FAQs

109

where dX 1 and dX 2 are correlated Brownian motions

with correlation ρ( S, σ , t).

Using Girsanov you can get the governing equation in

three steps:

1. Under a pricing measure Q, Girsanov plus the fact

that S is traded implies that

µ − r

dX 1 = d ˜ X 1 −

dt

σ

and

dX 2 = d ˜ X 2 − λ( S, σ , t) dt, where λ is the market price of volatility risk

2. Apply Itô’s formula to the discounted option price

V ( S, σ , t) = e− r( T− t) F ( S, σ , t), expanding under Q, using the formulæ for dS and dV obtained from the

Girsanov transformation

3. Since the option is traded, the coefficient of the dt

term in its Itô expansion must also be zero; this

yields the relevant equation

Girsanov and the idea of change of measure are par-

ticularly important in the fixed-income world where

practitioners often have to deal with many different

measures at the same time, corresponding to different

maturities. This is the reason for the popularity of the

BGM model and its ilk.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Lewis, A 2000 Option Valuation under Stochastic Volatility.

Finance Press

Neftci, S 1996 An Introduction to the Mathematics of Financial

Derivatives. Academic Press

110 Frequently Asked Questions In Quantitative Finance

What are the Greeks?

Short Answer

The ‘greeks’ are the sensitivities of derivatives prices

to underlyings, variables and parameters. They can be

calculated by differentiating option values with respect

to variables and/or parameters, either analytically, if

you have a closed-form formula, or numerically.

Example

Delta, = ∂V

∂ , is sensitivity of option price to the stock

S

price. Gamma, = ∂ 2 V , is the second derivative of the

∂S 2

option price to the underlying stock, it is the sensitivity

of the delta to the stock price. These two examples are

called greek because they are members of the Greek

alphabet. Some sensitivities, such as vega = ∂V

∂σ , are

still called ‘greek’ even though they aren’t in the Greek

alphabet.

Long Answer

Delta

The delta, , of an option or a portfolio of options

is the sensitivity of the option or portfolio to the under-

lying. It is the rate of change of value with respect to

the asset:

∂V

=

.

∂S

Speculators take a view on the direction of some quan-

tity such as the asset price and implement a strategy to

take advantage of their view. If they own options then

their exposure to the underlying is, to a first approxima-

tion, the same as if they own delta of the underlying.

Those who are not speculating on direction of the

underlying will hedge by buying or selling the under-

Chapter 2: FAQs

111

lying, or another option, so that the portfolio delta is

zero. By doing this they eliminate market risk.

Typically the delta changes as stock price and time

change, so to maintain a delta-neutral position the num-

ber of assets held requires continual readjustment by

purchase or sale of the stock. This is called rehedg-

ing or rebalancing the portfolio, and is an example of

dynamic hedging.

Sometimes going short the stock for hedging purposes

requires the borrowing of the stock in the first place.

(You then sell what you have borrowed, buying it back

later.) This can be costly, you may have to pay a repo

rate, the equivalent of an interest rate, on the amount

borrowed.

Gamma

The gamma, , of an option or a portfolio of

options is the second derivative of the position with

respect to the underlying:

∂ 2 V

=

.

∂S 2

Since gamma is the sensitivity of the delta to the under-

lying it is a measure of by how much or how often

a position must be rehedged in order to maintain a

delta-neutral position. If there are costs associated

with buying or selling stock, the bid-offer spread, for

example, then the larger the gamma the larger the cost

or friction caused by dynamic hedging.

Because costs can be large and because one wants

to reduce exposure to model error it is natural to try

to minimize the need to rebalance the portfolio too

frequently. Since gamma is a measure of sensitivity of

the hedge ratio to the movement in the underlying,

the hedging requirement can be decreased by a gamma-

112 Frequently Asked Questions In Quantitative Finance

neutral strategy. This means buying or selling more

options, not just the underlying.

Theta

The theta, , is the rate of change of the option

price with time.

∂V

=

.

∂t

The theta is related to the option value, the delta and

the gamma by the Black--Scholes equation.

Speed

The speed of an option is the rate of change of

the gamma with respect to the stock price.

∂ 3 V

Speed =

.

∂S 3

Traders use the gamma to estimate how much they will

have to rehedge by if the stock moves. The stock moves

by $1 so the delta changes by whatever the gamma is.

But that’s only an approximation. The delta may change

by more or less than this, especially if the stock moves

by a larger amount, or the option is close to the strike

and expiration. Hence the use of speed in a higher-order

Taylor series expansion.

Vega

The vega, sometimes known as zeta or kappa,

is a very important but confusing quantity. It is the

sensitivity of the option price to volatility.

∂V

Vega =

.

∂σ

This is completely different from the other greeks since

it is a derivative with respect to a parameter and not a

variable. This can be important. It is perfectly accept-

able to consider sensitivity to a variable, which does

vary, after all. However, it can be dangerous to measure

sensitivity to something, such as volatility, which is a

Chapter 2: FAQs

113

parameter and may, for example, have been assumed to

be constant. That would be internally inconsistent.

As with gamma hedging, one can vega hedge to reduce

sensitivity to the volatility. This is a major step towards

eliminating some model risk, since it reduces depen-

dence on a quantity that is not known very accurately.

There is a downside to the measurement of vega. It is

only really meaningful for options having single-signed

gamma everywhere. For example it makes sense to mea-

sure vega for calls and puts but not binary calls and

binary puts. The reason for this is that call and put

values (and options with single-signed gamma) have

values that are monotonic in the volatility: increase the

volatility in a call and its value increases everywhere.

Contracts with a gamma that changes sign may have

a vega measured at zero because as we increase the

volatility the price may rise somewhere and fall some-

where else. Such a contract is very exposed to volatility

risk but that risk is not measured by the vega.

Rho

ρ, is the sensitivity of the option value to the inter-

est rate used in the Black--Scholes formulæ:

∂V

ρ =

.

∂r

In practice one often uses a whole term structure of

interest rates, meaning a time-dependent rate r( t). Rho

would then be the sensitivity to the level of the rates

assuming a parallel shift in rates at all times.

Rho can also be sensitivity to dividend yield, or foreign

interest rate in a foreign exchange option.

Charm

The charm is the sensitivity of delta to time.

∂ 2 V .

∂S ∂t

114 Frequently Asked Questions In Quantitative Finance

This is useful for seeing how your hedge position will

change with time, for example up until the next time you

expect to hedge. This can be important near expiration.

Colour

The colour is the rate of change of gamma with

time.

∂ 3 V .

∂S 2 ∂t

Vanna

The Vanna is the sensitivity of delta to volatility.

∂ 2 V .

∂S ∂σ

This is used when testing sensitivity of hedge ratio to

volatility. It can be misleading at places where gamma

is small.

Vomma or Volga

The Vomma or Volga is the second

derivative of the option value with respect to volatility.

∂ 2 V .

∂σ 2

Because of Jensen’s Inequality, if volatility is stochastic

the Vomma/Volga measures convexity due to random

volatility and so gives you an idea of how much to add

(or subtract) from an option’s value.

Shadow greeks

The above greeks are defined in terms

of partial derivatives with respect to underlying, time,

volatility, etc. while holding the other variables/para-

meters fixed. That is the definition of a partial deriva-

tive.3 But, of course, the variables/parameters might, in

practice, move together. For example, a fall in the stock

3Here derivative has its mathematical meaning of that which is

differentiated not its financial meaning as an option.

Chapter 2: FAQs

115

price might be accompanied by an increase in volatility.

So one can measure sensitivity as both the underlying

and volatility move together. This is called a shadow

greek and is just like the concept of a total derivative

in, for example, fluid mechanics where one might follow

the path of a fluid particle.

References and Further Reading

Taleb, NN 1997 Dynamic Hedging. John Wiley & Sons

Wilmott, P 2001 Paul Wilmott Introduces Quantitative Finance.

John Wiley & Sons

116 Frequently Asked Questions In Quantitative Finance

Why Do Quants Like Closed-Form

Solutions?

Short Answer

Because they are fast to compute and easy to under-

stand.

Example

The Black–Scholes formulæ are simple and closed-form

and often used despite people knowing that they have

limitations, and despite being used for products for

which they were not originally intended.

Long Answer

There are various pressures on a quant when it comes

to choosing a model. What he’d really like is a model

that is

• robust: small changes in the random process for the

underlying don’t matter too much

• fast: prices and the greeks have to be quick to

compute for several reasons, so that the trade gets

done and you don’t lose out to a competitor, and so

that positions can be managed in real time as just

one small part of a large portfolio

• accurate: in a scientific sense the prices ought to be

good, perhaps matching historical data. This is

different from robust, of course

• easy to calibrate: banks like to have models that

match traded prices of simple contracts

There is some overlap in these. Fast may also mean

easy to calibrate, but not necessarily. Accurate and

robust might be similar, but again, not always.

From the scientific point of view the most important

of these is accuracy. The least important is speed. To

Chapter 2: FAQs

117

the scientist the question of calibration becomes one

concerning the existence of arbitrage. If you are a hedge

fund looking for prop trading opportunities with vanillas

then calibration is precisely what you don’t want to do.

And robustness would be nice, but maybe the financial

world is so unstable that models can never be robust.

To the practitioner he needs to be able to price quickly

to get the deal done and to manage the risk. If he is

in the business of selling exotic contracts then he will

invariably be calibrating, so that he can say that his

prices are consistent with vanillas. As long as the model

isn’t too inaccurate or sensitive, and he can add a suf-

ficient profit margin, then he will be content. So to the

practitioner speed and ability to calibrate to the market

are the most important.

The scientist and the practitioner have conflicting inter-

ests. And the practitioner usually wins.

And what could be faster than a closed-form solution?

This is why practitioners tend to favour closed forms.

They also tend to be easier to understand intuitively

than a numerical solution. The Black–Scholes formulæ

are perfect for this, having a simple interpretation in

terms of expectations, and using the cumulative distri-

bution function for the Gaussian distribution.

Such is the desire for simple formulæ that people often

use the formulæ for the wrong product. Suppose you

want to price certain Asian options based on an arith-

metic average. To do this properly in the Black–Scholes

world you would do this by solving a three-dimensional

partial differential equation or by Monte Carlo simula-

tion. But if you pretend that the averaging is geometric

and not arithmetic then often there are simple closed-

form solutions. So use those, even though they must

be wrong. The point is that they will probably be less

118 Frequently Asked Questions In Quantitative Finance

wrong than other assumptions you are making, such as

what future volatility will be.

Of course, the definition of closed form is to some

extent in the eye of the beholder. If an option can be

priced in terms of an infinite sum of hypergeometric

functions does that count? Some Asian options can be

priced that way. Or what about a closed form involving

a subtle integration in the complex plane that must

ultimately be done numerically? That is the Heston

stochastic volatility model.

If closed form is so appreciated, is it worth spending

much time seeking them out? Probably not. There are

always new products being invented and new pricing

models being devised, but they are unlikely to be of

the simple type that can be solved explicitly. Chances

are that either you will have to solve these numerically,

or approximate them by something not too dissimilar.

Approximations such as Black ’76 are probably your

best chance of finding closed-form solutions for new

products these days.

References and Further Reading

Black F 1976 The pricing of commodity contracts. Journal of

Financial Economics 3 167–79

Haug, EG 2003 Know your weapon, Parts 1 and 2. Wilmott

magazine, May and July

Haug, EG 2006 The complete Guide to Option Pricing Formulas.

McGraw-Hill

Lewis, A 2000 Option Valuation under Stochastic Volatility.

Finance Press

Chapter 2: FAQs

119

What are the Forward and Backward

Equations?

Short Answer

Forward and backward equations usually refer to the

differential equations governing the transition probabil-

ity density function for a stochastic process. They are

diffusion equations and must therefore be solved in the

appropriate direction in time, hence the names.

Example

An exchange rate is currently 1.88. What is the prob-

ability that it will be over 2 by this time next year? If

you have a stochastic differential equation model for

this exchange rate then this question can be answered

using the equations for the transition probability density

function.

Long Answer

Let us suppose that we have a random variable y evolv-

ing according to a quite general, one-factor stochastic

differential equation

dy = A( y, t) dt + B( y, t) dX.

Here A and B are both arbitrary functions of y and t.

Many common models can be written in this form,

including the lognormal asset random walk, and com-

mon spot interest rate models.

The transition probability density function p( y, t; y , t ) is the function of four variables defined by

Prob( a < y < b at time t | y at time t)

b

=

p( y, t; y , t ) dy .

a

120 Frequently Asked Questions In Quantitative Finance

This simply means the probability that the random

variable y lies between a and b at time t in the future, given that it started out with value y at time t. You can

think of y and t as being current or starting values with

y and t being future values.

The transition probability density function p( y, t; y , t ) satisfies two equations, one involving derivatives with

respect to the future state and time ( y and t ) and

called the forward equation, and the other involving

derivatives with respect to the current state and time

( y and t) and called the backward equation. These two

equations are parabolic partial differential equations not

dissimilar to the Black–Scholes equation.

The forward equation

Also known as the Fokker–Planck or

forward Kolmogorov equation this is

∂p

∂ 2

∂

= 1

( B( y , t )2 p) −

∂t

2 ∂y 2

∂y ( A( y , t ) p) .

This forward parabolic partial differential equation

requires initial conditions at time t and to be solved

for t > t.

Example:

An important example is that of the distri-

bution of equity prices in the future. If we have the

random walk

dS = µS dt + σ S dX

then the forward equation becomes

∂p

∂ 2

∂

= 1

( σ 2 S 2 p) −

∂t

2 ∂S 2

∂S ( µS p) .

A special solution of this representing a variable that

begins with certainty with value S at time t is

p( S, t; S , t )

2

− ln( S/S )+( µ− 1 σ 2)( t − t) / 2 σ 2( t − t)

=

1

e

2

.

σS 2 π( t − t)

This is plotted as a function of both S and t below.

Chapter 2: FAQs

121

Figure 2-6: The probability density function for the lognormal ran-

dom walk evolving through time.

The backward equation

Also known as the backward Kol-

mogorov equation this is

∂p

∂ 2 p

∂p

+ 1 B( y, t)2

+ A( y, t)

= 0 .

∂t

2

∂y 2

∂y

This must be solved backwards in t with specified final

data.

For example, if we wish to calculate the expected value

of some function F ( S) at time T we must solve this

equation for the function p( S, t) with

p( S, T ) = F ( S) .

Option prices

If we have the lognormal random walk for

S, as above, and we transform the dependent variable

using a discount factor according to

p( S, t) = er( T− t) V ( S, t), then the backward equation for p becomes an equation

for V which is identical to the Black–Scholes partial

differential equation. Identical but for one subtlety, the

122 Frequently Asked Questions In Quantitative Finance

equation contains a µ where Black–Scholes contains

r. We can conclude that the fair value of an option is

the present value of the expected payoff at expiration

under a risk-neutral random walk for the underlying.

Risk neutral here means replace µ with r.

References and Further Reading

Feller, W 1950 Probability Theory and Its Applications. Wiley,

New York

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

123

Which Numerical Method Should I Use

and When?

Short Answer

The three main numerical methods in common use are

Monte Carlo, finite difference and numerical quadrature.

(I’m including the binomial method as just a simplistic

version of finite differences.) Monte Carlo is great for

complex path dependency and high dimensionality,

and for problems which cannot easily be written in

differential equation form. Finite difference is best for

low dimensions and contracts with decision features

such as early exercise, ones which have a differen-

tial equation formulation. Numerical quadrature is for

when you can write the option value as a multiple

integral.

Example

You want to price a fixed-income contract using the

BGM model. Which numerical method should you use?

BGM is geared up for solution by simulation, so you

would use a Monte Carlo simulation.

You want to price an option which is paid for in instal-

ments, and you can stop paying and lose the option

at any time if you think it’s not worth keeping up the

payments. This may be one for finite-difference methods

since it has a decision feature.

You want to price a European, non path-dependent con-

tract on a basket of equities. This may be recast as a

multiple integral and so you would use a quadrature

method.

124 Frequently Asked Questions In Quantitative Finance

Long Answer

Finite-Difference methods

Finite-difference methods are designed for finding numer-

ical solutions of differential equations. Since we work

with a mesh, not unlike the binomial method, we will

find the contract value at all points is stock price-time

space. In quantitative finance that differential equation

is almost always of diffusion or parabolic type. The only

real difference between the partial differential equations

are the following:

• Number of dimensions;

• Functional form of coefficients;

• Boundary/final conditions;

• Decision features;

• Linear or non linear.

Number of dimensions:

Is the contract an option on a single

underlying or many? Is there any strong path depen-

dence in the payoff? Answers to these questions will

determine the number of dimensions in the problem. At

the very least we will have two dimensions: S or r, and

t. Finite-difference methods cope extremely well with

smaller number of dimensions, up to four, say. Above

that they get rather time consuming.

Functional form of coefficients:

The main difference between

an equity option problem and a single-factor interest

rate option problem is in the functional form of the

drift rate and the volatility. These appear in the gov-

erning partial differential equations as coefficients. The

standard model for equities is the lognormal model,

but there are many more ‘standard’ models in fixed

income. Does this matter? No, not if you are solving

Chapter 2: FAQs

125

the equations numerically, only if you are trying to find

a closed-form solution in which case the simpler the

coefficients the more likely you are to find a closed-

form solution.

Boundary/final conditions:

In a numerical scheme the differ-

ence between a call and a put is in the final condition.

You tell the finite-difference scheme how to start. And

in finite-difference schemes in finance we start at expira-

tion and work towards the present. Boundary conditions

are where we tell the scheme about things like knock-

out barriers.

Decision features:

Early exercise, instalment premiums,

chooser features, are all examples of embedded decisions

seen in exotic contracts. Coping with these numerically

is quite straightforward using finite-difference methods,

making these numerical techniques the natural ones for

such contracts. The difference between a European and

an American option is about three lines of code in a

finite-difference program and less than a minute’s coding.

Linear or non linear:

Almost all finance models are lin-

ear, so that you can solve for a portfolio of options

by solving each contract at a time and adding. Some

more modern models are non linear. Linear or non lin-

ear doesn’t make that much difference when you are

solving by finite-difference methods. So choosing this

method gives you a lot of flexibility in the type of model

you can use.

Efficiency

Finite differences are very good at coping with low

dimensions, and are the method of choice if you have

a contract with embedded decisions. They are excellent

for non-linear differential equations.

126 Frequently Asked Questions In Quantitative Finance

The time taken to price an option and calculate the

sensitivities to underlying(s) and time using the explicit

finite-difference method will be

O M −1− d/ 2 ,

where M is the number of different options in the port-

folio and we want an accuracy of , and d is the number

of dimensions other than time. So if we have a non-path-

dependent option on a single underlying then d = 1.

Note that we may need one piece of code per option,

hence M in the above.

Programme of study

If you are new to finite-difference methods and you

really want to study them, here is a suggested pro-

gramme of study.

• Explicit method/European calls, puts and binaries:

To get started you should learn the explicit method

as applied to the Black–Scholes equation for a

European option. This is very easy to programme and

you won’t make many mistakes.

• Explicit method/American calls, puts and binaries:

Not much harder is the application of the explicit

method to American options.

• Crank–Nicolson/European calls, puts and binaries:

Once you’ve got the explicit method under your belt

you should learn the Crank–Nicolson implicit method.

This is harder to program, but you will get a better

accuracy.

• Crank–Nicolson/American calls, puts and binaries:

There’s not much more effort involved in pricing

American-style options than in the pricing of

European-style options.

• Explicit method/path-dependent options: By now

you’ll be quite sophisticated and it’s time to price a

path-dependent contract. Start with an Asian option

Chapter 2: FAQs

127

with discrete sampling, and then try a

continuously-sampled Asian. Finally, try your hand at

lookbacks.

• Interest rate products: Repeat the above programme

for non-path-dependent and then path-dependent

interest rate products. First price caps and floors and

then go on to the index amortizing rate swap.

• Two-factor explicit: To get started on two-factor

problems price a convertible bond using an explicit

method, with both the stock and the spot interest

rate being stochastic.

• Two-factor implicit: The final stage is to implement

the implicit two-factor method as applied to the

convertible bond.

Monte Carlo methods

Monte Carlo methods simulate the random behaviour

underlying the financial models. So, in a sense they

get right to the heart of the problem. Always remem-

ber, though, that when pricing you must simulate the

risk-neutral random walk(s), the value of a contract is

then the expected present value of all cashflows. When

implementing a Monte Carlo method look out for the

following:

• Number of dimensions;

• Functional form of coefficients;

• Boundary/final conditions;

• Decision features;

• Linear or non linear.

again!

Number of dimensions:

For each random factor you will

have to simulate a time series. It will obviously take

longer to do this, but the time will only be proportional

to number of factors, which isn’t so bad. This makes

128 Frequently Asked Questions In Quantitative Finance

Monte Carlo methods ideal for higher dimensions when

the finite-difference methods start to crawl.

Functional form of coefficients:

As with the finite-difference

methods it doesn’t matter too much what the drift and

volatility functions are in practice, since you won’t be

looking for closed-form solutions.

Boundary/final conditions:

These play a very similar role

as in finite differences. The final condition is the payoff

function and the boundary conditions are where we

implement trigger levels etc.

Decision features:

When you have a contract with embed-

ded decisions the Monte Carlo method becomes cum-

bersome. This is easily the main drawback for simula-

tion methods. When we use the Monte Carlo method

we only find the option value at today’s stock price and

time. But to correctly price an American option, say, we

need to know what the option value would be at every

point in stock price-time space. We don’t typically find

this as part of the Monte Carlo solution.

Linear or non linear:

Simulation methods also cope poorly

with non-linear models. Some models just don’t have a

useful interpretation in terms of probabilities and expec-

tations so you wouldn’t expect them to be amenable to

solution by methods based on random simulations.

Efficiency

If we want an accuracy of and we have d underlyings

then the calculation time is

O d −3 .

It will take longer to price the greeks, but, on the pos-

itive side, we can price many options at the same time

for almost no extra time cost.

Chapter 2: FAQs

129

Programme of study

Here is a programme of study for the Monte Carlo path-

simulation methods.

• European calls, puts and binaries on a single equity:

Simulate a single stock path, the payoff for an option,

or even a portfolio of options, calculate the expected

payoff and present value to price the contract.

• Path-dependent option on a single equity: Price a

barrier, Asian, lookback, etc.

• Options on many stocks: Price a multi-asset contract

by simulating correlated random walks. You’ll see

how time taken varies with number of dimensions.

• Interest rate derivatives, spot rate model: This is not

that much harder than equities. Just remember to

present value along each realized path of rates before

taking the expectation across all paths.

• HJM model: Slightly more ambitious is the HJM

interest rate model. Use a single factor, then two

factors

etc.

• BGM model: A discrete version of HJM.

Numerical integration

Occasionally one can write down the solution of an

option-pricing problem in the form of a multiple integral.

This is because you can interpret the option value as an

expectation of a payoff, and an expectation of the payoff

is mathematically just the integral of the product of that

payoff function and a probability density function. This

is only possible in special cases. The option has to be

European, the underlying stochastic differential equation

must be explicitly integrable (so the lognormal random

walk is perfect for this) and the payoff shouldn’t usually

be path dependent. So if this is possible then pricing is

130 Frequently Asked Questions In Quantitative Finance

easy . . . you have a formula. The only difficulty comes

in turning this formula into a number. And that’s the

subject of numerical integration or quadrature. Look out

for the following.

• Can you write down the value of an option as an

integral?

That’s it in a nutshell.

Efficiency

There are several numerical quadrature methods. But

the two most common are based on random number

generation again. One uses normally distributed num-

bers and the other uses what are called low-discrepancy

sequences. The low-discrepancy numbers are clever in

that they appear superficially to be random but don’t

have the inevitable clustering that truly random num-

bers have.

Using the simple normal numbers, if you want an accu-

racy of and you are pricing M options the time taken

will be

O M −2 .

If you use the low-discrepancy numbers the time taken

will be

O M −1 .

You can see that this method is very fast, unfortunately

it isn’t often applicable.

Programme of study

Here is a programme of study for the numerical quadra-

ture methods.

Chapter 2: FAQs

131

• European calls, puts and binaries on a single equity

using normal numbers: Very simple. You will be

evaluating a single integral.

• European calls, puts and binaries on several

underlying lognormal equities, using normal

numbers: Very simple again. You will be evaluating a

multiple integral.

• Arbitrary European, non-path-dependent payoff, on

several underlying lognormal equities, using normal

numbers: You’ll only have to change a single

function.

• Arbitrary European, non-path-dependent payoff, on

several underlying lognormal equities, using

low-discrepancy numbers: Just change the source of

the random numbers in the previous code.

Summary

Subject

FD

MC

Quad.

Low dimensions

Good

Inefficient

Good

High dimensions

Slow

Excellent

Good

Path dependent

Depends

Excellent

Not good

Greeks

Excellent

Not good

Excellent

Portfolio

Inefficient Very good Very good

Decisions

Excellent

Poor

V. poor

Non linear

Excellent

Poor

V. poor

References and Further Reading

Ahmad, R 2007 Numerical and Computational Methods for

Derivative Pricing. John Wiley & Sons

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

132 Frequently Asked Questions In Quantitative Finance

What is Monte Carlo Simulation?

Short Answer

Monte Carlo simulations are a way of solving probabil-

istic problems by numerically ‘imagining’ many possible

scenarios or games so as to calculate statistical proper-

ties such as expectations, variances or probabilities of

certain outcomes. In finance we use such simulations to

represent the future behaviour of equities, exchange

rates, interest rates, etc. so as to either study the

possible future performance of a portfolio or to price

derivatives.

Example

We hold a complex portfolio of investments, we would

like to know the probability of losing money over the

next year since our bonus depends on us making a

profit. We can estimate this probability by simulating

how the individual components in our portfolio might

evolve over the next year. This requires us to have a

model for the random behaviour of each of the assets,

including the relationship or correlation between them,

if any.

Some problems which are completely deterministic can

also be solved numerically by running simulations, most

famously finding a value for π.

Long Answer

It is clear enough that probabilistic problems can be

solved by simulations. What is the probability of tossing

heads with a coin, just toss the coin often enough and

you will find the answer. More on this and its relevance

to finance shortly. But many deterministic problems

can also be solved this way, provided you can find a

probabilistic equivalent of the deterministic problem.

A famous example of this is Buffon’s needle, a problem

Chapter 2: FAQs

133

and solution dating back to 1777. Draw parallel lines

on a table one inch apart. Drop a needle, also one inch

long, onto this table. Simple trigonometry will show

you that the probability of the needle touching one of

the lines is 2 /π. So conduct many such experiments to

get an approximation to π. Unfortunately because of

the probabilistic nature of this method you will have

to drop the needle many billions of times to find π

accurate to half a dozen decimal places.

There can also be a relationship between certain types

of differential equation and probabilistic methods. Sta-

nislaw Ulam, inspired by a card game, invented this

technique while working on the Manhattan Project

towards the development of nuclear weapons. The name

‘Monte Carlo’ was given to this idea by his colleague

Nicholas Metropolis.

Monte Carlo simulations are used in financial problems

for solving two types of problems:

• Exploring the statistical properties of a portfolio of

investments or cashflows to determine quantities

such as expected returns, risk, possible downsides,

probabilities of making certain profits or losses, etc.

• Finding the value of derivatives by exploiting the

theoretical relationship between option values and

expected payoff under a risk-neutral random walk.

Exploring portfolio statistics

The most successful quantita-

tive models represent investments as random walks.

There is a whole mathematical theory behind these

models, but to appreciate the role they play in portfo-

lio analysis you just need to understand three simple

concepts.

First, you need an algorithm for how the most basic

investments evolve randomly. In equities this is often

134 Frequently Asked Questions In Quantitative Finance

the lognormal random walk. (If you know about the

real/risk-neutral distinction then you should know that

you will be using the real random walk here.) This can

be represented on a spreadsheet or in code as how

a stock price changes from one period to the next by

adding on a random return. In the fixed-income world

you may be using the BGM model to model how inter-

est rates of various maturities evolve. In credit you may

have a model that models the random bankruptcy of a

company. If you have more than one such investment

that you must model then you will also need to repre-

sent any interrelationships between them. This is often

achieved by using correlations.

Once you can perform such simulations of the basic

investments then you need to have models for more

complicated contracts that depend on them, these are

the options/derivatives/contingent claims. For this you

need some theory, derivatives theory. This the second

concept you must understand.

Finally, you will be able to simulate many thousands,

or more, future scenarios for your portfolio and use

the results to examine the statistics of this portfolio.

This is, for example, how classical Value at Risk can be

estimated, among other things.

Pricing derivatives

We know from the results of risk-

neutral pricing that in the popular derivatives theories

the value of an option can be calculated as the present

value of the expected payoff under a risk-neutral random

walk. And calculating expectations for a single contract

is just a simple example of the above-mentioned port-

folio analysis, but just for a single option and using

the risk-neutral instead of the real random walk. Even

though the pricing models can often be written as deter-

ministic partial differential equations they can be solved

in a probabilistic way, just as Stanislaw Ulam noted for

Chapter 2: FAQs

135

other, non-financial, problems. This pricing methodol-

ogy for derivatives was first proposed by the actuarially

trained Phelim Boyle in 1977.

Whether you use Monte Carlo for probabilistic or deter-

ministic problems the method is usually quite simple to

implement in basic form and so is extremely popular in

practice.

References and Further Reading

Boyle, P 1977 Options: a Monte Carlo approach. Journal of

Financial Economics 4 323–338

Glasserman, P 2003 Monte Carlo Methods in Financial Engineer-

ing. Springer Verlag

Jäckel, P 2002 Monte Carlo Methods in Finance. John Wiley &

Sons

136 Frequently Asked Questions In Quantitative Finance

What is the Finite-difference Method?

Short Answer

The finite-difference method is a way of approximating

differential equations, in continuous variables, into differ-

ence equations, in discrete variables, so that they may

be solved numerically. It is a method particularly useful

when the problem has a small number of dimensions,

that is, independent variables.

Example

Many financial problems can be cast as partial dif-

ferential equations. Usually these cannot be solved

analytically and so they must be solved numerically.

Long Answer

Financial problems starting from stochastic differential

equations as models for quantities evolving randomly,

such as equity prices or interest rates, are using the

language of calculus. In calculus we refer to gradients,

rates of change, slopes, sensitivities. These mathemati-

cal ‘derivatives’ describe how fast a dependent variable,

such as an option value, changes as one of the indepen-

dent variables, such as an equity price, changes. These

sensitivities are technically defined as the ratio of the

infinitesimal change in the dependent variable to the

infinitesimal change in the independent. And we need

an infinite number of such infinitesimals to describe an

entire curve. However, when trying to calculate these

slopes numerically, on a computer, for example, we can-

not deal with infinites and infinitesimals, and have to

resort to approximations.

Technically, a definition of the delta of an option is

∂V

V

=

=

( S + h, t) − V ( S − h, t) lim

∂S

h→0

2 h

Chapter 2: FAQs

137

where V ( S, t) is the option value as a function of stock price, S, and time, t. Of course, there may be other independent variables. The limiting procedure in the above

is the clue to how to approximate such derivatives

based on continuous variables by differences based

on discrete variables.

The first step in the finite-difference methods is to lay

down a grid, such as the one shown in Figure 2-7.

The grid typically has equally spaced asset points, and

equally spaced time steps. Although in more sophis-

S

i

k

Vi

k

t

Figure 2-7: The finite-difference grid.

138 Frequently Asked Questions In Quantitative Finance

ticated schemes these can vary. Our task will be to find

numerically an approximation to the option values at

each of the nodes on this grid.

The classical option pricing differential equations are

written in terms of the option function, V ( S, t), say, a single derivative with respect to time, ∂V

∂ , the option’s

t

theta, the first derivative with respect to the underlying,

∂V

∂ , the option’s delta, and the second derivative with

S

respect to the underlying, ∂ 2 V , the option’s gamma. I

∂S 2

am explicitly assuming we have an equity or exchange

rate as the underlying in these examples. In the world of

fixed income we might have similar equations but just

read interest rate, r, for underlying, S, and the ideas

carry over.

A simple discrete approximation to the partial derivative

for theta is

∂V

V

θ =

≈ ( S, t) − V( S, t − δt)

∂t

δt

where δt is the time step between grid points. Similarly,

∂V

V

=

≈ ( S + δS, t) − V( S − δS, t)

∂S

2 δS

where δS is the asset step between grid points. There

is a subtle difference between these two expressions.

Note how the time derivative has been discretized by

evaluating the function V at the ‘current’ S and t, and also one time step before. But the asset derivative uses

an approximation that straddles the point S, using S +

δS and S − δS. The first type of approximation is called a one-sided difference, the second is a central difference.

The reasons for choosing one type of approximation

over another are to do with stability and accuracy.

The central difference is more accurate than a one-

Chapter 2: FAQs

139

sided difference and tends to be preferred for the delta

approximation, but when used for the time derivative

it can lead to instabilities in the numerical scheme.

(Here I am going to describe the explicit finite-difference

scheme, which is the easiest such scheme, but is one

which suffers from being unstable if the wrong time

discretization is used.)

The central difference for the gamma is

∂ 2 V

V

=

≈ ( S + δS, t) − 2 V( S, t) + V( S − δS, t) .

∂S 2

δS 2

Slightly changing the notation so that V k is the option

i

value approximation at the i th asset step and k th time

step we can write

V k − V k−1

V k

− Vk

θ ≈ i

i

,

≈ i+1

i−1 and

δt

2 δS

V k

− 2 Vk + Vk

≈ i+1

i

i−1 .

δS 2

Finally, plugging the above, together with S = i δS, into

the Black–Scholes equation gives the following dis-

cretized version of the equation:

V k − V k−1

V k

− 2 Vk + Vk

i

i

+ 1 σ 2 i 2 δS 2 i+1

i

i−1

δt

2

δS 2

V k

− Vk

+ ri δS i+1

i−1 − rVk = 0 .

2 δS

i

This can easily be rearranged to give V k−1 in terms

i

of V k , V k and V k , as shown schematically in the

i+1

i

i−1

following figure.

In practice we know what the option value is as a func-

tion of S, and hence i, at expiration. And this allows us

140 Frequently Asked Questions In Quantitative Finance

S

This option

value

is calculated from

these three

t

Figure 2-8: The relationship between option values in the explicit

method.

to work backwards from expiry to calculate the option

value today, one time step at a time.

The above is the most elementary form of the finite-

difference methods, there are many other more sophis-

ticated versions.

The advantages of the finite-difference methods are in

their speed for low-dimensional problems, those with

Chapter 2: FAQs

141

up to three sources of randomness. They are also par-

ticularly good when the problem has decision features

such as early exercise because at each node we can

easily check whether the option price violates arbitrage

constraints.

References and Further Reading

Ahmad, R 2007 Numerical and Computational Methods for

Derivative Pricing. John Wiley & Sons

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

142 Frequently Asked Questions In Quantitative Finance

What is a Jump-Diffusion Model

and How Does It Affect Option Values?

Short Answer

Jump-diffusion models combine the continuous Brown-

ian motion seen in Black–Scholes models (the diffusion)

with prices that are allowed to jump discontinuously.

The timing of the jump is usually random, and this is

represented by a Poisson process. The size of the jump

can also be random. As you increase the frequency of

the jumps (all other parameters remaining the same),

the values of calls and puts increase. The prices of

binaries, and other options, can go either up or down.

Example

A stock follows a lognormal random walk. Every month

you roll a dice. If you roll a one then the stock price

jumps discontinuously. The size of this jump is decided

by a random number you draw from a hat. (This is

not a great example because the Poisson process is a

continuous process, not a monthly event.)

Long Answer

A Poisson process can be written as dq where dq is the

jump in a random variable q during time t to t + dt. dq is 0 with probability 1 − λ dt and 1 with probability λ dt.

Note how the probability of a jump scales with the time

period over which the jump may happen, dt. The scale

factor λ is known as the intensity of the process, the

larger λ the more frequent the jumps.

This process can be used to model a discontinuous

financial random variable, such as an equity price,

volatility or an interest rate. Although there have been

research papers on pure jump processes as financial

models it is more usual to combine jumps with classical

Chapter 2: FAQs

143

Brownian motion. The model for equities, for example,

is often taken to be

dS = µS dt + σ S dX + ( J − 1) S dq.

dq is as defined above, with intensity λ, J − 1 is the jump size, usually taken to be random as well. Jump-diffusion

models can do a good job of representing the real-life

phenomenon of discontinuity in variables, and capturing

the fat tails seen in returns data.

The model for the underlying asset results in a model

for option prices. This model will be an integro-

differential equation, typically, with the integral term

representing the probability of the stock jumping a

finite distance discontinuously. Unfortunately, markets

with jumps of this nature are incomplete, meaning that

options cannot be hedged to eliminate risk. In order to

derive option-pricing equations one must therefore make

some assumptions about risk preferences or introduce

more securities with which to hedge.

Robert Merton was the first to propose jump-diffusion

models. He derived the following equation for equity

option values

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ rS

− rV

∂t

2

∂S 2

∂S

∂

+ λ

V

E [ V ( JS, t) − V ( S, t)] − λ

SE [ J − 1] = 0 .

∂S

E[·] is the expectation taken over the jump size. In

probability terms this equation represents the expected

value of the discounted payoff. The expectation being

over the risk-neutral measure for the diffusion but the

real measure for the jumps.

There is a simple solution of this equation in the special

case that the logarithm of J is Normally distributed. If

144 Frequently Asked Questions In Quantitative Finance

the logarithm of J is Normally distributed with standard

deviation σ and if we write

k = E[ J − 1]

then the price of a European non-path-dependent option

can be written as

∞

1 e− λ ( T− t)( λ ( T − t)) nVBS( S, t; σn, rn) .

n!

n=0

In the above

nσ 2

λ = λ(1 + k), σ 2 = σ 2 +

n

T − t

and

n ln(1 + k)

rn = r − λk +

,

T − t

and VBS is the Black–Scholes formula for the option

value in the absence of jumps. This formula can be inter-

preted as the sum of individual Black–Scholes values

each of which assumes that there have been n jumps,

and they are weighted according to the probability that

there will have been n jumps before expiry.

Jump-diffusion models can do a good job of capturing

steepness in volatility skews and smiles for short-dated

option, something that other models, such as stochastic

volatility, have difficulties in doing.

References and Further Reading

Cox, J & Ross, S 1976 Valuation of Options for Alternative

Stochastic Processes. Journal of Financial Econometrics 3

Kingman, JFC 1995 Poisson Processes. Oxford Science Publica-

tions

Chapter 2: FAQs

145

Lewis, A Series of articles in Wilmott magazine September 2002

to August 2004

Merton, RC 1976 Option pricing when underlying stock returns

are discontinuous. Journal of Financial Economics 3 125–44

146 Frequently Asked Questions In Quantitative Finance

What is Meant by ‘‘Complete’’ and

‘‘Incomplete’’ Markets?

Short Answer

A complete market is one in which a derivative product

can be artificially made from more basic instruments,

such as cash and the underlying asset. This usually

involves dynamically rebalancing a portfolio of the

simpler instruments, according to some formula or algo-

rithm, to replicate the more complicated product, the

derivative. Obviously, an incomplete market is one in

which you can’t replicate the option with simpler instru-

ments.

Example

The classic example is replicating an equity option, a

call, say, by continuously buying or selling the equity so

that you always hold the amount

= e− D( T− t) N( d 1),

in the stock, where

x

N( x) =

1

√

e− 1 φ 2

2

dφ

2 π −∞

and

ln( S/E) + ( r − D + 1 σ 2)( T − t) d

2

1 =

√

.

σ T − t

Long Answer

A slightly more mathematical, yet still quite easily

understood, description is to say that a complete mar-

ket is one for which there exist the same number of

linearly independent securities as there are states of the

world in the future.

Chapter 2: FAQs

147

Consider, for example, the binomial model in which

there are two states of the world at the next time step,

and there are also two securities, cash and the stock.

That is a complete market. Now, after two time steps

there will be three possible states of the world, assum-

ing the binomial model recombines so that an up-down

move gets you to the same place as down-up. You might

think that you therefore need three securities for a com-

plete market. This is not the case because after the

first time step you get to change the quantity of stock

you are holding, this is where the dynamic part of the

replication comes in.

In the equity world the two most popular models for

equity prices are the lognormal, with a constant volatil-

ity, and the binomial. Both of these result in complete

markets, you can replicate other contracts in these

worlds.

In a complete market you can replicate derivatives with

the simpler instruments. But you can also turn this on

its head so that you can hedge the derivative with the

underlying instruments to make a risk-free instrument.

In the binomial model you can replicate an option from

stock and cash, or you can hedge the option with the

stock to make cash. Same idea, same equations, just

move terms to be on different sides of the ‘equals’ sign.

As well as resulting in replication of derivatives, or

the ability to hedge them, complete markets also have

a nice mathematical property. Think of the binomial

model. In this model you specify the probability of the

stock rising (and hence falling because the probabili-

ties must add to one). It turns out that this probability

does not affect the price of the option. This is a sim-

ple consequence of complete markets, since you can

148 Frequently Asked Questions In Quantitative Finance

hedge the option with the stock you don’t care whether

the stock rises or falls, and so you don’t care what the

probabilities are. People can therefore disagree on the

probability of a stock rising or falling but still agree on

the value of an option, as long as they share the same

view on the stock’s volatility.

In probabilistic terms we say that in a complete mar-

ket there exists a unique martingale measure, but for

an incomplete market there is no unique martingale

measure. The interpretation of this is that even though

options are risky instruments we don’t have to specify

our own degree of risk aversion in order to price them.

Enough of complete markets, where can we find in-

complete markets? The answer is ‘everywhere.’ In

practice, all markets are incomplete because of real-world

effects that violate the assumptions of the simple models.

Take volatility as an example. As long as we have a

lognormal equity random walk, no transaction costs,

continuous hedging, perfectly divisible assets, . . . , and

constant volatility then we have a complete market.

If that volatility is a known time-dependent function

then the market is still complete. It is even still com-

plete if the volatility is a known function of stock price

and time. But as soon as that volatility becomes ran-

dom then the market is no longer complete. This is

because there are now more states of the world than

there are linearly independent securities. In reality,

we don’t know what volatility will be in the future so

markets are incomplete.

We also get incomplete markets if the underlying follows

a jump-diffusion process. Again more possible states

than there are underlying securities.

Another common reason for getting incompleteness

is if the underlying or one of the variables governing

Chapter 2: FAQs

149

the behaviour of the underlying is random. Options

on terrorist acts cannot be hedged since terrorist acts

aren’t traded (to my knowledge at least).

We still have to price contracts even in incomplete

markets, so what can we do? There are two main ideas

here. One is to price the actuarial way, the other is to

try to make all option prices consistent with each other.

The actuarial way is to look at pricing in some average

sense. Even if you can’t hedge the risk from each option

it doesn’t necessarily matter in the long run. Because

in that long run you will have made many hundreds

or thousands of option trades, so all that really mat-

ters is what the average price of each contract should

be, even if it is risky. To some extent this relies on

results from the Central Limit Theorem. This is called

the actuarial approach because it is how the insurance

business works. You can’t hedge the lifespan of indi-

vidual policyholders but you can figure out what will

happen to hundreds of thousands of them on average

using actuarial tables.

The other way of pricing is to make options consis-

tent with each other. This is commonly used when we

have stochastic volatility models, for example, and is

also often seen in fixed-income derivatives pricing. Let’s

work with the stochastic volatility model to get inspira-

tion. Suppose we have a lognormal random walk with

stochastic volatility. This means we have two sources of

randomness (stock and volatility) but only one quan-

tity with which to hedge (stock). That’s like saying

that there are more states of the world than underly-

ing securities, hence incompleteness. Well, we know we

can hedge the stock price risk with the stock, leaving

us with only one source of risk that we can’t get rid

of. That’s like saying there is one extra degree of free-

dom in states of the world than there are securities.

150 Frequently Asked Questions In Quantitative Finance

Whenever you have risk that you can’t get rid of you

have to ask how that risk should be valued. The more

risk the more return you expect to make in excess of

the risk-free rate. This introduces the idea of the mar-

ket price of risk. Technically in this case it introduces

the market price of volatility risk. This measures the

excess expected return in relation to unhedgeable risk.

Now all options on this stock with the random volatility

have the same sort of unhedgeable risk, some may have

more or less risk than others but they are all exposed to

volatility risk. The end result is a pricing model which

explicitly contains this market price of risk parameter.

This ensures that the prices of all options are consistent

with each other via this ‘universal’ parameter. Another

interpretation is that you price options in terms of the

prices of other options.

References and Further Reading

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Merton, RC 1976 Option pricing when underlying stock returns

are discontinuous. Journal of Financial Economics 3 125–44

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

151

What is Volatility?

Short Answer

Volatility is annualized standard deviation of returns. Or

is it? Because that is a statistical measure, necessarily

backward looking, and because volatility seems to vary,

and we want to know what it will be in the future, and

because people have different views on what volatility

will be in the future, things are not that simple.

Example

Actual volatility is the σ that goes into the Black–Scholes

partial differential equation. Implied volatility is the

number in the Black–Scholes formula that makes a

theoretical price match a market price.

Long Answer

Actual volatility is a measure of the amount of random-

ness in a financial quantity at any point in time. It’s

what Desmond Fitzgerald calls the ‘bouncy, bouncy.’ It’s

difficult to measure, and even harder to forecast but it’s

one of the main inputs into option pricing models.

It’s difficult to measure since it is defined mathemati-

cally via standard deviations which requires historical

data to calculate. Yet actual volatility is not a historical

quantity but an instantaneous one.

Realized/historical volatilities are associated with a

period of time, actually two periods of time. We might

say that the daily volatility over the last sixty days has

been 27%. This means that we take the last sixty days’

worth of daily asset prices and calculate the volatility.

Let me stress that this has two associated timescales,

whereas actual volatility has none. This tends to be the

default estimate of future volatility in the absence of

any more sophisticated model. For example, we might

152 Frequently Asked Questions In Quantitative Finance

assume that the volatility of the next sixty days is the

same as over the previous sixty days. This will give us

an idea of what a sixty-day option might be worth.

Implied volatility is the number you have to put into

the Black–Scholes option-pricing equation to get the

theoretical price to match the market price. Often said

to be the market’s estimate of volatility.

Let’s recap. We have actual volatility which is the

instantaneous amount of noise in a stock price return.

It is sometimes modelled as a simple constant, some-

times as time dependent, sometimes as stock and time

dependent, sometimes as stochastic and sometimes

as a jump process, and sometimes as uncertain, that

is, lying within a range. It is impossible to measure

exactly, the best you can do is to get a statistical esti-

mate based on past data. But this is the parameter we

would dearly love to know because of its importance in

pricing derivatives. Some hedge funds believe that their

edge is in forecasting this parameter better than other

people, and so profit from options that are mispriced in

the market.

Since you can’t see actual volatility people often rely

on measuring historical or realized volatility. This is

a backward looking statistical measure of what volatil-

ity has been. And then one assumes that there is some

information in this data that will tell us what volatility

will be in the future. There are several models for mea-

suring and forecasting volatility and we will come back

to them shortly.

Implied volatility is the number you have to put into

the Black–Scholes option-pricing formula to get the

theoretical price to match the market price. This is

often said to be the market’s estimate of volatility.

More correctly, option prices are governed by supply

Chapter 2: FAQs

153

and demand. Is that the same as the market taking

a view on future volatility? Not necessarily because

most people buying options are taking a directional

view on the market and so supply and demand reflects

direction rather than volatility. But because people who

hedge options are not exposed to direction only volatil-

ity it looks to them as if people are taking a view on

volatility when they are more probably taking a view

on direction, or simply buying out-of-the-money puts

as insurance against a crash. For example, the market

falls, people panic, they buy puts, the price of puts

and hence implied volatility goes up. Where the price

stops depends on supply and demand, not on anyone’s

estimate of future volatility, within reason.

Implied volatility levels the playing field so you can

compare and contrast option prices across strikes and

expirations.

There is also forward volatility. The adjective ‘forward’ is

added to anything financial to mean values in the future.

So forward volatility would usually mean volatility, either

actual or implied, over some time period in the future.

Finally hedging volatility means the parameter that you

plug into a delta calculation to tell you how many of the

underlying to sell short for hedging purposes.

Since volatility is so difficult to pin down it is a natural

quantity for some interesting modelling. Here are some

of the approaches used to model or forecast volatility.

Econometric models:

These models use various forms of

time series analysis to estimate current and future

expected actual volatility. They are typically based on

some regression of volatility against past returns and

they may involve autoregressive or moving-average com-

ponents. In this category are the GARCH type of mod-

els. Sometimes one models the square of volatility, the

154 Frequently Asked Questions In Quantitative Finance

variance, sometimes one uses high-low-open-close data

and not just closing prices, and sometimes one models

the logarithm of volatility. The latter seems to be quite

promising because there is evidence that actual volatil-

ity is lognormally distributed. Other work in this area

decomposes the volatility of a stock into components,

market volatility, industry volatility and firm-specific

volatility. This is similar to CAPM for returns.

Deterministic models:

The simple Black–Scholes formulæ

assume that volatility is constant or time dependent.

But market data suggests that implied volatility varies

with strike price. Such market behaviour cannot be con-

sistent with a volatility that is a deterministic function

of time. One way in which the Black–Scholes world can

be modified to accommodate strike-dependent implied

volatility is to assume that actual volatility is a func-

tion of both time and the price of the underlying. This

is the deterministic volatility (surface) model. This is

the simplest extension to the Black–Scholes world that

can be made to be consistent with market prices. All it

requires is that we have σ ( S, t), and the Black–Scholes

partial differential equation is still valid. The interpre-

tation of an option’s value as the present value of the

expected payoff under a risk-neutral random walk also

carries over. Unfortunately the Black–Scholes closed-

form formulæ are no longer correct. This is a simple and

popular model, but it does not capture the dynamics of

implied volatility very well.

Stochastic volatility:

Since volatility is difficult to measure,

and seems to be forever changing, it is natural to model

it as stochastic. The most popular model of this type is

due to Heston. Such models often have several param-

eters which can either be chosen to fit historical data

or, more commonly, chosen so that theoretical prices

calibrate to the market. Stochastic volatility models

are better at capturing the dynamics of traded option

Chapter 2: FAQs

155

prices better than deterministic models. However, differ-

ent markets behave differently. Part of this is because

of the way traders look at option prices. Equity traders

look at implied volatility versus strike, FX traders look

at implied volatility versus delta. It is therefore natu-

ral for implied volatility curves to behave differently in

these two markets. Because of this there have grown

up the sticky strike, sticky delta, etc., models, which

model how the implied volatility curve changes as the

underlying moves.

Poisson processes:

There are times of low volatility and

times of high volatility. This can be modelled by volatil-

ity that jumps according to a Poisson process.

Uncertain volatility:

An elegant solution to the problem of

modelling the unseen volatility is to treat it as uncertain,

meaning that it is allowed to lie in a specified range but

whereabouts in that range it actually is, or indeed the

probability of being at any value, are left unspecified.

With this type of model we no longer get a single option

price, but a range of prices, representing worst-case

scenario and best-case scenario.

References and Further Reading

Avellaneda, M, Levy, A & Parás, A 1995 Pricing and hedging

derivative securities in markets with uncertain volatilities.

Applied Mathematical Finance 2 73–88

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7

(2) 32–39 (February)

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18–20

(January)

Heston, S 1993 A closed-form solution for options with stochas-

tic volatility with application to bond and currency options.

Review of Financial Studies 6 327–343

156 Frequently Asked Questions In Quantitative Finance

Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons

Lewis, A 2000 Option valuation under Stochastic Volatility.

Finance Press

Lyons, TJ 1995 Uncertain Volatility and the risk-free synthesis

of derivatives. Applied Mathematical Finance 2 117–133

Rubinstein, M 1994 Implied binomial trees. Journal of Finance

69 771–818

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

157

What is the Volatility Smile?

Short Answer

Volatility smile is the phrase used to describe how the

implied volatilities of options vary with their strikes. A

smile means that out-of-the-money puts and out-of-the-

money calls both have higher implied volatilities than

at-the-money options. Other shapes are possible as well.

A slope in the curve is called a skew. So a negative skew

would be a download sloping graph of implied volatility

versus strike.

Example

0.35

0.3

0.25

tility

0.2

lao

d v

0.15

plie

Im

0.1

0.05

0

1010

1020

1030

1040

1050

1060

1070

1080

1090

1100

Strike

Figure 2-9: The volatility ‘smile’ for one-month SP500 options,

February 2004.

158 Frequently Asked Questions In Quantitative Finance

Long Answer

Let us begin with how to calculate the implied volatil-

ities. Start with the prices of traded vanilla options,

usually the mid price between bid and offer, and all

other parameters needed in the Black–Scholes formulæ,

such as strikes, expirations, interest rates, dividends,

except for volatilities. Now ask the question, what volatil-

ity must be used for each option series so that the

theoretical Black–Scholes price and the market price

are the same?

Although we have the Black–Scholes formula for option

values as a function of volatility, there is no formula

for the implied volatility as a function of option value,

it must be calculated using some bisection, Newton–

Raphson, or other numerical technique for finding zeros

of a function. Now plot these implied volatilities against

strike, one curve per expiration. That is the implied

volatility smile. If you plot implied volatility against

both strike and expiration, as a three-dimensional plot,

that is the implied volatility surface. Often you will find

that the smile is quite flat for long-dated options, but

getting steeper for short-dated options.

Since the Black–Scholes formulæ assume constant

volatility (or with a minor change, time-dependent

volatility) you might expect a flat implied volatility plot.

This appears not to be the case from real option-price

data. How can we explain this? Here are some questions

to ask.

• Is volatility constant?

• Are the Black–Scholes formulæ correct?

• Do option traders use the Black–Scholes formulæ?

Volatility does not appear to be constant. By this we

mean that actual volatility is not constant, actual volatil-

ity being the amount of randomness in a stock’s return.

Chapter 2: FAQs

159

Actual volatility is something you can try to measure

from a stock price time series, and would exist even

if options didn’t exist. Although it is easy to say with

confidence that actual volatility is not constant it is alto-

gether much harder to estimate the future behaviour of

volatility. So that might explain why implied volatility is

not constant, people believe that volatility is constant.

If volatility is not constant then the Black–Scholes for-

mulæ are not correct. (Again, there is the small caveat

that the Black–Scholes formulæ can work if volatility is

a known deterministic function of time. But I think we

can also confidently dismiss this idea as well.)

Despite this, option traders do still use the

Black–Scholes formulæ for vanilla options. Of all the

models that have been invented, the Black–Scholes

model is still the most popular for vanilla contracts.

It is simple and easy to use, it has very few param-

eters, it is very robust. Its drawbacks are quite well

understood. But very often, instead of using models

without some of the Black–Scholes’ drawbacks, people

‘adapt’ Black–Scholes to accommodate those problems.

For example, when a stock falls dramatically we often

see a temporary increase in its volatility. How can that

be squeezed into the Black–Scholes framework? Easy,

just bump up the implied volatilities for option with

lower strikes. A low strike put option will be out of the

money until the stock falls, at which point it may be

at the money, and at the same time volatility might

rise. So, bump up the volatility of all of the out-of-

the-money puts. This deviation from the flat-volatility

Black–Scholes world tends to get more pronounced

closer to expiration.

A more general explanation for the volatility smile is

that it incorporates the kurtosis seen in stock returns.

160 Frequently Asked Questions In Quantitative Finance

Stock returns are not normal, stock prices are not log-

normal. Both have fatter tails than you would expect

from normally distributed returns. We know that, theo-

retically, the value of an option is the present value of

the expected payoff under a risk-neutral random walk.

If that risk-neutral probability density function has fat

tails then you would expect option prices to be higher

than Black–Scholes for very low and high strikes. Hence

higher implied volatilities, and the smile.

Another school of thought is that the volatility smile and

skew exist because of supply and demand. Option prices

come less from an analysis of probability of tail events

than from simple agreement between a buyer and a

seller. Out-of-the-money puts are a cheap way of buying

protection against a crash. But any form of insurance is

expensive, after all those selling the insurance also want

to make a profit. Thus out-of-the-money puts are rela-

tively over priced. This explains high implied volatility

for low strikes. At the other end, many people owning

stock will write out-of-the-money call options (so-called

covered call writing) to take in some premium, perhaps

when markets are moving sideways. There will therefore

be an oversupply of out-of-the-money calls, pushing the

prices down. Net result, a negative skew. Although the

simple supply/demand explanation is popular among

traders it does not sit comfortably with quants because

it does suggest that options are not correctly priced

and that there may be arbitrage opportunities.

While on the topic of arbitrage, it is worth mentioning

that there are constraints on the skew and the smile

that come from examining simple option portfolios. For

example, rather obviously, the higher the strike of a call

option, the lower its price. Otherwise you could make

money rather easily by buying the low strike call and

selling the higher strike call. This imposes a constraint

on the skew. Similarly, a butterfly spread has to have a

Chapter 2: FAQs

161

positive value since the payoff can never be negative.

This imposes a constraint on the curvature of the smile.

Both of these constraints are model independent. There

are many ways to build the volatility-smile effect into an

option-pricing model, and still have no arbitrage. The

most popular are, in order of complexity, as follows

• Deterministic volatility surface

• Stochastic volatility

• Jump diffusion

The deterministic volatility surface is the idea that

volatility is not constant, or even only a function of

time, but a known function of stock price and time,

σ( S, t). Here the word ‘known’ is a bit misleading. What

we really know are the market prices of vanillas options,

a snapshot at one instant in time. We must now figure

out the correct function σ ( S, t) such that the theoretical value of our options matches the market prices. This is

mathematically an inverse problem, essentially find the

parameter, volatility, knowing some solutions, market

prices. This model may capture the volatility surface

exactly at an instant in time, but it does a very poor job

of capturing the dynamics, that is, how the data change

with time.

Stochastic volatility models have two sources of ran-

domness, the stock return and the volatility. One of the

parameters in these models is the correlation between

the two sources of randomness. This correlation is typ-

ically negative so that a fall in the stock price is often

accompanied by a rise in volatility. This results in a

negative skew for implied volatility. Unfortunately, this

negative skew is not usually as pronounced as the real

market skew. These models can also explain the smile.

As a rule one pays for convexity. We see this in the

simple Black–Scholes world where we pay for gamma.

162 Frequently Asked Questions In Quantitative Finance

In the stochastic volatility world we can look at the sec-

ond derivative of option value with respect to volatility,

and if it is positive we would expect to have to pay for

this convexity, that is option values will be relatively

higher wherever this quantity is largest. For a call or

put in the world of constant volatility we have

∂ 2 V

d

/ 2

=

1 d 2 e− D( T − t) e− d 2

1

S T − t

√

.

∂σ 2

2 π σ

This function is plotted in Figure 2-10 for S = 100, T −

t = 1, σ = 0 . 2, r = 0 . 05 and D = 0. Observe how it is positive away from the money, and small at the money.

(Of course, this is a bit of a cheat because on one hand

I am talking about random volatility and yet using a

formula that is only correct for constant volatility.)

140

d^2 V/d vol^2

120

100

80

60

40

20

Strike

0

75

80

85

90

95

100

105

110

115

120

125

-20

Figure 2-10: ∂ 2 V /∂σ 2 versus strike.

Chapter 2: FAQs

163

Stochastic volatility models have greater potential for

capturing dynamics, but the problem, as always, is

knowing which stochastic volatility model to choose and

how to find its parameters. When calibrated to market

prices you will still usually find that supposed constant

parameters in your model keep changing. This is often

the case with calibrated models and suggests that the

model is still not correct, even though its complexity

seems to be very promising.

Jump-diffusion models allow the stock (and even the

volatility) to be discontinuous. Such models contain

so many parameters that calibration can be instan-

taneously more accurate (if not necessarily stable

through time).

References and Further Reading

Gatheral, J 2006 The Volatility Surface. John Wiley & Sons

Javaheri, A 2005 Inside Volatility Arbitrage. John Wiley & Sons

Taylor, SJ & Xu, X 1994 The magnitude of implied volatility

smiles: theory and empirical evidence for exchange rates.

The Review of Futures Markets 13

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

164 Frequently Asked Questions In Quantitative Finance

What is GARCH?

Short Answer

GARCH stands for Generalized Auto Regressive Con-

ditional Heteroscedasticity. This is an econometric

model used for modelling and forecasting time-depen-

dent variance, and hence volatility, of stock price re-

turns. It represents current variance in terms of past

variance(s).

Example

The simplest member of the GARCH family is GARCH(1, 1)

in which the variance, vn of stock returns at time step n

is modelled by

vn = (1 − α − β) w 0 + βvn−1 + αvn−1 B 2 , n−1

where w 0 is the long-term variance, α and β are posi-

tive parameters, with α + β < 1, and Bn are independent Brownian motions, that is, random numbers drawn from

a normal distribution. The latest variance, vn, can there-

fore be thought of as a weighted average of the most

recent variance, the latest square of returns, and the

long-term average.

Long Answer

What?

GARCH is one member of a large family of econo-

metric models used to model time-varying variance.

They are popular in quantitative finance because they

can be used for measuring and forecasting volatility.

It is clear from simple equity or index data that volatil-

ity is not constant. If it were then estimating it would be

very simple. After all, in finance we have what some-

times seems like limitless quantities of data. Since

volatility varies with time we would like at the very least

to know what it is right now. And, more ambitiously, we

Chapter 2: FAQs

165

would like to know what it is going to be in the future,

if not precisely then perhaps know its future expected

value. This requires a model.

The simplest popular model assumes that we can get

an estimate for volatility over the next N days, in the

future, by looking at volatility over the previous N days,

the past. This moving window volatility is initially

appealing but suffers from the problem that if there

was a one-off jump in the stock price it will remain in

the data with the same weight for the next N days and

then suddenly drop out. This leads to artificially inflated

volatility estimates for a while. One way around this is

to use the second most popular volatility model, the

exponentially weighted moving average (EWMA). This

takes the form

vn = βvn−1 + (1 − β) R 2 ,

n−1

where β is a parameter between zero and one, and the

R s are the returns, suitably normalized with the time

step. This models the latest variance as a weighted

average between the previous variance and the lat-

est square of returns. The larger β the more weight is

attached to the distant past and the less to the recent

past. This model is also simple and appealing, but it has

one drawback. It results in no time structure going into

the future. The expected variance tomorrow, the day

after, and every day in the future is just today’s vari-

ance. This is counterintuitive, especially at times when

volatility is at historical highs or lows.

And so we consider the third simplest model,

vn = (1 − α − β) w 0 + βvn−1 + αR 2 , n−1

the GARCH(1, 1) model. This adds a constant, long-term

variance, to the EWMA model. The expected variance, k

time steps in the future, then behaves like

E[ vn+ k] = w 0 + ( vn − w 0)( α + β) n.

166 Frequently Asked Questions In Quantitative Finance

Since α + β < 1 this is exponentially decay of the aver-

age to its mean. A much nicer, more realistic, time

dependence than we get from the EWMA model.

In GARCH( p, q) the ( p, q) refers to there being p past variances and q past returns in the estimate:

q

p

p

q

vn = 1 −

αi −

βi w 0 +

βivn− i +

αiR 2 .

n− i

i=1

i=1

i=1

i=1

Why?

Volatility is a required input for all classical

option-pricing models, it is also an input for many asset-

allocation problems and risk estimation, such as Value

at Risk. Therefore it is very important to have a method

for forecasting future volatility.

There is one slight problem with these econometric

models, however. The econometrician develops his

volatility models in discrete time, whereas the option-

pricing quant would ideally like a continuous-time

stochastic differential equation model. Fortunately,

in many cases the discrete-time model can be rein-

terpreted as a continuous-time model (there is weak

convergence as the time step gets smaller), and so both

the econometrician and the quant are happy. Still, of

course, the econometric models, being based on real

stock price data, result in a model for the real and not

the risk-neutral volatility process. To go from one to

the other requires knowledge of the market price of

volatility risk.

How?

The parameters in these models are usually deter-

mined by Maximum Likelihood Estimation applied to

the (log)likelihood function. Although this technique

is usually quite straightforward to apply there can be

Chapter 2: FAQs

167

difficulties in practice. These difficulties can be associ-

ated with

• having insufficient data;

• the (log)likelihood function being very ‘flat’ with

respect to the parameters, so that the maximum is

insensitive to the parameter values;

• estimating the wrong model, including having too

many parameters (the best model may be simpler

than you think).

Family Members

Here are some of the other members of the GARCH

family. New ones are being added all the time, they are

breeding like rabbits. In these models the ‘shocks’ can

typically either have a normal distribution, a Student’s

t-distribution or a Generalized Error distribution, the

latter two having the fatter tails.

NGARCH

√

2

vn = (1 − α − β) w 0 + βvn−1 + α Rn−1 − γ vn−1 .

This is similar to GARCH(1, 1) but the parameter γ

permits correlation between the stock and volatility

processes.

AGARCH

Absolute value GARCH. Similar to GARCH but

with the volatility (not the variance) being linear in the

absolute value of returns (instead of square of returns).

EGARCH

Exponential GARCH. This models the logarithm

of the variance. The model also accommodates asym-

metry in that negative shocks can have a bigger impact

on volatility than positive shocks.

168 Frequently Asked Questions In Quantitative Finance

REGARCH

Range-based Exponential GARCH. This mod-

els the low to high range of asset prices over a ‘day.’

IGARCH

Integrated GARCH. This is a type of GARCH

model with further constraints on the parameters.

FIGARCH

Fractionally Integrated GARCH. This model

uses the fractional differencing lag operator applied

to the variance. This adds an extra parameter to the

GARCH model, and is such that it includes GARCH and

IGARCH as extremes. This model has the long memory,

slow decay of volatility as seen in practice.

FIEGARCH

Fractionally Integrated Exponential GARCH.

This models the logarithm of variance and again has

the long memory, slow decay of volatility as seen in

practice.

TGARCH

Threshold GARCH. This is similar to GARCH

but includes an extra term that kicks in when the shock

is negative. This gives a realistic asymmetry to the

volatility model.

PARCH

Power ARCH. In this model the variance is

raised to a power other than zero (logarithm), one

(AGARCH) or two. This model can have the long mem-

ory, slow decay of volatility seen in practice.

CGARCH

Component GARCH. This models variance

as the sum of two or more ‘components.’ In a two-

component model, for example, one component is used

to capture short-term and another the long-term effects

of shocks. This model therefore has the long memory,

slow decay of volatility seen in practice.

Chapter 2: FAQs

169

References and Further Reading

Engle, R 1982 Autoregressive Conditional Heteroskedasticity

with Estimates of the Variance of United Kingdom Inflation.

Econometrica 5 987–1008

Bollerslev, T 1986 Generalised Autoregressive Conditional

Heteroskedasticity. Journal of Econometrics 31 307–27

170 Frequently Asked Questions In Quantitative Finance

How Do I Dynamically Hedge?

Short Answer

Dynamic hedging, or delta hedging, means the continu-

ous buying or selling of the underlying asset according

to some formula or algorithm so that risk is eliminated

from an option position. The key point in this is what

formula do you use, and, given that in practice you

can’t hedge continuously, how should you hedge dis-

cretely? First get your delta correct, and this means

use the correct formula and estimates for parameters,

such as volatility. Second decide when to hedge based

on the conflicting desires of wanting to hedge as often

as possible to reduce risk, but as little as possible to

reduce any costs associated with hedging.

Example

The implied volatility of a call option is 20% but you

think that is cheap, volatility is nearer 40%. Do you put

20% or 40% into the delta calculation? The stock then

moves, should you rebalance, incurring some inevitable

transactions costs, or wait a bit longer while taking the

risks of being unhedged?

Long Answer

There are three issues, at least, here. First, what is the

correct delta? Second, if I don’t hedge very often how

big is my risk? Third, when I do rehedge how big are

my transaction costs?

What is the correct delta?

Let’s continue with the above

example, implied volatility 20% but you believe volatility

will be 40%. Does 0.2 or 0.4 go into the Black–Scholes

delta calculation, or perhaps something else? First

let me reassure you that you won’t theoretically lose

Chapter 2: FAQs

171

money in either case (or even if you hedge using a

volatility somewhere in the 20 to 40 range) as long as

you are right about the 40% and you hedge continu-

ously. There will however be a big impact on your P&L

depending on which volatility you input.

If you use the actual volatility of 40% then you are guar-

anteed to make a profit that is the difference between

the Black–Scholes formula using 40% and the Black–

Scholes formula using 20%.

V ( S, t; σ ) − V ( S, t; ˜ σ ), where V ( S, t; σ ) is the Black–Scholes formula for the call option and σ denotes actual volatility and ˜ σ is implied

volatility.

That profit is realized in a stochastic manner, so that

on a marked-to-market basis your profit will be random

each day. This is not immediately obvious, neverthe-

less it is the case that each day you make a random

profit or loss, both equally likely, but by expiration your

total profit is a guaranteed number that was known at

the outset. Most traders dislike the potentially large

P&L swings that you get by hedging using the forecast

volatility that they hedge using implied volatility.

When you hedge with implied volatility, even though

it is wrong compared with your forecast, you will still

make money. But in this case the profit each day is

non negative and smooth, so much nicer than when

you hedge using forecast volatility. The downside is

that the final profit depends on the path taken by the

underlying. If the stock stays close to the strike then

you will make a lot of money. If the stock goes quickly

far into or out of the money then your profit will be

small. Hedging using implied volatility gives you a nice,

172 Frequently Asked Questions In Quantitative Finance

smooth, monotonically increasing P&L but at the cost of

not knowing how much money you will make.

The profit each time step is

1

σ 2 − ˜ σ 2 S 2 i dt,

2

where i is the Black–Scholes gamma using implied

volatility. You can see from this expression that as long

as actual volatility is greater than implied you will make

money from this hedging strategy. This means that you

do not have to be all that accurate in your forecast of

future actual volatility to make a profit.

How big is my hedging error?

In practice you cannot hedge

continuously. The Black–Scholes model, and the above

analysis, requires continuous rebalancing or your posi-

tion in the underlying. The impact of hedging discretely

is quite easy to quantify.

When you hedge you eliminate a linear exposure to the

movement in the underlying. Your exposure becomes

quadratic and depends on the gamma of your position.

If we use φ to denote a normally distributed random

variable with mean of zero and variance one, then the

profit you make over a time step δt due to the gamma

is simply

1 σ 2 S 2 δt φ 2 .

2

This is in an otherwise perfect Black–Scholes world.

The only reason why this is not exactly a Black–Scholes

world is because we are hedging at discrete time inter-

vals.

The Black–Scholes models prices in the expected value

of this expression. You will recognize the 1 σ 2 S 2 from 2

the Black–Scholes equation. So the hedging error is

Chapter 2: FAQs

173

simply

1 σ 2 S 2 δt ( φ 2 − 1) .

2

This is how much you make or lose between each rebal-

ancing.

We can make several important observations about

hedging error.

• It is large: it is O( δt) which is the same order of

magnitude as all other terms in the Black–Scholes

model. It is usually much bigger than interest

received on the hedged option portfolio

• On average it is zero: hedging errors balance out

• It is path dependent: the larger gamma, the larger the

hedging errors

• The total hedging error has standard deviation of

√ δt: total hedging error is your final error when you

get to expiration. If you want to halve the error you

will have to hedge four times as often.

• Hedging error is drawn from a chi-square distribution:

that’s what φ 2 is

• If you are long gamma you will lose money

approximately 68% of the time: this is chi-square

distribution in action. But when you make money it

will be from the tails, and big enough to give a mean

of zero. Short gamma you lose only 32% of the time,

but they will be large losses.

• In practice φ is not normally distributed: the fat tails,

high peaks we see in practice will make the above

observation even more extreme, perhaps a long

gamma position will lose 80% of the time and win

only 20%. Still the mean will be zero.

How much will transaction costs reduce my profit?

To reduce

hedging error we must hedge more frequently, but

the downside of this is that any costs associated with

174 Frequently Asked Questions In Quantitative Finance

trading the underlying will increase. Can we quantify

transaction costs? Of course we can.

If we hold a short position in delta of the underlying and

then rebalance to the new delta at a time δt later then

we will have had to have bought or sold whatever the

change in delta was. As the stock price changes by δS

then the delta changes by δS . If we assume that costs

are proportional to the absolute value of the amount of

the underlying bought or sold, such that we pay in costs

an amount κ times the value traded then the expected

cost each δt will be

√

κσ

2

S 2 δt

| |,

π

where the

2

π appears because we have to take the

expected value of the absolute value of a normal vari-

able. Since this happens every time step, we can adjust

the Black–Scholes equation by subtracting from it the

above divided by δt to arrive at

∂V

∂V

∂V

+

2

1 σ 2 S 2

+ rS

− rV − κσS 2

| | = 0 .

∂t

2

∂S 2

∂S

πδt

This equation is interesting for being non linear, so that

the value of a long call and a short call will be different.

The long call will be less than the Black–Scholes value

and a short call higher. The long position is worth less

because we have to allow for the cost of hedging. The

short position is even more of a liability because of costs.

Crucially we also see that the effect of costs grows

like the inverse of the square root of the time between

rehedges. As explained above if we want to halve hedg-

ing error we must hedge four times as often. But this

would double the effects of transaction costs.

In practice, people do not rehedge at fixed intervals,

except perhaps just before market close. There are

Chapter 2: FAQs

175

many other possible strategies involving hedging when

the underlying or delta moves a specified amount, or

even strategies involving utility theory.

References and Further Reading

Ahmad, R & Wilmott, P 2005 Which free lunch would you like

today, Sir? Wilmott magazine, November

Whalley, AE & Wilmott, P 1993 a Counting the costs. Risk

magazine 6 (10) 59–66 (October)

Whalley, AE & Wilmott, P 1993 b Option pricing with transac-

tion costs. MFG Working Paper, Oxford

Whalley, AE & Wilmott, P 1994 a Hedge with an edge. Risk

magazine 7 (10) 82–85 (October)

Whalley, AE & Wilmott, P 1994 b A comparison of hedging

strategies. Proceedings of the 7th European Conference on

Mathematics in Industry 427–434

Whalley, AE & Wilmott, P 1996 Key results in discrete hedging

and transaction costs. In Frontiers in Derivatives (Ed. Konishi,

A and Dattatreya, R.) 183–196

Whalley, AE & Wilmott, P 1997 An asymptotic analysis of an

optimal hedging model for option pricing with transaction

costs. Mathematical Finance 7 307–324

Wilmott, P 1994 Discrete charms. Risk magazine 7 (3) 48–51

(March)

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

176 Frequently Asked Questions In Quantitative Finance

What is Dispersion Trading?

Short Answer

Dispersion trading is a strategy involving the selling of

options on an index against buying a basket of options

on individual stocks. Such a strategy is a play on the

behaviour of correlations during normal markets and

during large market moves. If the individual assets

returns are widely dispersed then there may be little

movement in the index, but a large movement in the

individual assets. This would result in a large payoff on

the individual asset options but little to pay back on the

short index option.

Example

You have bought straddles on constituents of the SP500

index, and you have sold a straddle on the index itself.

On most days you don’t make much of a profit or loss

on this position, gains/losses on the equities balance

losses/gains on the index. But one day half of your

equities rise dramatically, and one half fall, with there

being little resulting move in the index. On this day you

make money on the equity options from the gammas,

and also make money on the short index option because

of time decay. That was a day on which the individual

stocks were nicely dispersed.

Long Answer

The volatility on an index, σI , can be approximated by

N

N

σ 2 =

w

I

iwj ρijσiσj,

i=1 j=1

where there are N constituent stocks, with volatilities σi,

weight wi by value and correlations ρij. (I say ‘approxi-

mate’ because technically we are dealing with a sum of

lognormals which is not lognormal, but this approxima-

tion is fine.)

Chapter 2: FAQs

177

If you know the implied volatilities for the individual

stocks and for the index option then you can back out

an implied correlation, amounting to an ‘average’ across

all stocks:

σ 2 − N w 2 σ 2

I

i=1

i

i

.

N

N

w

i=1

i = j=1

iwj ρijσiσj

Dispersion trading can be interpreted as a view on this

implied correlation versus one’s own forecast of where

this correlation ought to be, perhaps based on historical

analysis.

The competing effects in a dispersion trade are

• gamma profits versus time decay on each of the long

equity options

• gamma losses versus time decay (the latter a source

of profit) on the short index options

• the amount of correlation across the individual

equities

In the example above we had half of the equities increas-

ing in value, and half decreasing. If they each moved

more than their respective implied volatilities would

suggest then each would make a profit. For each stock

this profit would depend on the option’s gamma and the

implied volatility, and would be parabolic in the stock

move. The index would hardly move and the profit there

would also be related to the index option’s gamma.

Such a scenario would amount to there being an aver-

age correlation of zero and the index volatility being

very small.

But if all stocks were to move in the same direction the

profit from the individual stock options would be the

same but this profit would be swamped by the gamma

loss on the index options. This corresponds to a correla-

tion of one across all stocks and a large index volatility.

178 Frequently Asked Questions In Quantitative Finance

Why might dispersion trading be successful?

• Dynamics of markets are more complex than can be

captured by the simplistic concept of correlation.

• Index options might be expensive because of large

demand, therefore good to sell.

• You can choose to buy options on equities that are

predisposed to a high degree of dispersion. For

example, focus on stocks which move dramatically in

different directions during times of stress. This may

be because they are in different sectors, or because

they compete with one another, or because there

may be merger possibilities.

• Not all of the index constituents need to be bought.

You can choose to buy the cheaper equity options in

terms of volatility.

Why might dispersion trading be unsuccessful?

• It is too detailed a strategy to cope with large

numbers of contracts with bid-offer spreads.

• You should delta hedge the positions which could be

costly.

• You must be careful of downside during market

crashes.

References and Further Reading

Grace, D & Van der Klink, R 2005 Dispersion Trading Project.

Technical Report, École Polytechnique Fédérale de Lausanne

Chapter 2: FAQs

179

What is Bootstrapping Using Discount

Factors?

Short Answer

Bootstrapping means building up a forward interest-

rate curve that is consistent with the market prices of

common fixed-income instruments such as bonds and

swaps. The resulting curve can then be used to value

other instruments, such as bonds that are not traded.

Example

You know the market prices of bonds with one, two

three, five years to maturity. You are asked to value a

four-year bond. How can you use the traded prices so

that your four-year bond price is consistent?

Long Answer

Imagine that you live in a world where interest rates

change in a completely deterministic way, no random-

ness at all. Interest rates may be low now, but rising

in the future, for example. The spot interest rate is the

interest you receive from one instant to the next. In this

deterministic interest-rate world this spot rate can be

written as a function of time, r( t). If you knew what this

function was you would be able to value fixed-coupon

bonds of all maturities by using the discount factor

T

exp −

r( τ ) dτ ,

t

to present value a payment at time T to today, t.

Unfortunately you are not told what this r function is.

Instead you only know, by looking at market prices of

various fixed-income instruments, some constraints on

this r function.

180 Frequently Asked Questions In Quantitative Finance

As a simple example, suppose you know that a zero-

coupon bond, principal $100, maturing in one year, is

worth $95 today. This tells us that

t+1

exp −

r( τ ) dτ

= 0 . 95 .

t

Suppose a similar two-year zero-coupon bond is worth

$92, then we also know that

t+2

exp −

r( τ ) dτ

= 0 . 92 .

t

This is hardly enough information to calculate the entire

r( t) function, but it is similar to what we have to deal

with in practice. In reality, we have many bonds of dif-

ferent maturity, some without any coupons but most

with, and also very liquid swaps of various maturities.

Each such instrument is a constraint on the r( t) func-

tion.

Bootstrapping is backing out a deterministic spot rate

function, r( t), also called the (instantaneous) forward rate curve that is consistent with all of these liquid

instruments.

Note that usually only the simple ‘linear’ instruments

are used for bootstrapping. Essentially this means

bonds, but also includes swaps since they can be de-

composed into a portfolio of bonds. Other contracts

such as caps and floors contain an element of optional-

ity and therefore require a stochastic model for interest

rates. It would not make financial sense to assume a

deterministic world for these instruments, just as you

wouldn’t assume a deterministic stock price path for an

equity option.

Because the forward rate curve is not uniquely deter-

mined by the finite set of constraints that we encounter

Chapter 2: FAQs

181

in practice, we have to impose some conditions on the

function r( t).

• Forward rates should be positive, or there will be

arbitrage opportunities

• Forward rates should be continuous (although this is

commonsense rather than because of any financial

argument)

• Perhaps the curve should also be smooth

Even with these desirable characteristics the forward

curve is not uniquely defined.

Finding the forward curve with these properties

amounts to deciding on a way of interpolating ‘between

the points,’ the ‘points’ meaning the constraints on

the integrals of the r function. There have been many

proposed interpolation techniques such as

• linear in discount factors

• linear in spot rates

• linear in the logarithm of rates

• piecewise linear continuous forwards

• cubic splines

• Bessel cubic spline

• monotone-preserving cubic spline

• quartic splines

and others.

Finally, the method should result in a forward rate func-

tion that is not too sensitive to the input data, the bond

prices and swap rates, it must be fast to compute and

must not be too local in the sense that if one input is

changed it should only impact on the function nearby.

And, of course, it should be emphasized that there is no

‘correct’ way to join the dots.

182 Frequently Asked Questions In Quantitative Finance

Because of the relative liquidity of the instruments it

is common to use deposit rates in the very short term,

bonds and FRAs for the medium term and swaps for

longer end of the forward curve.

Finally, because the bootstrapped forward curve is

assumed to come from deterministic rates it is danger-

ous to use it to price instruments with convexity since

such instruments require a model for randomness, as

explained by Jensen’s Inequality.

References and Further Reading

Hagan, P & West, G Interpolation methods for curve con-

struction. http://www.riskworx.com/insights/interpolation/

interpolation.pdf

Jones, J 1995 private communication

Ron, U 2000 A practical guide to swap curve construction.

Technical Report 17, Bank of Canada

Walsh, O 2003 The art and science of curve building. Wilmott

magazine November 8–10

Chapter 2: FAQs

183

What is the LIBOR Market Model and

Its Principal Applications in Finance?

Short Answer

The LIBOR Market Model (LMM), also known as the

BGM or BGM/J model, is a model for the stochastic

evolution of forward interest rates. Its main strength

over other interest rate models is that it describes the

evolution of forward rates that exist, at market-traded

maturities, as opposed to theoretical constructs such as

the spot interest rate.

Example

In the LMM the variables are a set of forward rates

for traded, simple fixed-income instruments. The para-

meters are volatilities of these and correlations between

them. From no arbitrage we can find the risk-neutral

drift rates for these variables. The model is then used

to price other instruments.

Long Answer

The history of interest-rate modelling begins with deter-

ministic rates, and the ideas of yield to maturity, dur-

ation, etc. The assumption of determinism is not at all

satisfactory for pricing derivatives however, because of

Jensen’s Inequality.

In 1976 Fischer Black introduced the idea of treat-

ing bonds as underlying assets so as to use the

Black–Scholes equity option formulæ for fixed-income

instruments. This is also not entirely satisfactory since

there can be contradictions in this approach. On one

hand bond prices are random, yet on the other hand

interest rates used for discounting from expiration to

the present are deterministic. An internally consistent

stochastic rates approach was needed.

184 Frequently Asked Questions In Quantitative Finance

The first step on the stochastic interest rate path used

a very short-term interest rate, the spot rate, as the

random factor driving the entire yield curve. The math-

ematics of these spot-rate models was identical to that

for equity models, and the fixed-income derivatives sat-

isfied similar equations as equity derivatives. Diffusion

equations governed the prices of derivatives, and deriva-

tives prices could be interpreted as the risk-neutral

expected value of the present value of all cashflows as

well. And so the solution methods of finite-difference

methods for solving partial differential equations, trees

and Monte Carlo simulation carried over. Models of this

type are Vasicek, Cox, Ingersoll & Ross, Hull & White.

The advantage of these models is that they are easy to

solve numerically by many different methods. But there

are several aspects to the downside. First, the spot rate

does not exist, it has to be approximated in some way.

Second, with only one source of randomness the yield

curve is very constrained in how it can evolve, essen-

tially parallel shifts. Third, the yield curve that is output

by the model will not match the market yield curve. To

some extent the market thinks of each maturity as being

semi independent from the others, so a model should

match all maturities otherwise there will be arbitrage

opportunities.

Models were then designed to get around the second

and third of these problems. A second random factor

was introduced, sometimes representing the long-term

interest rate (Brennan & Schwartz), and sometimes the

volatility of the spot rate (Fong & Vasicek). This allowed

for a richer structure for yield curves. And an arbitrary

time-dependent parameter (or sometimes two or three

such) was allowed in place of what had hitherto been

constant(s). The time dependence allowed for the yield

curve (and other desired quantities) to be instanta-

neously matched. Thus was born the idea of calibration,

the first example being the Ho & Lee model.

Chapter 2: FAQs

185

The business of calibration in such models was rarely

straightforward. The next step in the development of

models was by Heath, Jarrow & Morton (HJM) who

modelled the evolution of the entire yield curve directly

so that calibration simply became a matter of specifying

an initial curve. The model was designed to be easy to

implement via simulation. Because of the non-Markov

nature of the general HJM model it is not possible to

solve these via finite-difference solution of partial dif-

ferential equations, the governing partial differential

equation would generally be in an infinite number of

variables, representing the infinite memory of the gen-

eral HJM model. Since the model is usually solved by

simulation it is straightforward having any number of

random factors and so a very, very rich structure for

the behaviour of the yield curve. The only downside

with this model, as far as implementation is concerned,

is that it assumes a continuous distribution of maturities

and the existence of a spot rate.

The LIBOR Market Model (LMM) as proposed by Mil-

tersen, Sandmann, Sondermann, Brace, Gatarek, Musiela

and Jamshidian in various combinations and at var-

ious times, models traded forward rates of different

maturities as correlated random walks. The key advan-

tage over HJM is that only prices which exist in the

market are modelled, the LIBOR rates. Each traded for-

ward rate is represented by a stochastic differential

equation model with a drift rate and a volatility, as well

as a correlation with each of the other forward rate

models. For the purposes of pricing derivatives we work

as usual in a risk-neutral world. In this world the drifts

cannot be specified independently of the volatilities and

correlations. If there are N forward rates being modelled

then there will be N volatility functions to specify and

N( N − 1) / 2 correlation functions, the risk-neutral drifts are then a function of these parameters.

186 Frequently Asked Questions In Quantitative Finance

Again, the LMM is solved by simulation with the yield

curve ‘today’ being the initial data. Calibration to the

yield curve is therefore automatic. The LMM can also

be made to be consistent with the standard approach

for pricing caps, floors and swaptions using Black 1976.

Thus calibration to volatility- and correlation-dependent

liquid instruments can also be achieved.

Such a wide variety of interest models have been

suggested because there has not been a universally

accepted model. This is in contrast to the equity world

in which the lognormal random walk is a starting point

for almost all models. Whether the LMM is a good model

in terms of scientific accuracy is another matter, but

its ease of use and calibration and its relationship with

standard models make it very appealing to practitioners.

References and Further Reading

Brace, A, Gatarek, D & Musiela, M 1997 The market model of

interest rate dynamics. Mathematical Finance 7 127–154

Brennan, M & Schwartz, E 1982 An equilibrium model of bond

pricing and a test of market efficiency. Journal of Financial

and Quantitative Analysis 17 301–329

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term struc-

ture of interest rates. Econometrica 53 385–467

Fong, G & Vasicek, O 1991, Interest rate volatility as a stochas-

tic factor. Working Paper

Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the

term structure of interest rates: a new methodology. Econo-

metrica 60 77–105

Ho, T & Lee, S 1986 Term structure movements and pric-

ing interest rate contingent claims. Journal of Finance 42

1129–1142

Chapter 2: FAQs

187

Hull, JC & White, A 1990 Pricing interest rate derivative secu-

rities. Review of Financial Studies 3 573–592

Rebonato, R 1996 Interest-rate Option Models. John Wiley & Sons

Vasicek, OA 1977 An equilibrium characterization of the term

structure. Journal of Financial Economics 5 177–188

188 Frequently Asked Questions In Quantitative Finance

What is Meant by the ‘Value’ of a

Contract?

Short Answer

Value usually means the theoretical cost of building

up a new contract from simpler products, such as

replicating an option by dynamically buying and selling

stock.

Example

Wheels cost $10 each. A soapbox is $20. How much is a

go-cart? The value is $60.

Long Answer

To many people the value of a contract is what they

see on a screen or comes out of their pricing software.

Matters are actually somewhat more subtle than this.

Let’s work with the above soapbox example.

To the quant the value of the go-cart is simply $60, the

cost of the soapbox and four wheels, ignoring nails and

suchlike, and certainly ignoring the cost of manpower

involved in building it.

Are you going to sell the go-cart for $60? I don’t think

so. You’d like to make a profit, so you sell it for $80.

That is the price of the go-cart.

Why did someone buy it from you for $80? Clearly the

$80 must be seen by them as being a reasonable amount

to pay. Perhaps they are going to enter a go-carting

competition with a first prize of $200. Without the go-

cart they can’t enter, and they can’t win the $200. The

possibility of winning the prize money means that the

go-cart is worth more to them than the $80. Maybe they

would have gone as high as $100.

Chapter 2: FAQs

189

This simple example illustrates the subtlety of the whole

valuation/pricing process. In many ways options are like

go-carts and valuable insight can be gained by thinking

on this more basic level.

The quant rarely thinks like the above. To him value

and price are the same, the two words often used inter-

changeably. And the concept of worth does not crop up.

When a quant has to value an exotic contract he looks

to the exchange-traded vanillas to give him some insight

into what volatility to use. This is calibration. A vanilla

trades at $10, say. That is the price. The quant then

backs out from a Black–Scholes valuation formula the

market’s implied volatility. By so doing he is assuming

that price and value are identical.

Related to this topic is the question of whether a math-

ematical model explains or describes a phenomenon.

The equations of fluid mechanics, for example, do both.

They are based on conservation of mass and momen-

tum, two very sound physical principles. Contrast this

with the models for derivatives.

Prices are dictated in practice by supply and demand.

Contracts that are in demand, such as out-of-the-money

puts for downside protection, are relatively expensive.

This is the explanation for prices. Yet the mathematical

models we use for pricing have no mention of supply

or demand. They are based on random walks for the

underlying with an unobservable volatility parameter,

and the assumption of no arbitrage. The models try

to describe how the prices ought to behave given a

volatility. But as we know from data, if we plug in our

own forecast of future volatility into the option-pricing

formulæ we will get values that disagree with the market

prices. Either our forecast is wrong and the market

knows better, or the model is incorrect, or the market

190 Frequently Asked Questions In Quantitative Finance

is incorrect. Commonsense says all three are to blame.

Whenever you calibrate your model by backing out

volatility from supply-demand driven prices using a

valuation formula you are mixing apples and oranges.

To some extent what the quant is trying to do is the

same as the go-cart builder. The big difference is that

the go-cart builder does not need a dynamic model

for the prices of wheels and soapboxes, his is a static

calculation. One go-cart equals one soapbox plus four

wheels. It is rarely so simple for the quant. His calcula-

tions are inevitably dynamic, his hedge changes as the

stock price and time change. It would be like a go-cart

for which you had to keep buying extra wheels during

the race, not knowing what the price of wheels would

be before you bought them. This is where the math-

ematical models come in, and errors, confusion, and

opportunities appear.

And worth? That is a more subjective concept. Quan-

tifying it might require a utility approach. As Oscar

Wilde said ‘‘A cynic is a man who knows the price of

everything but the value of nothing.’’

References and Further Reading

Wilde, The Complete Works of Oscar Wilde. Harper Perennial

Chapter 2: FAQs

191

What is Calibration?

Short Answer

Calibration means choosing parameters in your model

so that the theoretical prices for exchange-traded con-

tracts output from your model match exactly, or as

closely as possible, the market prices at an instant in

time. In a sense it is the opposite of fitting parameters

to historical time series. If you match prices exactly

then you are eliminating arbitrage opportunities, and

this is why it is popular.

Example

You have your favourite interest rate model, but you

don’t know how to decide what the parameters in the

model should be. You realize that the bonds, swaps and

swaptions markets are very liquid, and presumably very

efficient. So you choose your parameters in the model

so that your model’s theoretical output for these simple

instruments is the same as their market prices.

Long Answer

Almost all financial models have some parameter(s)

that can’t be measured accurately. In the simplest non-

trivial case, the Black–Scholes model, that parameter is

volatility. If we can’t measure that parameter how can

we decide on its value? For if we don’t have an idea of

its value then the model is useless.

Two ways spring to mind. One is to use historical data,

the other is to use today’s price data.

Let’s see the first method in action. Examine, perhaps,

equity data to try and estimate what volatility is. The

problem with that is that it is necessarily backward

looking, using data from the past. This might not be

relevant to the future. Another problem with this is

192 Frequently Asked Questions In Quantitative Finance

that it might give prices that are inconsistent with the

market. For example, you are interested in buying a

certain option. You think volatility is 27%, so you use

that number to price the option, the price you get is

$15. However, the market price of that option is $19.

Are you still interested in buying it? You can either

decide that the option is incorrectly priced or that your

volatility estimate is wrong.

The other method is to assume, effectively, that there is

information in the market prices of traded instruments.

In the above example we ask what volatility must we

put into a formula to get the ‘correct’ price of $19. We

then use that number to price other instruments. In this

case we have calibrated our model to an instantaneous

snapshot of the market at one moment in time, rather

than to any information from the past.

Calibration is common in all markets, but is usually

more complicated than in the simple example above.

Interest rate models may have dozens of parameters or

even entire functions to be chosen by matching with the

market.

Calibration can therefore often be time consuming. Cal-

ibration is an example of an inverse problem, in which

we know the answer (the prices of simple contracts)

and want to find the problem (the parameters). Inverse

problems are notoriously difficult, for example being

very sensitive to initial conditions.

Calibration can be misleading, since it suggests that

your prices are correct. For example if you calibrate a

model to a set of vanilla contracts, and then calibrate

a different model to the same set of vanillas, how do

you know which model is better? Both correctly price

vanillas today. But how will they perform tomorrow?

Will you have to recalibrate? If you use the two different

Chapter 2: FAQs

193

models to price an exotic contract how do you know

which price to use? How do you know which gives the

better hedge ratios? How will you even know whether

you have made money or lost it?

References and Further Reading

Schoutens, W, Simons, E & Tistaert, J 2004 A perfect calibra-

tion! Now what? Wilmott magazine March 66–78

194 Frequently Asked Questions In Quantitative Finance

What is the Market Price of Risk?

Short Answer

The market price of risk is the return in excess of the

risk-free rate that the market wants as compensation for

taking risk.

Example

Historically a stock has grown by an average of 20% per

annum when the risk-free rate of interest was 5%. The

volatility over this period was 30%. Therefore, for each

unit of risk this stock returns on average an extra 0.5

return above the risk-free rate. This is the market price

of risk.

Long Answer

In classical economic theory no rational person would

invest in a risky asset unless they expect to beat the

return from holding a risk-free asset. Typically risk is

measured by standard deviation of returns, or volatility.

The market price of risk for a stock is measured by

the ratio of expected return in excess of the risk-free

interest rate divided by standard deviation of returns.

Interestingly, this quantity is not affected by leverage.

If you borrow at the risk-free rate to invest in a risky

asset both the expected return and the risk increase,

such that the market price of risk is unchanged. This

ratio, when suitably annualized, is also the Sharpe ratio.

If a stock has a certain value for its market price of risk

then an obvious question to ask is what is the market

price of risk for an option on that stock? In the famous

Black–Scholes world in which volatility is deterministic

and you can hedge continuously and costlessly then the

market price of risk for the option is the same as that

for the underlying equity. This is related to the concept

of a complete market in which options are redundant

because they can be replicated by stock and cash.

Chapter 2: FAQs

195

In derivatives theory we often try to model quantities

as stochastic, that is, random. Randomness leads to

risk, and risk makes us ask how to value risk, that is,

how much return should we expect for taking risk. By

far the most important determinant of the role of this

market price of risk is the answer to the question, is

the quantity you are modelling traded directly in the

market?

If the quantity is traded directly, the obvious example

being a stock, then the market price of risk does not

appear in the Black–Scholes option pricing model. This

is because you can hedge away the risk in an option

position by dynamically buying and selling the under-

lying asset. This is the basis of risk-neutral valuation.

Hedging eliminates exposure to the direction that the

asset is going and also to its market price of risk. You

will see this if you look at the Black–Scholes equation.

There the only parameter taken from the stock random

walk is its volatility, there is no appearance of either its

growth rate or its price of risk.

On the other hand, if the modelled quantity is not

directly traded then there will be an explicit reference

in the option-pricing model to the market price of risk.

This is because you cannot hedge away associated risk.

And because you cannot hedge the risk you must know

how much extra return is needed to compensate for

taking this unhedgeable risk. Indeed, the market price

of risk will typically appear in classical option-pricing

models any time you cannot hedge perfectly. So expect

it to appear in the following situations:

• When you have a stochastic model for a quantity that

is not traded. Examples: stochastic volatility; interest

rates (this is a subtle one, the spot rate is not

traded); risk of default.

196 Frequently Asked Questions In Quantitative Finance

• When you cannot hedge. Examples: jump models;

default models; transaction costs.

When you model stochastically a quantity that is not

traded then the equation governing the pricing of deriva-

tives is usually of diffusion form, with the market price

of risk appearing in the ‘drift’ term with respect to the

non-traded quantity. To make this clear, here is a gen-

eral example.

Suppose that the price of an option depends on the

value of a quantity of a substance called phlogiston.

Phlogiston is not traded but either the option’s payoff

depends on the value of phlogiston, or the value of phlo-

giston plays a role in the dynamics of the underlying

asset. We model the value of phlogiston as

d = µdt + σdX.

The market price of phlogiston risk is λ. In the classical

option-pricing models we will end up with an equation

for an option with the following term

∂V

. . . + (µ − λσ)

+ . . . = 0 .

∂

The dots represent all the other terms that one usually

gets in a Black–Scholes-type of equation. Observe that

the expected change in the value of phlogiston, µ, has

been adjusted to allow for the market price of phlogis-

ton risk. We call this the risk-adjusted or risk-neutral

drift. Conveniently, because the governing equation is

still of diffusive type we can continue to use Monte

Carlo simulation methods for pricing. Just remember to

simulate the risk-neutral random walk

d = (µ − λσ) dt + σdX.

and not the real one.

You can imagine estimating the real drift and volatility

for any observable financial quantity simply by looking

Chapter 2: FAQs

197

at a times series of the value of that quantity. But how

can you estimate its market price of risk? Market price

of risk is only observable through option prices. This is

the point at which practice and elegant theory start to

part company. Market price of risk sounds like a way of

calmly assessing required extra value to allow for risk.

Unfortunately there is nothing calm about the way that

markets react to risk. For example, it is quite simple to

relate the slope of the yield curve to the market price

of interest rate risk. But evidence from this suggests

that market price of risk is itself random, and should

perhaps also be modelled stochastically.

Note that when you calibrate a model to market prices

of options you are often effectively calibrating the

market price of risk. But that will typically be just a

snapshot at one point in time. If the market price of risk

is random, reflecting people’s shifting attitudes from

fear to greed and back again, then you are assuming

fixed something which is very mobile, and calibration

will not work.

There are some models in which the market price of

risk does not appear because they typically involve

using some form of utility theory approach to find a

person’s own price for an instrument rather than the

market’s.

References and Further Reading

Ahn, H & Wilmott, P 2003b Stochastic volatility and mean-

variance analysis. Wilmott magazine November 2003 84–90

Markowitz, H 1959 Portfolio Selection: efficient diversification of

investment. John Wiley & Sons

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

198 Frequently Asked Questions In Quantitative Finance

What is the Difference Between the

Equilibrium Approach and the

No-Arbitrage Approach to Modelling?

Short Answer

Equilibrium models balance supply and demand, they

require knowledge of investor preferences and prob-

abilities. No-arbitrage models price one instrument by

relating it to the prices of other instruments.

Example

The Vasicek interest rate model can be calibrated to

historical data. It can therefore be thought of as a rep-

resentation of an equilibrium model. But it will rarely

match traded prices. Perhaps it would therefore be a

good trading model. The BGM model matches market

prices each day and therefore suggests that there are

never any trading opportunities.

Long Answer

Equilibrium models represent a balance of supply and

demand. As with models of equilibria in other, non-

financial, contexts there may be a single equilibrium

point, or multiple, or perhaps no equilibrium possible

at all. And equilibrium points may be stable such that

any small perturbation away from equilibrium will be

corrected (a ball in a valley), or unstable such that a

small perturbation will grow (a ball on the top of a hill).

The price output by an equilibrium model is supposedly

correct in an absolute sense.

Genuine equilibrium models in economics usually

require probabilities for future outcomes, and a rep-

resentation of the preferences of investors. The latter

perhaps quantified by utility functions. In practice nei-

ther of these is usually available, and so the equilibrium

Chapter 2: FAQs

199

models tend to be of more academic than practical

interest.

No-arbitrage, or arbitrage-free, models represent the

point at which there aren’t any arbitrage profits to be

made. If the same future payoffs and probabilities can

be made with two different portfolios then the two port-

folios must both have the same value today, otherwise

there would be an arbitrage. In quantitative finance

the obvious example of the two portfolios is that of an

option on the one hand and a cash and dynamically

rebalanced stock position on the other. The end result

being the pricing of the option relative to the price of

the underlying asset. The probabilities associated with

future stock prices falls out of the calculation and pref-

erences are never needed. When no-arbitrage pricing is

possible it tends to be used in practice. The price out-

put by a no-arbitrage model is supposedly correct in a

relative sense.

For no-arbitrage pricing to work we need to have mar-

kets that are complete, so that we can price one con-

tract in terms of others. If markets are not complete and

we have sources of risk that are unhedgeable then we

need to be able to quantify the relevant market price

of risk. This is a way of consistently relating prices of

derivatives with the same source of unhedgeable risk, a

stochastic volatility for example.

Both the equilibrium and no-arbitrage models suffer

from problems concerning parameter stability.

In the fixed-income world, examples of equilibrium

models are Vasicek, CIR, Fong & Vasicek. These have

parameters which are constant, and which can be esti-

mated from time series data. The problem with these

is that they permit very simple arbitrage because the

prices that they output for bonds will rarely match

200 Frequently Asked Questions In Quantitative Finance

traded prices. Now the prices may be correct based on

the statistics of the past but are they correct going for-

ward? The models of Ho & Lee and Hull & White are a

cross between the equilibrium models and no-arbitrage

models. Superficially they look very similar to the for-

mer but by making one or more of the parameters time

dependent they can be calibrated to market prices and

so supposedly remove arbitrage opportunities. But still,

if the parameters, be they constant or functions, are

not stable then we will have arbitrage. But the question

is whether that arbitrage is foreseeable. The interest

rate models of HJM and BGM match market prices each

day and are therefore even more in the no-arbitrage

camp.

References and Further Reading

Brace, A, Gatarek, D & Musiela, M 1997 The market model of

interest rate dynamics. Mathematical Finance 7 127–154

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term struc-

ture of interest rates. Econometrica 53 385–467

Fong, G & Vasicek, O 1991, Interest rate volatility as a stochas-

tic factor. Working Paper

Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the

term structure of interest rates: a new methodology. Econo-

metrica 60 77–105

Ho, T & Lee, S 1986 Term structure movements and pric-

ing interest rate contingent claims. Journal of Finance 42

1129–1142

Hull, JC & White, A 1990 Pricing interest rate derivative secu-

rities. Review of Financial Studies 3 573–592

Vasicek, OA 1977 An equilibrium characterization of the term

structure. Journal of Financial Economics 5 177–188

Chapter 2: FAQs

201

How Good is the Assumption of Normal

Distributions for Financial Returns?

Short Answer

The answer has to be ‘it depends.’ It depends on the

timescale over which returns are measured. For stocks

over very short timescales, intraday to several days,

the distributions are not normal, they have fatter tails

and higher peaks than normal. Over longer periods

they start to look more normal, but then over years or

decades they look lognormal.

It also depends on what you mean by ‘good.’ They are

very good in the sense that they are simple distributions

to work with, and also, thanks to the Central Limit

Theorem, sensible distributions to work with since there

are sound reasons why they might appear. They are also

good in that basic stochastic calculus and Itô’s lemma

assume normal distributions and those concepts are

bricks and mortar to the quant.

Example

In Figure 2-11 is the probability density function for

the daily returns on the S&P index since 1980, scaled

to have zero mean and standard deviation of one, and

also the standardized normal distribution. The empirical

peak is higher than the normal distribution and the tails

are both fatter.

On 19th October 1987 the SP500 fell 20.5%. What is the

probability of a 20% one-day fall in the SP500? Since we

are working with over 20 years of daily data, we could

argue that empirically there will be a 20% fall in the

SPX index every 20 years or so. To get a theoretical

estimate, based on normal distributions, we must first

estimate the daily standard deviation for SPX returns.

202 Frequently Asked Questions In Quantitative Finance

0.7

PDF

SPX Returns

Normal

0.6

0.5

0.4

0.3

0.2

0.1

0

-4

-3

-2

-1

0

1

2

3

4

Scaled return

Figure 2-11: The standardized probability density functions for SPX

returns and the Normal distribution.

Over that period it was 0.0106, equivalent to an average

volatility of 16.9%. What is the probability of a 20% or

more fall when the standard deviation is 0.0106? This is

a staggeringly small 1.8 10−79. That is just once every

2 1076 years. Empirical answer: Once every 20 years.

Theoretical answer: Once every 2 1076 years. That’s how

bad the normal-distribution assumption is in the tails.

Long Answer

Asset returns are not normally distributed according to

empirical evidence. Statistical studies show that there

is significant kurtosis (fat tails) and some skewness

Chapter 2: FAQs

203

(asymmetry). Whether this matters or not depends on

several factors:

• Are you holding stock for speculation or are you

hedging derivatives?

• Are the returns independent and identically

distributed (i.i.d.), albeit non normally?

• Is the variance of the distribution finite?

• Can you hedge with other options?

Most basic theory concerning asset allocation, such

as Modern Portfolio Theory, assumes that returns are

normally distributed. This allows a great deal of analyti-

cal progress to be made since adding random numbers

from normal distributions gives you another normal

distribution. But speculating in stocks, without hedg-

ing, exposes you to asset direction; you buy the stock

since you expect it to rise. Assuming that this stock

isn’t your only investment then your main concern is

for the expected stock price in the future, and not so

much its distribution. On the other hand, if you are

hedging options then you largely eliminate exposure to

asset direction. That’s as long as you aren’t hedging too

infrequently.

If you are hedging derivatives then your exposure is

to the range of returns, not the direction. That means

you are exposed to variance, if the asset moves are

small, or to the sizes and probabilities of discontinuous

jumps. Asset models can be divided roughly speaking

into those for which the variance of returns is finite,

and those for which it is not.

If the variance is finite then it doesn’t matter too much

whether or not the returns are normal. No, more impor-

tant is whether they are i.i.d. The ‘independent’ part is

also not that important since if there is any relation-

ship between returns from one period to the next it

204 Frequently Asked Questions In Quantitative Finance

tends to be very small in practice. The real question is

about variance, is it constant? If it is constant, and we

are hedging frequently, then we may as well work with

normal distributions and the Black–Scholes constant

volatility model. However, if it is not constant then

we may want to model this more accurately. Typical

approaches include the deterministic or local volatility

models, in which volatility is a function of asset and

time, σ ( S, t), and stochastic volatility models, in which we represent volatility by another stochastic process.

The latter models require a knowledge or specification

of risk preferences since volatility risk cannot be hedged

just with the underlying asset.

If the variance of returns is infinite, or there are jumps

in the asset, then normal distributions and Black–Scholes

are less relevant. Models capturing these effects also

require a knowledge or specification of risk preferences.

It is theoretically even harder to hedge options in these

worlds than in the stochastic volatility world.

To some extent the existence of other traded options

with which one can statically hedge a portfolio of deriva-

tives can reduce exposure to assumptions about distri-

butions or parameters. This is called hedging model

risk. This is particularly important for market makers.

Indeed, it is instructive to consider the way market

makers reduce risk.

• The market maker hedges one derivative with

another one, one sufficiently similar as to have

similar model exposure.

• As long as the market maker has a positive

expectation for each trade, although with some model

risk, having a large number of positions he will

reduce exposure overall by diversification. This is

more like an actuarial approach to model risk.

Chapter 2: FAQs

205

• If neither of the above is possible then he could

widen his bid-ask spreads. He will then only trade

with those people who have significantly different

market views from him.

References and Further Reading

Mandelbrot, B & Hudson, R 2004 The (Mis)Behaviour of Mar-

kets: A Fractal View of Risk, Ruin and Reward. Profile Books

206 Frequently Asked Questions In Quantitative Finance

How Robust is the Black–Scholes

Model?

Short Answer

Very robust. You can drop quite a few of the assump-

tions underpinning Black–Scholes and it won’t fall over.

Example

Transaction costs? Simply adjust volatility. Time-

dependent volatility? Use root-mean-square-average

volatility instead. Interest rate derivatives? Black ’76

explains how to use the Black–Scholes formulæ in situa-

tions where it wasn’t originally intended.

Long Answer

Here are some assumptions that seems crucial to the

whole Black–Scholes model, and what happens when

you drop those assumptions.

Hedging is continuous:

If you hedge discretely it turns out

that Black–Scholes is right on average. In other words

sometimes you lose because of discrete hedging, some-

times you win, but on average you break even. And

Black–Scholes still applies.

There are no transaction costs:

If there is a cost associated

with buying and selling the underlying for hedging this

can be modelled by a new term in the Black–Scholes

equation that depends on gamma. And that term is

usually quite small. If you rehedge at fixed time inter-

vals then the correction is proportional to the absolute

value of the gamma, and can be interpreted as simply

a correction to volatility in the standard Black–Scholes

formulæ. So instead of pricing with a volatility of 20%,

say, you might use 17% and 23% to represent the bid-

offer spread dues to transaction costs.

Chapter 2: FAQs

207

Volatility is constant:

If volatility is time dependent then

the Black–Scholes formulæ are still valid as long as you

plug in the ‘average’ volatility over the remaining life of

the option. Here average means the root-mean-square

average since volatilities can’t be added but variances

can.

Even if volatility is stochastic we can still use basic

Black–Scholes formulæ provided the volatility process is

independent of, and uncorrelated with, the stock price.

Just plug the average variance over the option’s lifetime,

conditional upon its current value, into the formulæ.

There are no arbitrage opportunities:

Even if there are arbi-

trage opportunities because implied volatility is different

from actual volatility you can still use the Black–Scholes

formulæ to tell you how much profit you can expect

to make, and use the delta formulæ to tell you how to

hedge. Moreover, if there is an arbitrage opportunity

and you don’t hedge properly, it probably won’t have

that much impact on the profit you expect to make.

The underlying is lognormally distributed:

The Black–Scholes

model is often used for interest-rate products which are

clearly not lognormal. But this approximation is often

quite good, and has the advantage of being easy to

understand. This is the model commonly referred to as

Black ’76.

There are no costs associated with borrowing stock for going short:

Easily accommodated within a Black–Scholes model, all

you need to do is make an adjustment to the risk-neutral

drift rate, rather like when you have a dividend.

Returns are normally distributed:

Thanks to near-continuous

hedging and the Central Limit Theorem all you really

208 Frequently Asked Questions In Quantitative Finance

need is for the returns distribution to have a finite vari-

ance, the precise shape of that distribution, its skew

and kurtosis, don’t much matter.

Black–Scholes is a remarkably robust model.

References and Further Reading

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 2: FAQs

209

Why is the Lognormal Distribution

Important?

Short Answer

The lognormal distribution is often used as a model

for the distribution of equity or commodity prices,

exchange rates and indices. The normal distribution

is often used to model returns.

Example

The stochastic differential equation commonly used to

represent stocks,

dS = µS dt + σ S dX

results in a lognormal distribution for S, provided µ and

σ are not dependent on stock price.

Long Answer

A quantity is lognormally distributed if its logarithm is

normally distributed, that is the definition of lognormal.

The probability density function is

1

√

exp − 1 ( ln( x) − a) 2

x ≥ 0,

2 π bx

2 b 2

where the parameters a and b > 0 represent location

and scale. The distribution is skewed to the right,

extending to infinity and bounded below by zero. (The

left limit can be shifted to give an extra parameter, and

it can be reflected in the vertical axis so as to extend to

minus infinity instead.)

If we have the stochastic differential equation above

then the probability density function for S in terms of

time and the parameters is

2

1

− ln( S/S

σ 2) t / 2 σ 2 t

√

0)−( µ− 1

e

2

,

σS 2 πt

where S 0 is the value of S at time t = 0.

210 Frequently Asked Questions In Quantitative Finance

You would expect equity prices to follow a random

walk around an exponentially growing average. So take

the logarithm of the stock price and you might expect

that to be normal about some mean. That is the non-

mathematical explanation for the appearance of the

lognormal distribution.

More mathematically we could argue for lognormality

via the Central Limit Theorem. Using Ri to represent

the random return on a stock price from day i − 1 to

day i we have

S 1 = S 0(1 + R 1),

the stock price grows by the return from day zero, its

starting value, to day 1. After the second day we also

have

S 2 = S 1(1 + R 2) = S 0(1 + R 1)(1 + R 2) .

After n days we have

n

Sn = S 0

(1 + Ri),

i=1

Figure 2-12: The probability density function for the lognormal ran-

dom walk evolving through time.

Chapter 2: FAQs

211

the stock price is the initial value multiplied by n fac-

tors, the factors being one plus the random returns.

Taking logarithms of this we get

n

ln( Sn) = ln( S 0) +

ln(1 + Ri),

i=1

the logarithm of a product being the sum of the loga-

rithms.

Now think Central Limit Theorem. If each Ri is ran-

dom, then so is ln(1 + Ri). So the expression for ln( Sn)

is just the sum of a large number of random num-

bers. As long as the Ri are independent and identically

distributed and the mean and standard deviation of

ln(1 + Ri) are finite then we can apply the CLT and con-

clude that ln( Sn) must be normally distributed. Thus

Sn is normally distributed. Since here n is number of

‘days’ (or any fixed time period) the mean of ln( Sn) is

going to be linear in n, i.e., will grow linearly with time,

and the standard deviation will be proportional to the

square root of n, i.e., will grow like the square root of

time.

References and Further Reading

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

212 Frequently Asked Questions In Quantitative Finance

What are Copulas and How are They

Used in Quantitative Finance?

Short Answer

Copulas are used to model joint distribution of multiple

underlyings. They permit a rich ‘correlation’ structure

between underlyings. They are used for pricing, for risk

management, for pairs trading, etc., and are especially

popular in credit derivatives.

Example

You have a basket of stocks which during normal days

exhibit little relationship with each other. We might

say that they are uncorrelated. But on days when the

market moves dramatically they all move together. Such

behaviour can be modelled by copulas.

Long Answer

The technique now most often used for pricing credit

derivatives when there are many underlyings is that of

the copula. The copula4 function is a way of simplifying

the default dependence structure between many under-

lyings in a relatively transparent manner. The clever

trick is to separate the distribution for default for each

individual name from the dependence structure between

those names. So you can rather easily analyze names

one at a time, for calibration purposes, for example, and

then bring them all together in a multivariate distribu-

tion. Mathematically, the copula way of representing the

dependence (one marginal distribution per underlying,

and a dependence structure) is no different from speci-

fying a multivariate density function. But it can simplify

the analysis.

4From the Latin for ‘join.’

Chapter 2: FAQs

213

The copula approach in effect allows us to readily go

from a single-default world to a multiple-default world

almost seamlessly. And by choosing the nature of the

dependence, the copula function, we can explore models

with rich ‘correlation’ structure. For example, having a

higher degree of dependence during big market moves

is quite straightforward.

Take N uniformly distributed random variables U 1, U 2,

. . . , UN, each defined on [0, 1]. The copula function is

defined as

C( u 1, u 2, . . . , uN ) = Prob( U 1 ≤ u 1, U 2 ≤ u 2, . . . , UN ≤ uN ) .

Clearly we have

C( u 1, u 2, . . . , 0, . . . , uN ) = 0, and

C(1, 1, . . . , ui, . . . , 1) = ui.

That is the copula function. The way it links many

univariate distributions with a single multivariate dis-

tribution is as follows.

Let x 1, x 2, . . . , xN be random variables with cumulative distribution functions (so-called marginal distributions)

of F 1( x 1), F 2( x 2), . . . , FN ( xN ). Combine the F s with the copula function,

C( F 1( x 1), F 2( x 2), . . . , FN ( xN )) = F ( x 1, x 2, . . . , xN ) and it’s easy to show that this function F ( x 1, x 2, . . . , xN ) is the same as

Prob( X 1 ≤ x 1, X 2 ≤ x 2, . . . , XN ≤ xN ) .

In pricing basket credit derivatives we would use the

copula approach by simulating default times of each of

the constituent names in the basket. And then perform

many such simulations in order to be able to analyze

214 Frequently Asked Questions In Quantitative Finance

the statistics, the mean, standard deviation, distribution,

etc., of the present value of resulting cashflows.

Here are some examples of bivariate copula functions.

They are readily extended to the multivariate case.

Bivariate Normal:

C( u, v) = N 2 N−1( u), N−1( v), ρ , −1 ≤ ρ ≤ 1, 1

1

where N 2 is the bivariate Normal cumulative distribution

function, and N−1 is the inverse of the univariate Normal

1

cumulative distribution function.

Frank:

C( u, v) = 1 ln 1 + ( eαu − 1)( eαv − 1) , −∞ < α < ∞ .

α

eα − 1

Fréchet–Hoeffding upper bound:

C( u, v) = min( u, v) .

Gumbel–Hougaard:

1 /θ

C( u, v) = exp − (− ln u) θ + (− ln v) θ

,

1 ≤ θ < ∞ .

This copula is good for representing extreme value dis-

tributions.

Product:

C( u, v) = uv

One of the simple properties to examine with each of

these copulas, and which may help you decide which is

best for your purposes, is the tail index. Examine

C

λ

( u, u)

( u) =

.

u

Chapter 2: FAQs

215

This is the probability that an event with probability

less than u occurs in the first variable given that at the

same time an event with probability less than u occurs

in the second variable. Now look at the limit of this as

u → 0,

λ

C( u, u)

L = lim

.

u→0

u

This tail index tells us about the probability of both

extreme events happening together.

References and Further Reading

Li, D 2000 On Default Correlation: A Copula Function Approach.

RiskMetrics Working Paper

Nelsen, RB 1999 An Introduction to Copulas. Springer Verlag

216 Frequently Asked Questions In Quantitative Finance

What is Asymptotic Analysis and How

is It Used in Financial Modelling?

Short Answer

Asymptotic analysis is about exploiting a large or small

parameter in a problem to find simple(r) equations or

even solutions. You may have a complicated integral

that is much nicer if you approximate it. Or a par-

tial differential equation that can be solved if you can

throw away some of the less important terms. Some-

times these are called approximate solutions. But the

word ‘approximate’ does not carry the same technical

requirements as ‘asymptotic.’

Example

The SABR model is a famous model for a forward rate

and its volatility that exploits low volatility of volatility

in order for closed-form solutions for option prices to

be found. Without that parameter being small we would

have to solve the problem numerically.

Long Answer

Asymptotic analysis is about exploiting a large or small

parameter to find simple(r) solutions/expressions. Out-

side finance asymptotic analysis is extremely common,

and useful. For example almost all problems in fluid

mechanics use it to make problems more tractable. In

fluid mechanics there is a very important non-dimensional

parameter called the Reynolds number. This quantity is

given by

ρUL

Re =

,

µ

where ρ is the density of the fluid, U is a typical velocity

in the flow, L is a typical lengthscale, and µ is the fluid’s

Chapter 2: FAQs

217

viscosity. This parameter appears in the Navier–Stokes

equation which, together with the Euler equation for

conservation of mass, governs the flow of fluids. And

this means the flow of air around an aircraft, and the

flow of glass. These equations are generally difficult to

solve. In university lectures they are solved in special

cases, perhaps special geometries. In real life during the

design of aircraft they are solved numerically. But these

equations can often be simplified, essentially approxi-

mated, and therefore made easier to solve, in special

‘regimes.’ The two distinct regimes are those of high

Reynolds number and low Reynolds number. When Re

is large we have fast flows, which are essentially invis-

cid to leading order. Assuming that Re 1 means that

the Navier–Stokes equation dramatically simplifies, and

can often be solved analytically. On the other hand if

we have a problem where Re 1 then we have slow

viscous flow. Now the Navier–Stokes equation simpli-

fies again, but in a completely different way. Terms

that were retained in the high Reynolds number case

are thrown away as being unimportant, and previously

ignored terms become crucial.

Remember we are looking at what happens when a

parameter gets small, well, let’s denote it by . (Equiv-

alently we also do asymptotic analysis for large para-

meters, but the we can just define the large parameter

to be 1 / .) In asymptotic analysis we use the following

symbols a lot: O(·), o(·) and ∼. These are defined as

follows.

We say that f ( ) = O g( )

as → 0

f ( )

if

lim

is finite.

→0 g( )

We say that f ( ) = o g( )

as → 0

f ( )

if

lim

→ 0 .

→0 g( )

218 Frequently Asked Questions In Quantitative Finance

We say that f ( ) ∼ g( ) as → 0

f ( )

if

lim

= 1 .

→0 g( )

In finance there have been several examples of asymp-

totic analysis.

Transactions costs:

Transaction costs are usually a small

percentage of a trade. There are several models for the

impact that these costs have on option prices and in

some cases these problems can be simplified by per-

forming an asymptotic analysis as this cost parameter

tends to zero. These costs models are invariably non

linear.

SABR:

This model for forward rates and their volatil-

ity is a two-factor model. It would normally have to

be solved numerically but as long as the volatility of

volatility parameter is small then closed-form asymp-

totic solutions can be found. Since the model requires

small volatility of volatility it is best for interest rate

derivatives.

Fast drift and high volatility in stochastic volatility models:

These

are a bit more complicated, singular perturbation prob-

lems. Now the parameter is large, representing both

fast reversion of volatility to its mean and large volatil-

ity of volatility. This model is more suited to the more

dramatic equity markets which exhibit this behaviour.

References and Further Reading

Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Man-

aging smile risk. Wilmott magazine, September

Chapter 2: FAQs

219

Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of

stochastic volatility models. In New Directions in Mathe-

matical Finance, Ed. Wilmott, P & Rasmussen, H, John Wiley

& Sons

Whalley, AE & Wilmott, P 1997 An asymptotic analysis of an

optimal hedging model for option pricing with transaction

costs. Mathematical Finance 7 307–324

220 Frequently Asked Questions In Quantitative Finance

What is a Free-boundary Problem and

What is the Optimal-Stopping Time

for an American Option?

Short Answer

A boundary-value problem is typically a differential

equation with specified solution on some domain. A

free-boundary problem is one for which that boundary

is also to be found as part of the solution. When to

exercise an American option is an example of a free-

boundary problem, the boundary representing the time

and place at which to exercise. This is also called an

optimal-stopping problem, the ‘stopping’ here referring

to exercise.

Example

Allow a box of ice cubes to melt. As they do there

will appear a boundary between the water and the

ice, the free boundary. As the ice continues to melt

so the amount of water increases and the amount of ice

decreases.

Waves on a pond is another example of a free boundary.

Long Answer

In a boundary-value problem the specification of the

behaviour of the solution on some domain is to pin

down the problem so that is has a unique solution.

Depending on the type of equation being solved we

must specify just the right type of conditions. Too few

conditions and the solution won’t be unique. Too many

and there may not be any solution. In the diffusion

equations found in derivatives valuation we must specify

a boundary condition in time. This would be the final

payoff, and it is an example of a final condition. We

Chapter 2: FAQs

221

must also specify two conditions in the asset space. For

example, a put option has zero value at infinite stock

price and is the discounted strike at zero stock price.

These are examples of boundary conditions. These

three are just the right number and type of conditions

for there to exist a unique solution of the Black–Scholes

parabolic partial differential equation.

In the American put problem it is meaningless to specify

the put’s value when the stock price is zero because the

option would have been exercised before the stock ever

got so low. This is easy to see because the European put

value falls below the payoff for sufficiently small stock.

If the American option price were to satisfy the same

equation and boundary conditions as the European then

it would have the same solution, and this solution would

permit arbitrage.

The American put should be exercised when the stock

falls sufficiently low. But what is ‘sufficient’ here?

To determine when it is better to exercise than to hold

we must abide by two principles.

• The option value must never fall below the payoff,

otherwise there will be an arbitrage opportunity.

• We must exercise so as to give the option its highest

value.

The second principle is not immediately obvious. The

explanation is that we are valuing the option from the

point of view of the writer. He must sell the option for

the most it could possibly be worth, for if he under-

values the contract he may make a loss if the holder

exercises at a better time. Having said that, we must

also be aware that we value from the viewpoint of a

delta-hedging writer. He is not exposed to direction of

222 Frequently Asked Questions In Quantitative Finance

the stock. However the holder is probably not hedg-

ing and is therefore very exposed to stock direction.

The exercise strategy that is best for the holder will

probably not be what the writer thinks is best. More of

this anon.

The mathematics behind finding the optimal time to

exercise, the optimal-stopping problem, is rather tech-

nical. But its conclusion can be stated quite succinctly.

At the stock price at which it is optimal to exercise we

must have

• the option value and the payoff function must be

continuous as functions of the underlying,

• the delta, the sensitivity of the option value with

respect to the underlying, must also be continuous as

functions of the underlying.

This is called the smooth-pasting condition since it rep-

resents the smooth joining of the option value function

to its payoff function. (Smooth meaning function and its

first derivative being continuous.)

This is now a free-boundary problem. On a fixed, pre-

scribed boundary we would normally impose one condi-

tion. (For example, the above example of the put’s value

at zero stock price.) But now we don’t know where the

boundary actually is. To pin it down uniquely we impose

two conditions, continuity of function and continuity of

gradient. Now we have enough conditions to find the

unknown solution.

Free-boundary problems such as these are non linear.

You cannot add two together to get another solution.

For example, the problem for an American straddle is

not the same as the sum of the American call and the

American put.

Chapter 2: FAQs

223

Although the fascinating mathematics of free-boundary

problems can be complicated, and difficult or impossible

to solve analytically, they can be easy to solve by finite-

difference methods. For example, if in a finite-difference

solution we find that the option value falls below the

payoff then we can just replace it with the payoff. As

long as we do this each time step before moving on to

the next time step then we should get convergence to

the correct solution.

As mentioned above, the option is valued by maximizing

the value from the point of view of the delta-hedging

writer. If the holder is not delta hedging but speculating

on direction he may well find that he wants to exit his

position at a time that the writer thinks is suboptimal.

In this situation there are three ways to exit:

• sell the option;

• delta hedge to expiration;

• exercise the option.

The first of these is to be preferred because the option

may still have market value in excess of the payoff.

The second choice is only possible if the holder can

hedge at low cost. If all else fails, he can always close

his position by exercising. This is of most relevance

in situations where the option is an exotic, over the

counter, contract with an early-exercise feature when

selling or delta hedging may not be possible.

There are many other contracts with decision features

that can be treated in a way similar to early exercise

as free-boundary problems. Obvious examples are con-

version of a convertible bond, callability, shout options,

choosers.

224 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Ahn, H & Wilmott, P 2003 On exercising American options:

the risk of making more money than you expected. Wilmott

magazine March 2003 52–63

Crank, JC 1984 Free and moving Boundary Value Problems.

Oxford

Chapter 2: FAQs

225

What are Low Discrepancy Numbers?

Short Answer

Low-discrepancy sequences are sequences of num-

bers that cover a space without clustering and without

gaps, in such a way that adding another number to the

sequence also avoids clustering and gaps. They give the

appearance of randomness yet are deterministic. They

are used for numerically estimating integrals, often in

high dimensions. The best known sequences are due to

Faure, Halton, Hammersley, Niederreiter and Sobol’.

Example

You have an option that pays off the maximum of 20

exchange rates on a specified date. You know all the

volatilities and correlations. How can you find the value

of this contract? If we assume that each exchange rate

follows a lognormal random walk, then this problem

can be solved as a 20-dimensional integral. Such a high-

dimensional integral must be evaluated by numerical

quadrature, and an efficient way to do this is to use

low-discrepancy sequences.

Long Answer

Some financial problems can be recast as integrations,

sometimes in a large number of dimensions. For ex-

ample, the value of a European option on lognormal

random variables can be written as the present value

of the risk-neutral expected payoff. The expected payoff

is an integral of the product of the payoff function and

the probability density function for the underlying(s) at

expiration. If there are n underlyings then there is typ-

ically an n-dimensional integral to be calculated. If the

number of dimensions is small then there are simple

efficient algorithms for performing this calculation. In

one dimension, for example, divide the domain of inte-

gration up into uniform intervals and use the trapezium

226 Frequently Asked Questions In Quantitative Finance

rule. This means evaluating the integrand at a number

of points, and as the number of function evaluations

increases so the accuracy of the method improves.

Unfortunately, in higher dimensions evaluating the

function at uniformly spaced points becomes compu-

tationally inefficient.

If the domain of integration is a unit hypercube (and, of

course, it can always be transformed into one) then the

value of the integral is the same as the average of the

function over that domain:

1

1

N

. . .

f (x) dx ≈ 1

f (x i) .

0

0

N i=1

Where the x i are uniformly distributed. This suggests

that an alternative method of numerical evaluation of

the integral is to select the points in the hypercube

from a uniform random distribution and then compute

their average. If N function evaluations are performed

then the method converges like O( N−1 / 2). This is the

Monte Carlo method of numerical integration. Although

very simple to perform it suffers from problems associ-

ated with the inevitable clustering and gapping that will

happen with randomly chosen numbers.

Clearly we would like to use a sequence of numbers that

do not suffer from the gapping/clustering problem. This

is where low-discrepancy sequences come in.

Low-discrepancy numbers exploit the Koksma-Hlawka

inequality which puts a bound on the error in the

above averaging method for an arbitrary sets of sam-

pling points x i. The Koksma-Hlawka inequality says that

if f (x) is of bounded variation V ( f ) then

1

1

N

. . .

f (x) dx − 1

f (x i) ≤ V ( f ) D∗ (x

N

1, . . . , (x) N )

0

0

N i=1

Chapter 2: FAQs

227

where D∗ (x

N

1, . . . , (x) N ) is the discrepancy of the se-

quence. (This discrepancy measures the deviation from

a uniform distribution. It is calculated by looking at

how many of the sampling points can be found in sub

intervals compared with how many there would be for

a uniform distribution and then taking the worst case.)

Rather than the details, the important point concerning

this result is that the bound is a product of one term

specific to the function (its variation, which is indepen-

dent of the set of sampling points) and a term specific

to the set of sampling points (and independent of the

function being sampled). So once you have found a set

of points that is good, of low discrepancy, then it will

work for all integrands of bounded variation.

The popular low-discrepancy sequences mentioned

above have

(ln N) n

D∗ < C

N

N

where C is a constant. Therefore convergence of this

quasi Monte Carlo numerical quadrature method is

faster than genuinely random Monte Carlo.

Another advantage of these low-discrepancy sequences

is that if you collapse the points onto a lower dimension

(for example, let all of the points in a two-dimensional

plot fall down onto the horizontal axis) they will not be

repeated, they will not fall on top of one another. This

means that if there is any particularly strong depen-

dence on one of the variables over the others then the

method will still give an accurate answer because it will

distribute points nicely over lower dimensions.

Unfortunately, achieving a good implementation of

some low-discrepancy sequences remains tricky. Some

practitioners prefer to buy off-the-shelf software for

generating quasi-random numbers.

228 Frequently Asked Questions In Quantitative Finance

References and Further Reading

Barrett, JW, Moore, G & Wilmott, P 1992 Inelegant efficiency.

Risk magazine 5 (9) 82–84

Cheyette, O 1990 Pricing options on multiple assets. Adv. Fut.

Opt. Res. 4 68–91

Faure, H 1969 Résultat voisin d’un théreme de Landau sur le

nombre de points d’un réseau dans une hypersphere. C. R.

Acad. Sci. Paris S´

er. A 269 383–386

Halton, JH 1960 On the efficiency of certain quasi-random

sequences of points in evaluating multi-dimensional inte-

grals. Num. Maths. 2 84–90

Hammersley, JM & Handscomb, DC 1964 Monte Carlo Methods.

Methuen, London

Haselgrove, CB 1961 A method for numerical integration. Math-

ematics of Computation 15 323–337

Jäckel, P 2002 Monte Carlo Methods in Finance. John Wiley &

Sons

Niederreiter, H 1992 Random Number Generation and Quasi-

Monte Carlo Methods. SIAM

Ninomiya, S & Tezuka, S 1996 Toward real-time pricing of

complex financial derivatives. Applied Mathematical Finance

3 1–20

Paskov 1996 New methodologies for valuing derivatives. In

Mathematics of Derivative Securities (Eds Pliska, SR and

Dempster, M)

Paskov, SH & Traub, JF 1995 Faster valuation of financial

derivatives. Journal of Portfolio Management Fall 113–120

Press, WH, Flannery, BP, Teukolsky, SA & Vetterling, WT 1992

Numerical Recipes in C. Cambridge University Press

Sloan, IH & Walsh, L 1990 A computer search of rank two

lattice rules for multidimensional quadrature. Mathematics of

Computation 54 281–302

Chapter 2: FAQs

229

Sobol’, IM 1967 On the distribution of points in cube and the

approximate evaluation of integrals. USSR Comp. Maths and

Math. Phys. 7 86–112

Traub, JF & Wozniakowski, H 1994 Breaking intractability.

Scientific American Jan 102–107

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 3

The Most Popular

Probability

Distributions and Their

Uses in Finance

232 Frequently Asked Questions In Quantitative Finance

R andom variables can be continuous or discrete

(the latter denoted below by ∗). Or a combination.

New distributions can also be made up using random

variables from two or more distributions.

Here is a list of distributions seen in finance (mostly),

and some words on each.

Normal or Gaussian

This distribution is unbounded below

and above, and is symmetrical about its mean. It has

two parameters: a, location; b > 0 scale. Its probability

density function is given by

1

√

− ( x− a)2

e

2 b 2

.

2 π b

This distribution is commonly used to model equity

returns, and, indeed, the changes in many financial

quantities. Errors in observations of real phenomena

are often normally distributed. The normal distribution

is also common because of the Central Limit Theorem.

Mean

a.

Variance

b 2 .

Lognormal

Bounded below, unbounded above. It has

two parameters: a, location; b > 0 scale. Its probability

density function is given by

1

√

exp − 1 ( ln( x) − a) 2

x ≥ 0 .

2 π bx

2 b 2

This distribution is commonly used to model equity

prices. Lognormality of prices follows from the assumption

of normally distributed returns.

Chapter 3: The Most Popular Probability Distributions

233

0.45

Normal

a = 0

0.4

b = 1

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

4

1

Lognormal

a = 0.4

0.9

b = 0.3

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1

1.5

2

2.5

3

3.5

4

234 Frequently Asked Questions In Quantitative Finance

Mean

ea+ 1 b 2

2

.

Variance

e 2 a+ b 2 ( eb 2 − 1) .

Poisson∗

The random variables take non-negative integer

values only. The distribution has one parameter: a > 0.

Its probability density function is given by

e− aax , x = 0,1,2,3, ....

x!

This distribution is used in credit risk modeling, repre-

senting the number of credit events in a given time.

Mean

a.

0.3

Poisson

a = 2

0.25

0.2

0.15

0.1

0.05

0

0

1

2

3

4

5

6

7

Chapter 3: The Most Popular Probability Distributions

235

Variance

a.

Chi square

Bounded below and unbounded above. It has

two parameters a ≥ 0, the location; ν, an integer, the

degrees of freedom. Its probability density function is

given by

∞

e−( x+ a) / 2

xi−1+ ν/ 2 ai

x ≥ 0,

2 ν/ 2

22 ij! ( i + ν/ 2)

i=0

where (·) is the Gamma function. The chi-square distri-

bution comes from adding up the squares of ν normally

distributed random variables. The chi-square distribu-

tion with one degree of freedom is the distribution of

the hedging error from an option that is hedged only

discretely. It is therefore a very important distribution

in option practice, if not option theory.

0.3

Chi Square

a = 0

0.25

b = 3

0.2

0.15

0.1

0.05

0

0

0.5

1

1.5

2

2.5

3

3.5

4

236 Frequently Asked Questions In Quantitative Finance

Mean

ν + a.

Variance

2( ν + 2 a) .

Gumbel

Unbounded above and below. It has two param-

eters: a, location; b > 0 scale. Its probability density

function is given by

1

a− x

a− x

e

b

b

e− e

.

b

The Gumbel distribution is useful for modelling extreme

values, representing the distribution of the maximum

value out of a large number of random variables drawn

from an unbounded distribution.

0.4

Gumbel

a = -1

0.35

b = 1

0.3

0.25

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

4

Chapter 3: The Most Popular Probability Distributions

237

Mean

a + γ b,

where γ is Euler’s constant, 0.577216 . . . .

Variance

1 π 2 b 2 .

6

Weibull

Bounded below and unbounded above. It has

three parameters: a, location; b > 0, scale; c > 0, shape.

Its probability density function is given by

c

x − a c−1

x − a c

exp −

,

x > a.

b

b

b

The Weibull distribution is also useful for modelling

extreme values, representing the distribution of the max-

imum value out of a large number of random variables

1

Weibull

a = 1

0.9

b = 1

0.8

c = 2

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

1

1.5

2

2.5

3

3.5

4

238 Frequently Asked Questions In Quantitative Finance

drawn from a bounded distribution. (The figure shows

a ‘humped’ Weibull, but depending on parameter values

the distribution can be monotonic.)

Mean

c + 1

a + b

.

c

Variance

c + 2

c + 1 2

b 2

−

.

c

c

Where (·) is the Gamma function.

Student’s t

Unbounded above and below. It has three

parameters: a, location; b > 0, scale; c > 0, degrees of freedom. Its probability density function is given by

− c+1

c+1

x− a 2

2

2

√

1 +

b

,

b πc c

c

2

where (·) is the Gamma function. This distribution

represents small-sample drawings from a normal distri-

bution. It is also used for representing equity returns.

Mean

a.

Variance

c

b 2 .

c − 2

Note that the n th moment only exists if c > n.

Pareto

Bounded below, unbounded above. It has two

parameters: a > 0, scale; b > 0 shape. Its probability

density function is given by

bab

x ≥ a.

xb+1

Chapter 3: The Most Popular Probability Distributions

239

0.4

Student's t

a = -1

0.35

b = 1

0.3

c = 2

0.25

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

4

Pareto

a = 1

1

b = 1

0.8

0.6

0.4

0.2

0

1

1.5

2

2.5

3

3.5

4

240 Frequently Asked Questions In Quantitative Finance

Commonly used to describe the distribution of wealth,

this is the classical power-law distribution.

Mean

ab .

b − 1

Variance

a 2 b

.

( b − 2)( b − 1)2

Note that the n th moment only exists if b > n.

Uniform

Bounded below and above. It has two location

parameters, a and b. Its probability density function is

given by

1

,

a < x < b.

b − a

1.2

Uniform

a = 1

1

b = 2

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Chapter 3: The Most Popular Probability Distributions

241

Mean

a + b .

2

Variance

( b − a)2 .

12

Inverse normal

Bounded below, unbounded above. It has

two parameters: a > 0, location; b > 0 scale. Its proba-

bility density function is given by

b

2

− b x− a

e 2 x

a

x ≥ 0 .

2 πx 3

This distribution models the time taken by a Brownian

Motion to cover a certain distance.

1.2

Inverse normal

a = 1

1

b = 4

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

242 Frequently Asked Questions In Quantitative Finance

Mean

a.

Variance

a 3 .

b

Gamma

Bounded below, unbounded above. It has three

parameters: a, location; b > 0 scale; c > 0 shape. Its probability density function is given by

1

x − a c−1 a− x

e b ,

x ≥ a,

b( c)

b

where (·) is the Gamma function. When c = 1 this is

the exponential distribution and when a = 0 and b = 2

this is the chi-square distribution with 2 c degrees of

freedom.

0.8

Gamma

a = 0

0.7

b = 0.5

0.6

c = 2

0.5

0.4

0.3

0.2

0.1

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Chapter 3: The Most Popular Probability Distributions

243

Mean

a + bc.

Variance

b 2 c.

Logistic

This distribution is unbounded below and above.

It has two parameters: a, location; b > 0 scale. Its prob-

ability density function is given by

x− a

1

e b

.

b

x− a

2

1 + e b

The logistic distribution models the mid value of highs

and lows of a collection of random variables, as the

number of samples becomes large.

Mean

a.

0.3

Logistic

a = -2

0.25

b = 1

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

4

244 Frequently Asked Questions In Quantitative Finance

Variance

1 π 2 b 2 .

3

Laplace

This distribution is unbounded below and above.

It has two parameters: a, location; b > 0 scale. Its prob-

ability density function is given by

1 e−| x− a|

b

.

2 b

Errors in observations are usually either normal or

Laplace.

Mean

a.

Variance

2 b 2 .

0.18

Laplace

a = 1

0.16

b = 3

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

-4

-3

-2

-1

0

1

2

3

4

Chapter 3: The Most Popular Probability Distributions

245

Cauchy

This distribution is unbounded below and above.

It has two parameters: a, location; b > 0 scale. Its prob-

ability density function is given by

1

.

π

2

b 1 + x− a

b

This distribution is rarely used in finance. It does not

have any finite moments, but its mode and median are

both a.

Beta

This distribution is bounded below and above.

It has four parameters: a, location of lower limit; b >

a location of upper limit; c > 0 and d > 0 shape. Its probability density function is given by

( c + d)

(x − a)c−1 (b − x)d−1 , a ≤ x ≤ b,

( c) ( d)( b − a) c+ d−1

0.35

Cauchy

a = 0.5

0.3

b = 1

0.25

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

4

246 Frequently Asked Questions In Quantitative Finance

1.2

Beta

a = 1

1

b = 3

c = 2

0.8

d = 4

0.6

0.4

0.2

0

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

where (·) is the Gamma function. This distribution is

rarely used in finance.

Mean

ad + bc .

c + d

Variance

cd( b − a)2

.

( c + d + 1)( c + d)2

Exponential

Bounded below, unbounded above. It has

two parameters: a, location; b > 0 scale. Its probability

density function is given by

1 a− x

e b

x ≥ a.

b

This distribution is rarely used in finance.

Chapter 3: The Most Popular Probability Distributions

247

Exponential

a = 0

1

b = 1

0.8

0.6

0.4

0.2

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Mean

a + b.

Variance

b 2 .

Lévy

Unbounded below and above. It has four param-

eters: µ, a location (mean); 0 < α ≤ 2, the peakedness;

−1 < β < 1, the skewness; ν > 0, a spread. (Conven-

tional notation is used here.) This distribution has been

saved to last because its probability density function

does not have a simple closed form. Instead it must be

written in terms of its characteristic function. If P( x)

is the probability density function then the moment

generating function is given by

∞

M ( z) =

eizxP( x) dx,

−∞

248 Frequently Asked Questions In Quantitative Finance

0.4

Levy

µ = 0

0.35

α = 0.5

0.3

β = 0

0.25

ν = 1

0.2

0.15

0.1

0.05

0

-4

-3

-2

-1

0

1

2

3

√

where i =

−1. For the Lévy distribution

ln( M ( z)) = iµz − να| z| α ( 1 − iβ sgn( z) tan( πa/ 2) ) , for α = 1

or

ln( M ( z)) = iµz − ν| z| 1 + 2 iβ sgn( z) ln(| z|) , for α = 1 .

π

The normal distribution is a special case of this with

α = 2 and β = 0, and with the parameter ν being one

half of the variance. The Lévy distribution, or Pareto

Lévy distribution, is increasingly popular in finance

because it matches data well, and has suitable fat tails.

It also has the important theoretical property of being

a stable distribution in that the sum of independent

random numbers drawn from the Lévy distribution will

itself be Lévy. This is a useful property for the distribu-

tion of returns. If you add up n independent numbers

from the Lévy distribution with the above parameters

Chapter 3: The Most Popular Probability Distributions

249

then you will get a number from another Lévy distri-

bution with the same α and β but with mean of n 1 /αµ

and spread n 1 /αν. The tail of the distribution decays like

| x|−1− α.

Mean

µ.

Variance

infinite, unless α = 2, when it is 2 ν.

References and further reading

Spiegel, MR, Schiller, JJ, Srinivasan, RA 2000 Schaum’s Outline

of Probability and Statistics. McGraw–Hill

Chapter 4

Ten Different Ways

to Derive

Black–Scholes

252 Frequently Asked Questions In Quantitative Finance

T hetendifferentwaysofderivingtheBlack–Scholes

equation or formulæ that follow use different types

of mathematics, with different amounts of complexity

and mathematical baggage. Some derivations are useful

in that they can be generalized, and some are very

specific to this one problem. Naturally we will spend

more time on those derivations that are most useful or

give the most insight. The first eight ways of deriving

the Black–Scholes equation/formulæ are taken from the

excellent paper by Jesper Andreason, Bjarke Jensen and

Rolf Poulsen (1998).

In most cases we work within a framework in which

the stock path is continuous, the returns are normally

distributed, there aren’t any dividends, or transac-

tion costs, etc. To get the closed-form formulæ (the

Black–Scholes formulæ) we need to assume that volatil-

ity is constant, or perhaps time dependent, but for

the derivations of the equations relating the greeks

(the Black–Scholes equation) the assumptions can be

weaker, if we don’t mind not finding a closed-form

solution.

In many cases, some assumptions can be dropped.

The final derivation, Black–Scholes for accountants,

uses perhaps the least amount of formal mathematics

and is easy to generalize. It also has the advantage

that it highlights one of the main reasons why the

Black–Scholes model is less than perfect in real life. I

will spend more time on that derivation than most of

the others.

I am curious to know which derivation(s) readers prefer.

Please email your comments to paul@wilmott.com. Also if

you know of other derivations please let me know.

Chapter 4: Ten Different Ways to Derive Black–Scholes

253

Hedging and the Partial Differential

Equation

The original derivation of the Black–Scholes partial

differential equation was via stochastic calculus, Itô’s

lemma and a simple hedging argument (Black & Scholes,

1973).

Assume that the underlying follows a lognormal ran-

dom walk

dS = µS dt + σ S dX.

Use to denote the value of a portfolio of one long

option position and a short position in some quantity

of the underlying:

= V( S, t) − S.

(4.1)

The first term on the right is the option and the second

term is the short asset position.

Ask how the value of the portfolio changes from time t

to t + dt. The change in the portfolio value is due partly

to the change in the option value and partly to the

change in the underlying:

d = dV − dS.

From Itô’s lemma we have

∂V

∂V

∂ 2 V

d =

dt +

dS + 1 σ 2 S 2

dt − dS.

∂t

∂S

2

∂S 2

The right-hand side of this contains two types of terms,

the deterministic and the random. The deterministic

terms are those with the dt, and the random terms are

those with the dS. Pretending for the moment that we

254 Frequently Asked Questions In Quantitative Finance

know V and its derivatives then we know everything

about the right-hand side except for the value of dS,

because this is random.

These random terms can be eliminated by choosing

∂V

=

.

∂S

After choosing the quantity , we hold a portfolio

whose value changes by the amount

∂V

∂ 2 V

d =

+ 1 σ 2 S 2

dt.

∂t

2

∂S 2

This change is completely riskless. If we have a com-

pletely risk-free change d in the portfolio value

then it must be the same as the growth we would get

if we put the equivalent amount of cash in a risk-free

interest-bearing account:

d = r dt.

This is an example of the no arbitrage principle.

Putting all of the above together to eliminate and

in favour of partial derivatives of V gives

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ rS

− rV = 0,

∂t

2

∂S 2

∂S

the Black–Scholes equation.

Solve this quite simple linear diffusion equation with the

final condition

V ( S, T ) = max( S − K, 0)

and you will get the Black–Scholes call option formula.

This derivation of the Black–Scholes equation is perhaps

the most useful since it is readily generalizable (if not

necessarily still analytically tractable) to different under-

lyings, more complicated models, and exotic contracts.

Chapter 4: Ten Different Ways to Derive Black–Scholes

255

Martingales

The martingale pricing methodology was formalized

by Harrison and Kreps (1979) and Harrison and Pliska

(1981).1

We start again with

dSt = µS dt + σ S dWt

The Wt is Brownian motion with measure P. Now intro-

duce a new equivalent martingale measure Q such that

˜

Wt = Wt + ηt,

where η = ( µ − r) /σ .

Under Q we have

dSt = rS dt + σ S d ˜

Wt.

Introduce

Gt = e− r( T− t) E Q

t [max( ST − K , 0)] .

The quantity er( T− t) Gt is a Q-martingale and so

d er( T− t) Gt = αter( T− t) Gt d ˜

Wt

for some process αt. Applying Itô’s lemma,

dGt = ( r + αη) Gtdt + αGt dWt.

This stochastic differential equation can be rewritten

as one representing a strategy in which a quantity

αGt/σS of the stock and a quantity ( G − αGt/σ) er( T− t) 1If my notation changes, it is because I am using the notation

most common to a particular field. Even then the changes are

minor, often just a matter of whether one puts a subscript t on

a dW for example.

256 Frequently Asked Questions In Quantitative Finance

of a zero-coupon bond maturing at time T are bought:

αGt

G − αGt S

dG

σ S

t =

dS +

d( e− r( T− t)) .

σS

e− r( T− t)

Such a strategy is self financing because the values of

the stock and bond positions add up to G. Because

of the existence of such a self-financing strategy and

because at time t = T we have that GT is the call payoff we must have that Gt is the value of the call before

expiration. The role of the self-financing strategy is to

ensure that there are no arbitrage opportunities.

Thus the price of a call option is

e− r( T− t) E Q

t [max( ST − K , 0)] .

The interpretation is simply that the option value is the

present value of the expected payoff under a risk-neutral

random walk.

For other options simply put the payoff function inside

the expectation.

This derivation is most useful for showing the link

between option values and expectations, as it is the

theoretical foundation for valuation by Monte Carlo

simulation.

Now that we have a representation of the option value

in terms of an expectation we can formally calculate this

quantity and hence the Black–Scholes formulæ. Under Q

the logarithm of the stock price at expiration is normally

distributed with mean m = ln( St) + r − 1 σ 2 ( T − t) 2

and variance v 2 = σ 2( T − t). Therefore the call option value is

∞

e− 1 x 2

2

e− r( T− t)

( em+ vx − K) √

dx.

ln K− m

2 π

v

Chapter 4: Ten Different Ways to Derive Black–Scholes

257

A simplification of this using the cumulative distribution

function for the standardized normal distribution results

in the well-known call option formula.

Change of Numeraire

The following is a derivation of the Black–Scholes call

(or put) formula, not the equation, and is really just a

trick for simplifying some of the integration.

It starts from the result that the option value is

e− r( T− t) E Q

t [max( ST − K , 0)] .

This can also be written as

e− r( T− t) E Q

t [( ST − K ) H( S − K )],

where H( S − K) is the Heaviside function, which is zero

for S < K and 1 for S > K.

Now define another equivalent martingale measure Q

such that

˜

W = W

t

t + ηt − σ t.

The option value can then be written as

( ST − K) H( S − K)

StE Q

.

t

ST

where

dSt = ( r + σ 2) S dt + σ Sd ˜

W .

t

It can also be written as a combination of the two

expressions,

ST H( S − K)

StE Q

−

t

Ke− r( T− t) E Q

S

t [ H( S − K )] .

T

Notice that the same calculation is to be performed,

an expectation of H( S − K), but under two different

258 Frequently Asked Questions In Quantitative Finance

measures. The end result is the Black–Scholes formula

for a call option.

This method is most useful for simplifying valuation

problems, perhaps even finding closed-form solutions,

by using the most suitable traded contract to use for

the numeraire.

The relationship between the change of numeraire result

and the partial differential equation approach is very

simple, and informative.

First let us make the comparison between the risk-

neutral expectation and the Black–Scholes equation

as transparent as possible. When we write

e− r( T− t) E Q

t [max( ST − K , 0)]

we are saying that the option value is the present

value of the expected payoff under the risk-neutral ran-

dom walk

dS = rS dt + σ S d ˜

Wt.

The partial differential equation

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ rS

− rV = 0

∂t

2

∂S 2

∂S

means exactly the same because of the relationship

between it and the Fokker–Planck equation. In this

equation the diffusion coefficient is always just one half

of the square of the randomness in dS. The coefficient

of ∂V /∂S is always the risk-neutral drift rS and the coef-

ficient of V is always minus the interest rate, − r, and

represents the present valuing from expiration to now.

If we write the option value as V = SV then we can

think of V as the number of shares the option is equiv-

alent to, in value terms. It is like using the stock as the

Chapter 4: Ten Different Ways to Derive Black–Scholes

259

unit of currency. But if we rewrite the Black–Scholes

equation in terms of V using

∂V

∂V

∂V

∂V

= S

,

= S

+ V ,

∂t

∂t

∂S

∂S

∂ 2 V

∂ 2 V

∂V

and

= S

+ 2 S

,

∂S 2

∂S 2

∂S

then we have

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ ( r + σ 2) S

= 0 .

∂t

2

∂S 2

∂S

The function V can now be interpreted, using the

same comparison with the Fokker–Planck equation,

as an expectation, but this time with respect to the

random walk

dS = ( r + σ 2) S dt + σ S d ˜

W .

t

And there is no present valuing to be done. Since at

expiration we have for the call option

max( ST − K, 0)

ST

we can write the option value as

( ST − K) H( S − K)

StE Q

.

t

ST

where

dSt = ( r + σ 2) S dt + σ Sd ˜

W .

t

Change of numeraire is no more than a change of depen-

dent variable.

Local Time

The most obscure of the derivations is the one involving

the concept from stochastic calculus known as ‘local

time.’ Local time is a very technical idea involving the

time a random walk spends in the vicinity of a point.

260 Frequently Asked Questions In Quantitative Finance

The derivation is based on the analysis of a stop-loss

strategy in which one attempts to hedge a call by selling

one share short if the stock is above the present value

of the strike, and holding nothing if the stock is below

the present value of the strike. Although at expiration the

call payoff and the stock position will cancel each other

exactly, this is not a strategy that eliminates risk. Na¨ıvely

you might think that this strategy would work, after all

when you sell short one of the stock as it passes through

the present value of the strike you will neither make nor

lose money (assuming there are no transaction costs). But

if that were the case then an option initially with strike

above the forward stock price should have zero value. So

clearly something is wrong here.

To see what goes wrong you have to look more closely

at what happens as the stock goes through the present

value of the strike. In particular, look at discrete moves

in the stock price.

As the forward stock price goes from K to K + sell one

share and buy K bonds. And then every time the stock

falls below the present value of the strike you reverse

this. Even in the absence of transaction costs, there will

be a slippage in this process. And the total slippage

will depend on how often the stock crosses this point.

Herein lies the rub. This happens an infinite number of

times in continuous Brownian motion.

If U ( ) is the number of times the forward price moves

from K to K + , which will be finite since is finite,

then the financing cost of this strategy is

U( ) .

Now take the limit as → 0 and this becomes the

quantity known as local time. This local-time term is

what explains the apparent paradox with the above

example of the call with zero value.

Chapter 4: Ten Different Ways to Derive Black–Scholes

261

Now we go over to the risk-neutral world to value

the local-time term, ending up, eventually, with the

Black–Scholes formula.

It is well worth simulating this strategy on a spread-

sheet, using a finite time step and let this time step get

smaller and smaller.

Parameters as Variables

The next derivation is rather novel in that it involves

differentiating the option value with respect to the

parameters strike, K, and expiration, T , instead of the

more usual differentiation with respect to the variables

S and t. This will lead to a partial differential equation

that can be solved for the Black–Scholes formulæ. But

more importantly, this technique can be used to deduce

the dependence of volatility on stock price and time,

given the market prices of options as functions of strike

and expiration. This is an idea due to Dupire (1993)

(also see Derman & Kani, 1993, and Rubinstein, 1993, for

related work done in a discrete setting) and is the basis

for deterministic volatility models and calibration.

We begin with the call option result from above

V = e− r( T− t) E Q

t [max( ST − K , 0)],

that the option value is the present value of the risk-

neutral expected payoff. This can be written as

∞

V ( K, T ) = e− r( T− t∗)

max( S − K, 0) p( S∗, t∗; S, T ) dS

0

∞

= e− r( T− t∗)

( S − K) p( S∗, t∗; S, T ) dS, K

where p( S∗, t∗; S, T ) is the transition probability density function for the risk-neutral random walk with S∗ being

262 Frequently Asked Questions In Quantitative Finance

today’s asset price and t∗ today’s date. Note that here

the arguments of V are the ‘variables’ strike, K, and

expiration, T .

If we differentiate this with respect to K we get

∂V

∞

= − e− r( T− t∗)

p( S∗, t∗; S, T ) dS.

∂K

K

After another differentiation, we arrive at this equation

for the probability density function in terms of the

option prices

∂ 2 V

p( S∗, t∗; K, T ) = er( T− t∗)

.

∂K 2

We also know that the forward equation for the tran-

sition probability density function, the Fokker–Planck

equation, is

∂p

∂ 2

∂

= 1

( σ 2 S 2 p) −

( rSp) .

∂T

2 ∂S 2

∂S

Here σ ( S, t) is evaluated at t = T . We also have

∂V

∞

∂

= −

p

rV + e− r( T− t∗)

( S − K)

dS.

∂T

K

∂T

This can be written as

∂V

∞

∂ 2

∂

= −

( σ 2 S 2 p)

( rSp)

rV + e− r( T− t∗)

1

−

∂T

2

K

∂S 2

∂S

× ( S − K) dS.

using the forward equation. Integrating this by parts

twice we get

∂V

∞

= − rV + 1 e− r( T− t∗) σ 2 K 2 p + re− r( T− t∗) Sp dS.

∂T

2

K

In this expression σ ( S, t) has S = K and t = T . After some simple manipulations we get

∂V

∂ 2

∂

=

V

V

1 σ 2 K 2

− rK

.

∂T

2

∂K 2

∂K

Chapter 4: Ten Different Ways to Derive Black–Scholes

263

This partial differential equation can now be solved for

the Black–Scholes formulæ.

This method is not used in practice for finding these

formulæ, but rather, knowing the traded prices of van-

illas as a function of K and T we can turn this equation

around to find σ , since the above analysis is still valid

even if volatility is stock and time dependent.

Continuous-time Limit of the

Binomial Model

Some of our ten derivations lead to the Black–Scholes

partial differential equation, and some to the idea of the

option value as the present value of the option payoff

under a risk-neutral random walk. The following simple

model does both.

In the binomial model the asset starts at S and over a

time step δt either rises to a value u × S or falls to a value v × S, with 0 < v < 1 < u. The probability of a rise is p and so the probability of a fall is 1 − p.

We choose the three constants u, v and p to give the

binomial walk the same drift, µ, and volatility, σ , as the

asset we are modelling. This choice is far from unique

and here we use the choices that result in the simplest

formulæ:

√

u = 1 + σ δt,

√

v = 1 − σ δt

and

√

µ δt

p = 1 +

.

2

2 σ

Having defined the behaviour of the asset we are ready

to price options.

264 Frequently Asked Questions In Quantitative Finance

uS

Probability of rise = p

S

δ t

vS

Figure 4-1: The model.

Suppose that we know the value of the option at the

time t + δt. For example this time may be the expiration

of the option. Now construct a portfolio at time t con-

sisting of one option and a short position in a quantity

of the underlying. At time t this portfolio has value

= V − S,

where the option value V is for the moment unknown.

At time t + δt the option takes one of two values, de-

pending on whether the asset rises or falls

V + or V − .

Chapter 4: Ten Different Ways to Derive Black–Scholes

265

At the same time the portfolio of option and stock

becomes either

V + − uS or V − − vS.

Having the freedom to choose , we can make the value

of this portfolio the same whether the asset rises or

falls. This is ensured if we make

V + − uS = V − − vS.

This means that we should choose

V + − V −

= ( u − v) S

for hedging. The portfolio value is then

u( V + − V −)

V + − uS = V + −

= V− − vS

( u − v)

v

=

( V + − V −)

V − −

.

( u − v)

Let’s denote this portfolio value by

+ δ.

This just means the original portfolio value plus the

change in value. But we must also have δ = r δt

to avoid arbitrage opportunities. Bringing all of these

expressions together to eliminate , and after some

rearranging, we get

V =

1

p V + + (1 − p ) V − ,

1 + r δt

where

√ δt

p = 1 + r

.

2

2 σ

This is an equation for V given V +, and V −, the option values at the next time step, and the parameters r

and σ .

The right-hand side of the equation for V can be inter-

preted, rather clearly, as the present value of the expected

266 Frequently Asked Questions In Quantitative Finance

future option value using the probabilities p for an up

move and 1 − p for a down.

Again this is the idea of the option value as the present

value of the expected payoff under a risk-neutral random

walk. The quantity p is the risk-neutral probability, and

it is this that determines the value of the option not

the real probability. By comparing the expressions for

p and p we see that this is equivalent to replacing the

real asset drift µ with the risk-free rate of return r.

We can examine the equation for V in the limit as δt → 0.

We write

V = V ( S, t), V + = V ( uS, t + δt) and V − = V ( vS, t + δt) .

Expanding these expressions in Taylor series for small

δt we find that

∂V

∼

as δt → 0,

∂S

and the binomial pricing equation for V becomes

∂V

∂ 2

∂

+

V

V

1 σ 2 S 2

+ rS

− rV = 0 .

∂t

2

∂S 2

∂S

This is the Black–Scholes equation.

CAPM

This derivation, originally due to Cox & Rubinstein

(1985) starts from the Capital Asset Pricing Model in

continuous time. In particular it uses the result that

there is a linear relationship between the expected

return on a financial instrument and the covariance

of the asset with the market. The latter term can be

thought of as compensation for taking risk. But the

asset and its option are perfectly correlated, so the

Chapter 4: Ten Different Ways to Derive Black–Scholes

267

compensation in excess of the risk-free rate for taking

unit amount of risk must be the same for each.

For the stock, the expected return (dividing by dt) is µ.

Its risk is σ .

From Itô we have

∂V

∂ 2 V

∂V

dV =

dt + 1 σ 2 S 2

dt +

dS.

∂t

2

∂S 2

∂S

Therefore the expected return on the option is

1

∂V

∂ 2

∂

+

V

V

1 σ 2 S 2

+ µS

V

∂t

2

∂S 2

∂S

and the risk is

1

∂V

σS

.

V

∂S

Since both the underlying and the option must have the

same compensation, in excess of the risk-free rate, for

unit risk

∂

µ −

1

V

r

+ 1 σ 2 S 2 ∂ 2 V + µS ∂V

= V ∂t

2

∂S 2

∂S .

σ

1 σ S ∂V

V

∂S

Now rearrange this. The µ drops out and we are left

with the Black–Scholes equation.

Utility Theory

The utility theory approach is probably one of the least

useful of the ten derivation methods, requiring that we

value from the perspective of an investor with a utility

function that is a power law. This idea was introduced

by Rubinstein (1976).

268 Frequently Asked Questions In Quantitative Finance

The steps along the way to finding the Black–Scholes

formulæ are as follows. We work within a single-period

framework, so that the concept of continuous hedging,

or indeed anything continuous at all, is not needed.

We assume that the stock price at the terminal time

(which will shortly also be an option’s expiration) and

the consumption are both lognormally distributed with

some correlation. We choose a utility function that is

a power of the consumption. A valuation expression

results. For the market to be in equilibrium requires

a relationship between the stock’s and consumption’s

expected growths and volatilities, the above-mentioned

correlation and the degree of risk aversion in the utility

function. Finally, we use the valuation expression for an

option, with the expiration being the terminal date. This

valuation expression can be interpreted as an expecta-

tion, with the usual and oft-repeated interpretation.

A Diffusion Equation

The penultimate derivation of the Black–Scholes partial

differential equation is rather unusual in that it uses just

pure thought about the nature of Brownian motion and

a couple of trivial observations. It also has a very neat

punchline that makes the derivation helpful in other

modelling situations.

It goes like this.

Stock prices can be modelled as Brownian motion, the

stock price plays the role of the position of the ‘pollen

particle’ and time is time. In mathematical terms Brow-

nian motion is just an example of a diffusion equation.

So let’s write down a diffusion equation for the value

of an option as a function of space and time, i.e. stock

price and time, that’s V ( S, t). What’s the general linear

Chapter 4: Ten Different Ways to Derive Black–Scholes

269

diffusion equation? It is

∂V

∂ 2 V

∂V

+ a

+ b

+ cV = 0 .

∂t

∂S 2

∂S

Note the coefficients a, b and c. At the moment these

could be anything.

Now for the two trivial observations.

First, cash in the bank must be a solution of this

equation. Financial contracts don’t come any simpler

than this. So plug V = ert into this diffusion equation

to get

rert + 0 + 0 + cert = 0 .

So c = − r.

Second, surely the stock price itself must also be a

solution? After all, you could think of it as being a call

option with zero strike. So plug V = S into the general

diffusion equation. We find

0 + 0 + b + cS = 0 .

So b = − cS = rS.

Putting b and c back into the general diffusion equation

we find

∂V

∂ 2 V

∂V

+ a

+ rS

− rV = 0 .

∂t

∂S 2

∂S

This is the risk-neutral Black–Scholes equation. Two

of the coefficients (those of V and ∂V /∂S) have been

pinned down exactly without any modelling at all. Ok,

so it doesn’t tell us what the coefficient of the second

derivative term is, but even that has a nice interpreta-

tion. It means at least a couple of interesting things.

First, if we do start to move outside the Black–Scholes

world then chances are it will be the diffusion coefficient

270 Frequently Asked Questions In Quantitative Finance

that we must change from its usual 1 σ 2 S 2 to accommo-

2

date new models.

Second, if we want to fudge our option prices, to mas-

sage them into line with traded prices for example, we

can only do so by fiddling with this diffusion coeffi-

cient, i.e. what we now know to be the volatility. This

derivation tells us that our only valid fudge factor is the

volatility.

Black–Scholes for Accountants

The final derivation of the Black–Scholes equation

requires very little complicated mathematics, and

doesn’t even need assumptions about Gaussian returns,

all we need is for the variance of returns to be finite.

The Black–Scholes analysis requires continuous hedging,

which is possible in theory but impossible, and even

undesirable, in practice. Hence one hedges in some

discrete way. Let’s assume that we hedge at equal time

periods, δt. And consider the value changes associated

with a delta-hedged option.

• We start with zero cash

• We buy an option

• We sell some stock short

• Any cash left (positive or negative) is put into a

risk-free account.

We start by borrowing some money to buy the option.

This option has a delta, and so we sell delta of the

underlying stock in order to hedge. This brings in some

money. The cash from these transactions is put in the

bank. At this point in time our net worth is zero.

Chapter 4: Ten Different Ways to Derive Black–Scholes

271

S

Long option

Short stock

Hedged portfolio

Figure 4-2: How our portfolio depends on S.

Our portfolio has a dependence on S as shown in

Figure 4-2.

We are only concerned with small movements in the

stock over a small time period, so zoom in on the cur-

rent stock position. Locally the curve is approximately

a parabola, see Figure 4-3.

Now think about how our net worth will change from

now to a time δt later. There are three reasons for our

total wealth to change over that period.

1. The option price curve changes

2. There is an interest payment on the money in the bank

3. The stock moves

The option curve falls by the time value, the theta mul-

tiplied by the time step:

× δt.

272 Frequently Asked Questions In Quantitative Finance

Today

Figure 4-3: The curve is approximately quadratic.

To calculate how much interest we received we need to

know how much money we put in the bank. This was

× S

from the stock sale and

− V

from the option purchase. Therefore the interest we

receive is

r( S − V ) δt.

Finally, look at the money made from the stock move.

Since gamma is positive, any stock price move is good

for us. The larger the move the better.

The curve in Figure 4-3 is locally quadratic, a parabola

with coefficient 1 . The stock move over a time period

2

δt is proportional to three things:

• the volatility σ

• the stock price S

• the square root of the time step

Multiply these three together, square the result because

the curve is parabolic and multiply that by 1 and you

2

Chapter 4: Ten Different Ways to Derive Black–Scholes

273

get the profit made from the stock move as

1 σ 2 S 2 δt.

2

Put these three value changes together (ignoring the δt

term which multiplies all of them) and set the resulting

expression equal to zero, to represent no arbitrage, and

you get

+ 1 σ 2 S 2 + r( S − V) = 0, 2

the Black–Scholes equation.

Now there was a bit of cheating here, since the stock

price move is really random. What we should have said

is that

1 σ 2 S 2 δt

2

is the profit made from the stock move on average.

Crucially all we need to know is that the variance of

returns is

σ 2 S 2 δt,

we don’t even need the stock returns to be normally

distributed. There is a difference between the square

of the stock price moves and its average value and this

gives rise to hedging error, something that is always

seen in practice. If you hedge discretely, as you must,

then Black–Scholes only works on average. But as you

hedge more and more frequently, going to the limit

δt = 0, then the total hedging error tends to zero, so

justifying the Black–Scholes model.

References and Further Reading

Andreason, J, Jensen, B & Poulsen, R 1998 Eight Valuation

Methods in Financial Mathematics: The Black–Scholes For-

mula as an Example. Math. Scientist 23 18–40

274 Frequently Asked Questions In Quantitative Finance

Black, F & Scholes, M 1973 The pricing of options and

corporate liabilities. Journal of Political Economy 81 637–59

Cox, J & Rubinstein, M 1985 Options Markets. Prentice–Hall

Derman, E & Kani, I 1994 Riding on a smile. Risk magazine 7

(2) 32–39

Dupire, B 1994 Pricing with a smile. Risk magazine 7 (1) 18–20

Harrison, JM & Kreps, D 1979 Martingales and arbitrage in

multiperiod securities markets. Journal of Economic Theory

20 381–408

Harrison, JM & Pliska, SR 1981 Martingales and stochastic

integrals in the theory of continuous trading. Stochastic

Processes and their Applications 11 215–260

Joshi, M 2003 The Concepts and Practice of Mathematical

Finance. CUP

Rubinstein, M 1976 The valuation of uncertain income streams

and the pricing of options. Bell J. Econ. 7 407–425

Rubinstein, M 1994 Implied binomial trees. Journal of Finance

69 771–818

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 5

Models and Equations

276 Frequently Asked Questions In Quantitative Finance

Equity, Foreign Exchange and

Commodities

The lognormal random walk

The most common and simplest model is the lognormal

random walk:

dS + µS dt + σ S dX.

The Black–Scholes hedging argument leads to the fol-

lowing equation for the value of non-path-dependent

contracts,

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ ( r − D) S

− rV = 0 .

∂t

2

∂S 2

∂S

The parameters are volatility σ , dividend yield D and

risk-free interest rate r. All of these can be functions of

S and/or t, although it wouldn’t make much sense for

the risk-free rate to be S dependent.

This equation can be interpreted probabilistically. The

option value is

T

e− t r( τ) dτ E Q

t [Payoff( ST )] ,

where ST is the stock price at expiry, time T , and

the expectation is with respect to the risk-neutral ran-

dom walk

dS = r( t) S dt + σ ( S, t) S dX.

When σ , D and r are only time dependent we can write

down an explicit formula for the value of any non-path-

dependent option without early exercise (and without

any decision feature) as

Chapter 5: Models and Equations

277

e− r( T− t)

∞

σ 2 π( T − t) 0

2

− ln( S/S )+ r− D− 1 σ 2 ( T− t) / 2 σ 2( T− t) dS

e

2

Payoff( S ) S ,

where

T

σ =

1

σ( τ)2 dτ,

T − t t

T

D =

1

D( τ ) dτ

T − t t

and

T

r =

1

r( τ ) dτ.

T − t t

The . parameters represent the ‘average’ of the para-

meters from the current time to expiration. For the

volatility parameter the relevant average is the root-

mean-square average, since variances can be summed

but standard deviations (volatilities) cannot.

The above is a very general formula which can be

greatly simplified for European calls, puts and

binaries.

Multi-dimensional lognormal random

walks

There is a formula for the value of a European non-path-

dependent option with payoff of Payoff( S 1, . . . , Sd) at time T :

278 Frequently Asked Questions In Quantitative Finance

V = e− r( T− t) ( 2 π( T − t) )− d/ 2 (Det )−1 / 2( σ 1 · · · σd)−1

∞

∞ Payoff( S ··· S )

· · ·

1

d

0

0

S · · · S

1

d

× exp − 1 αT −1 α dS · · · dS

2

1

d

where

S

σ 2

α

i

i

i =

1

ln

+ r − D

( T − t) ,

σ

i −

i( T − t)1 / 2

S

2

i

is the correlation matrix and there is a continuous

dividend yield of Di on each asset.

Stochastic volatility

If the risk-neutral volatility is modelled by

dσ = ( p − λq) dt + q dX 2,

where λ is the market price of volatility risk, with the

stock model still being

dS = µS dt + σ S dX 1,

with correlation between them of ρ, then the option-

pricing equation is

∂V

∂ 2

∂ 2

∂ 2

∂

+

V

V

V

V

1 σ 2 S 2

+ ρσSq

+ 1 q 2

+ rS

∂t

2

∂S 2

∂S ∂σ

2

∂σ 2

∂S

∂V

+ ( p − λq)

− rV = 0 .

∂σ

This pricing equation can be interpreted as representing

the present value of the expected payoff under risk-neutral

random walks for both S and σ . So for a call option, for

example, we can price via the expected payoff

V ( S, σ , t) = e− r( T− t) E Q

t [max( ST − K , 0)] .

Chapter 5: Models and Equations

279

For other contracts replace the maximum function with

the relevant, even path-dependent, payoff function.

Hull & White (1987)

Hull & White considered both

general and specific volatility models. They showed that

when the stock and the volatility are uncorrelated and

the risk-neutral dynamics of the volatility are unaffected

by the stock (i.e. p − λq and q are independent of S) then the fair value of an option is the average of the

Black–Scholes values for the option, with the average

taken over the distribution of σ 2.

Square-root model/Heston (1993)

In Heston’s model

√

dv = ( a − bv) dt + c v dX 2,

where v = σ 2. This has arbitrary correlation between

the underlying and its volatility. This is popular because

there are closed-form solutions for European options.

3/2 model

dv = ( av − bv 2) dt + cv 3 / 2 dX 2, where v = σ 2. Again, this has closed-form solutions.

GARCH-diffusion

In stochastic differential equation form

the GARCH(1,1) model is

dv = ( a − bv) dt + cv dX 2 .

Here v = σ 2.

Ornstein–Uhlenbeck process

With y = log v, v = σ 2,

dy = ( a − by) dt + c dX 2 .

This model matches real, as opposed to risk-neutral,

data well.

280 Frequently Asked Questions In Quantitative Finance

Asymptotic analysis

If the volatility of volatility is large

and the speed of mean reversion is fast in a stochastic

volatility model,

p − λq

q

dS = rS dt + σ S dX 1 and dσ =

dt + √ dX

2

with a correlation ρ, then closed-form approximate

solutions (asymptotic solutions) of the pricing equation

can be found for simple options for arbitrary functions

p − λq and q. In the above model the represents a

small parameter. The asymptotic solution is then a

power series in 1 / 2.

Schönbucher’s stochastic implied volatility

Schönbucher begins

with a stochastic model for implied volatility and

then finds the actual volatility consistent, in a no-

arbitrage sense, with these implied volatilities. This

model calibrates to market prices by definition.

Jump diffusion

Given the jump-diffusion model

dS = µS dt + σ S dX + ( J − 1) S dq,

the equation for an option is

∂V

∂ 2 V

∂V

+ 1 σ 2 S 2

+ rS

− rV

∂t

2

∂S 2

∂S

∂V

+ λE [ V( JS, t) − V( S, t)] − λ

SE [ J − 1] = 0 .

∂S

E[·] is the expectation taken over the jump size.

If the logarithm of J is Normally distributed with

standard deviation σ then the price of a European

non-path-dependent option can be written as

∞

1 e− λ ( T− t)( λ ( T − t)) nVBS( S, t; σn, rn), n!

n=0

Chapter 5: Models and Equations

281

where

nσ 2

k = E[ J − 1], λ = λ(1 + k), σ 2 = σ 2 +

n

T − t

and

n ln(1 + k)

rn = r − λk +

,

T − t

and VBS is the Black–Scholes formula for the option

value in the absence of jumps.

Fixed Income

In the following we use the continuously compounded

interest convention. So that one dollar put in the bank

at a constant rate of interest r would grow exponen-

tially, ert. This is the convention used outside the fixed-

income world. In the fixed-income world where interest

is paid discretely, the convention is that money grows

according to

n

1 + r τ

,

where n is the number of interest payments, τ is the

time interval between payments (here assumed con-

stant) and r is the annualized interest rate.

To convert from discrete to continuous use

r = 1 ln(1 + r τ ) .

τ

The yield to maturity (YTM) or internal rate of return (IRR)

Suppose that we have a zero-coupon bond maturing

at time T when it pays one dollar. At time t it has a

value Z( t; T ). Applying a constant rate of return of y between t and T , then one dollar received at time T has a present value of Z( t; T ) at time t, where

Z( t; T ) = e− y( T− t) .

282 Frequently Asked Questions In Quantitative Finance

It follows that

y = − ln Z .

T − t

Suppose that we have a coupon-bearing bond. Discount

all coupons and the principal to the present by using

some interest rate y. The present value of the bond, at

time t, is then

N

V = Pe− y( T− t) +

Cie− y( ti− t),

i=1

where P is the principal, N the number of coupons,

Ci the coupon paid on date ti. If the bond is a traded

security then we know the price at which the bond can

be bought. If this is the case then we can calculate the

yield to maturity or internal rate of return as the value

y that we must put into the above to make V equal to

the traded price of the bond. This calculation must be

performed by some trial and error/iterative procedure.

The plot of yield to maturity against time to maturity is

called the yield curve.

Duration

Since we are often interested in the sensitivity

of instruments to the movement of certain underlying

factors it is natural to ask how does the price of a bond

vary with the yield, or vice versa. To a first approxima-

tion this variation can be quantified by a measure called

the duration.

By differentiating the value function with respect to y

we find that

N

dV

= −( T − t) Pe− y( T− t) −

Ci( ti − t) e− y( ti− t) .

dy

i=1

Chapter 5: Models and Equations

283

This is the slope of the price/yield curve. The quantity

− 1 dV

V dy

is called the Macaulay duration. (The modified dura-

tion is similar but uses the discretely compounded

rate.) The Macaulay duration is a measure of the aver-

age life of the bond.

For small movements in the yield, the duration gives a

good measure of the change in value with a change in the

yield. For larger movements we need to look at higher

order terms in the Taylor series expansion of V ( y).

Convexity

The Taylor series expansion of V gives

dV

dV

d 2 V

= 1

δy + 1

( δy)2 + · · · ,

V

V dy

2 V dy 2

where δy is a change in yield. For very small movements

in the yield, the change in the price of a bond can be

measured by the duration. For larger movements we

must take account of the curvature in the price/yield

relationship.

The dollar convexity is defined as

N

d 2 V

= ( T − t)2 Pe− y( T− t) +

Ci( ti − t)2 e− y( ti− t) .

dy 2

i=1

and the convexity is

1 d 2 V .

V dy 2

Yields are associated with individual bonds. Ideally we

would like a consistent interest rate theory that can be

used for all financial instruments simultaneously. The

284 Frequently Asked Questions In Quantitative Finance

simplest of these assumes a deterministic evolution of a

spot rate.

The spot rate and forward rates

The interest rate we consider

will be what is known as a short-term interest rate or

spot interest rate r( t). This means that the rate r( t) is to apply at time t. Interest is compounded at this rate at

each moment in time but this rate may change, generally

we assume it to be time dependent.

Forward rates are interest rates that are assumed to

apply over given periods in the future for all instruments.

This contrasts with yields which are assumed to apply

from the present up to maturity, with a different yield

for each bond.

Let us suppose that we are in a perfect world in which

we have a continuous distribution of zero-coupon bonds

with all maturities T . Call the prices of these at time t,

Z( t; T ). Note the use of Z for zero-coupon.

The implied forward rate is the curve of a time-dependent

spot interest rate that is consistent with the market price

of instruments. If this rate is r( τ ) at time τ then it satisfies

T

Z( t; T ) = e− t r( τ) dτ .

On rearranging and differentiating this gives

∂

r( T ) = −

(ln Z( t; T )) .

∂T

This is the forward rate for time T as it stands today,

time t. Tomorrow the whole curve (the dependence of r

on the future) may change. For that reason we usually

denote the forward rate at time t applying at time T in

the future as F ( t; T ) where

∂

F ( t; T ) = −

(ln Z( t; T )) .

∂T

Chapter 5: Models and Equations

285

Writing this in terms of yields y( t; T ) we have

Z( t; T ) = e− y( t; T)( T− t) and so

∂y

F ( t; T ) = y( t; T ) +

.

∂T

This is the relationship between yields and forward

rates when everything is differentiable with respect to

maturity.

In the less-than-perfect real world we must do with only

a discrete set of data points. We continue to assume

that we have zero-coupon bonds but now we will only

have a discrete set of them. We can still find an implied

forward rate curve as follows. (In this I have made the

simplifying assumption that rates are piecewise con-

stant. In practice one uses other functional forms to

achieve smoothness.)

Rank the bonds according to maturity, with the shortest

maturity first. The market prices of the bonds will be

denoted by ZM where i is the position of the bond in

i

the ranking.

Using only the first bond, ask the question ‘What inter-

est rate is implied by the market price of the bond?’

The answer is given by y 1, the solution of

ZM = e− r 1( T 1− t),

1

i.e.

ln( ZM )

r

1

1 = −

.

T 1 − t

This rate will be the rate that we use for discounting

between the present and the maturity date T 1 of the

first bond. And it will be applied to all instruments

whenever we want to discount over this period.

286 Frequently Asked Questions In Quantitative Finance

Now move on to the second bond having maturity date

T 2. We know the rate to apply between now and time

T 1, but at what interest rate must we discount between

dates T 1 and T 2 to match the theoretical and market

prices of the second bond? The answer is r 2 which

solves the equation

ZM = e− r 1( T 1− t) e− r 2( T 2− T 1), 2

i.e.

ln ZM /ZM

r

2

1

2 = −

.

T 2 − T 1

By this method of bootstrapping we can build up the

forward rate curve. Note how the forward rates are

applied between two dates, for which period I have

assumed they are constant.

This method can easily be extended to accommodate

coupon-bearing bonds. Again rank the bonds by their

maturities, but now we have the added complexity that

we may only have one market value to represent the

sum of several cashflows. Thus one often has to make

some assumptions to get the right number of equations

for the number of unknowns.

To price non-linear instruments, options, we need a

model that captures the randomness in rates.

Black 1976

Market practice with fixed-income derivatives is often

to treat them as if there is an underlying asset that

is lognormal. This is the methodology proposed by

Black (1976).

Bond options

A simple example of Black ’76 would be a

European option on a bond, as long as the maturity of

Chapter 5: Models and Equations

287

the bond is significantly greater than the expiration of

the option. The relevant formulæ are, for a call option

e− r( T− t) (FN( d 1) − KN( d 2) ) , and for a put

e− r( T− t) (− FN(− d 1) + KN( d−2) ) , where

ln( F /K) + 1 σ 2( Ti − t)

d

2

1 =

√

,

σ Ti − t

ln( F /K) − 1 σ 2( Ti − t)

d

2

2 =

√

.

σ Ti − t

Here F is the forward price of the underlying bond at

the option maturity date T . The volatility of this forward

price is σ . The interest rate r is the rate applicable to

the option’s expiration and K is the strike.

Caps and floors

A cap is made up of a string of caplets

with a regular time interval between them. The payoff

for the i th caplet is max( ri − K, 0) at time Ti+1 where ri is the interest rate applicable from ti to ti+1 and K is the strike.

Each caplet is valued under Black ’76 as

e− r( Ti+1− t) (FN( d 1) − KN( d 2) ) , where r is the continuously compounded interest rate

applicable from t to Ti+1, F is the forward rate from time Ti to time Ti+1, K the strike and

ln( F /K) + 1 σ 2( Ti − t)

d

2

1 =

√

,

σ Ti − t

ln( F /K) − 1 σ 2( Ti − t)

d

2

2 =

√

,

σ Ti − t

where σ is the volatility of the forward rate.

288 Frequently Asked Questions In Quantitative Finance

The floorlet can be thought of in a similar way in terms

of a put on the forward rate and so its formula is

e− r( Ti+1− t) (KN(− d 2) − FN(− d 1) ) .

Swaptions

A payer swaption, which is the right to pay

fixed and receive floating, can be modelled as a call on

the forward rate of the underlying swap. Its formula

is then

1 −

1

τm

1+ F

m

e− r( T− t) (FN( d 1) − KN( d 2) ) , F

where r is the continuously compounded interest rate

applicable from t to T , the expiration, F is the forward swap rate, K the strike and

ln( F /K) + 1 σ 2( T − t)

d

2

1 =

√

,

σ T − t

ln( F /K) − 1 σ 2( T − t)

d

2

2 =

√

,

σ T − t

where σ is the volatility of the forward swap rate. τ is

the tenor of the swap and m the number of payments

per year in the swap.

The receiver swaption is then

1 −

1

τm

1+ F

m

e− r( T− t) (KN(− d 2) − FN(− d 1) ) .

F

Spot rate models

The above method for pricing derivatives is not entirely

internally consistent. For that reason there have been

developed other interest rate models that are internally

consistent.

Chapter 5: Models and Equations

289

In all of the spot rate models below we have

dr = u( r, t) dt + w( r, t) dX

as the real process for the spot interest rate. The risk-

neutral process which governs the value of fixed-income

instruments is

dr = ( u − λw) dt + w dX

where λ is the market price of interest rate risk. In each

case the stochastic differential equation we describe is

for the risk-neutral spot rate process, not the real.

The differential equation governing the value of non-

path-dependent contracts is

∂V

∂ 2

∂

+

V

V

1 w 2

+ ( u − λw)

− rV = 0 .

∂t

2

∂r 2

∂r

The value of fixed-income derivatives can also be inter-

preted as

E Q

t

Present value of cashflows ,

where the expectation is with respect to the risk-neutral

process

Vasicek

In this model the risk-neutral process is

dr = ( a − br) dt + c dX,

with a, b and c being constant. It is possible for r to go negative in this model.

There is a solution for bonds of the form exp( A( t; T ) −

B( t; T ) r).

Cox, Ingersoll and Ross

In this model the risk-neutral pro-

cess is

dr = ( a − br) dt + cr 1 / 2 dX, 290 Frequently Asked Questions In Quantitative Finance

with a, b and c being constant. As long as a is sufficiently large this process cannot go negative.

There is a solution for bonds of the form exp( A( t; T ) −

B( t; T ) r).

Ho and Lee

In this model the risk-neutral process is

dr = a( t) dt + c dX,

with c being constant. It is possible for r to go negative

in this model.

There is a solution for bonds of the form exp( A( t; T ) −

B( t; T ) r).

The time-dependent parameter a( t) is chosen so that

the theoretical yield curve matches the market yield

curve initially. This is calibration.

Hull and White

There are Hull and White versions of the

above models. They take the forms

dr = (a( t) − b( t) r) dt + c( t) dX, or

dr = (a( t) − b( t) r) dt + c( t) r 1 / 2 dX.

The functions of time allow various market data to be

matched or calibrated.

There are solutions for bonds of the form exp( A( t; T ) −

B( t; T ) r).

Black–Karasinski

In this model the risk-neutral spot-rate

process is

d(ln r) = (a( t) − b( t) ln r) dt + c( t) dX.

There are no closed-form solutions for simple bonds.

Chapter 5: Models and Equations

291

Two-factor models

In the two-factor models there are two sources of ran-

domness, allowing a much richer structure of theoretical

yield curves than can be achieved by single-factor mod-

els. Often, but not always, one of the factors is still the

spot rate.

Brennan and Schwartz

In the Brennan & Schwartz model

the risk-neutral spot rate process is

dr = ( a 1 + b 1( l − r)) dt + σ 1 r dX 1

and the long rate satisfies

dl = l( a 2 − b 2 r + c 2 l ) dt + σ 2 l dX 2 .

Fong and Vasicek

Fong & Vasicek consider the following

model for risk-neutral variables

dr = a( r − r) dt + ξ dX 1

and

dξ = b( ξ − ξ ) dt + c ξ dX 2 .

Thus they model the spot rate, and ξ the square root of

the volatility of the spot rate.

Longstaff and Schwartz

Longstaff & Schwartz consider the

following model for risk-neutral variables

√

dx = a( x − x) dt +

x dX 1

and

√

dy = b( y − y) dt +

y dX 2,

where the spot interest rate is given by

r = cx + dy.

292 Frequently Asked Questions In Quantitative Finance

Hull and White

The risk-neutral model

dr = ( η( t) − u − γ r) dt + c dX 1

and

du = − au dt + b dX 2

is a two-factor version of the one-factor Hull & White.

The function η( t) is used for fitting the initial yield

curve.

All of the above, except for the Brennan & Schwartz

model, have closed-form solutions for simple bonds in

terms of the exponential of a linear function of the two

variables.

The market price of risk as a random factor

Suppose that we

have the two real random walks

dr = u dt + w dX 1

and

dλ = p dt + q dX 2,

where λ is the market price of r risk. The zero-coupon

bond pricing equation is then

∂Z

∂ 2 Z

∂ 2 Z

∂ 2 Z

+ 1 w 2

+ ρwq

+ 1 q 2

∂t

2

∂r 2

∂r∂λ

2

∂λ 2

∂

∂

+

Z

Z

( u − λw)

+ ( p − λ

− rZ = 0 .

∂

λq)

r

∂λ

Since the market price of risk is related to the slope

of the yield curve as the short end, there is only one

unobservable in this equation, λλ.

SABR

The SABR (stochastic, α, β, ρ) model by Hagan, Kumar,

Lesniewski & Woodward (2002) is a model for a forward

Chapter 5: Models and Equations

293

rate, F , and its volatility, α, both of which are stochastic:

dF = αF β dX 1 and dα = να dX 2 .

There are three parameters, β, ν and a correlation ρ.

The model comes into its own because it is designed

for the special case where the volatility α and volatility

of volatility, ν, are both small. In this case there are

relatively simple closed-form approximations (asymp-

totic solutions). The model is therefore most relevant

for markets such as fixed income, rather than equity.

Equity markets typically have large volatility making the

model unsuitable.

The models calibrates well to simple fixed-income instru-

ments of specified maturity, and if the parameters are

allowed to be time dependent then a term structure can

also be fitted.

Heath, Jarrow and Morton

In the Heath, Jarrow & Morton (HJM) model the evo-

lution of the entire forward curve is modelled. The

risk-neutral forward curve evolves according to

dF ( t; T ) = m( t, T ) dt + ν( t, T ) dX.

Zero-coupon bonds then have value given by

T

Z( t; T ) = e− t F( t; s) ds, the principal at maturity is here scaled to $1. A hedging

argument shows that the drift of the risk-neutral process

for F cannot be specified independently of its volatility

and so

T

m( t, T ) = ν( t, T )

ν( t, s) ds.

t

This is equivalent to saying that the bonds, which are

traded, grow at the risk-free spot rate on average.

294 Frequently Asked Questions In Quantitative Finance

A multi-factor version of this results in the following

risk-neutral process for the forward rate curve

N

T

N

dF ( t, T ) =

νi( t, T)

νi( t, s) ds dt +

νi( t, T) dXi.

t

i=1

i=1

In this the dXi are uncorrelated with each other.

Brace, Gatarek and Musiela

The Brace, Gatarek & Musiela (BGM) model is a discrete

version of HJM where only traded bonds are modelled

rather than the unrealistic entire continuous yield curve.

If Zi( t) = Z( t; Ti) is the value of a zero-coupon bond, maturing at Ti, at time t, then the forward rate applicable between Ti and Ti+1 is given by

Zi

Fi = 1

− 1 ,

τ Zi+1

where τ = Ti+1 − Ti. Assuming equal time period between

all maturities we have the risk-neutral process for the

forward rates are given by





i

σjFjτρij

dF





i =

σiFi dt + σiFi dXi.

1 + τ Fj

j=1

Modelling interest rates is then a question of the func-

tional forms for the volatilities of the forward rates σi

and the correlations between them ρij.

Prices as expectations

For all of the above models the value of fixed-income

derivatives can be interpreted as

E Q

t

Present value of cashflows ,

Chapter 5: Models and Equations

295

where the expectation is with respect to the risk-neutral

process(es). The ‘present value’ here is calculated path-

wise. If performing a simulation for valuation purposes

you must discount cashflows for each path using the

relevant discount factor for that path.

Credit

Credit risk models come in two main varieties, the struc-

tural and the reduced form.

Structural models

Structural models try to model the behaviour of the firm

so as to represent the default or bankruptcy of a com-

pany in as realistic a way as possible. The classical work

in this area was by Robert Merton who showed how to

think of a company’s value as being a call option on its

assets. The strike of the option being the outstanding

debt.

Merton assumes that the assets of the company A follow

a random walk

dA = µA dt + σ A dX.

If V is the current value of the outstanding debt, allow-

ing for risk of default, then the value of the equity

equals assets less liabilities:

S = A − V .

Here S is the value of the equity. At maturity of this

debt

S( A, T ) = max( A − D, 0) and V ( A, T ) = min( D, A),

296 Frequently Asked Questions In Quantitative Finance

where D is the amount of the debt, to be paid back at

time T .

If we can hedge the debt with a dynamically changing

quantity of equity then the Black–Scholes hedging argu-

ment applies and we find that the current value of the

debt, V , satisfies

∂V

∂ 2 V

∂V

+ 1 σ 2 A 2

+ rA

− rA = 0

∂t

2

∂A 2

∂A

subject to

V ( A, T ) = min( D, A)

and exactly the same partial differential equation for the

equity of the firm S but with

S( A, T ) = max( A − D, 0) .

The problem for S is exactly that for a call option, but

now we have S instead of the option value, the under-

lying variable is the asset value A and the strike is

D, the debt. The formula for the equity value is the

Black–Scholes value for a call option.

Reduced form

The more popular approach to the modelling of credit

risk is to use an instantaneous risk of default or hazard

rate, p. This means that if at time t the company has not

defaulted then the probability of default between times t

and t + dt is p dt. This is just the same Poisson process seen in jump-diffusion models. If p is constant then this

results in the probability of a company still being in

existence at time T , assuming that it wasn’t bankrupt at

time t, being simply

e− p( T− t) .

Chapter 5: Models and Equations

297

If the yield on a risk-free, i.e. government bond, with

maturity T is r then its value is

e− r( T− t) .

If we say that an equivalent bond on the risky company

will pay off 1 if the company is not bankrupt and zero

otherwise then the present value of the expected payoff

comes from multiplying the value of a risk-free bond by

the probability that the company is not in default to get

e− r( T− t) × e− p( T− t) = e−( r+ p)( T− t) .

So to represent the value of a risky bond just add a

credit spread of p to the yield on the equivalent risk-free

bond. Or, conversely, knowing the yields on equivalent

risk-free and risky bonds one can estimate p the implied

risk of default.

This is a popular way of modelling credit risk because

it is so simple and the mathematics is identical to that

for interest rate models.

References and Further Reading

Black F 1976 The pricing of commodity contracts. Journal of

Financial Economics 3 167–79

Black, F & Scholes, M 1973 The pricing of options and corpo-

rate liabilities. Journal of Political Economy 81 637–59

Brace, A, Gatarek, D & Musiela, M 1997 The market model of

interest rate dynamics. Mathematical Finance 7 127–154

Cox, J, Ingersoll, J & Ross, S 1985 A theory of the term struc-

ture of interest rates. Econometrica 53 385–467

Hagan, P, Kumar, D, Lesniewski, A & Woodward, D 2002 Man-

aging smile risk. Wilmott magazine, September

Haug, EG 1997 The Complete Guide to Option Pricing Formulas.

McGraw–Hill

298 Frequently Asked Questions In Quantitative Finance

Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the

term structure of interest rates: a new methodology. Econo-

metrica 60 77–105

Heston, S 1993 A closed-form solution for options with stochas-

tic volatility with application to bond and currency options.

Review of Financial Studies 6 327–343

Ho, T & Lee, S 1986 Term structure movements and pric-

ing interest rate contingent claims. Journal of Finance 42

1129–1142

Hull, JC & White, A 1987 The pricing of options on assets with

stochastic volatilities. Journal of Finance 42 281–300

Hull, JC & White, A 1990 Pricing interest rate derivative secu-

rities. Review of Financial Studies 3 573–592

Lewis, A 2000 Option valuation under Stochastic Volatility.

Finance Press

Merton, RC 1973 Theory of rational option pricing. Bell Journal

of Economics and Management Science 4 141–83

Merton, RC 1974 On the pricing of corporate debt: the risk

structure of interest rates. Journal of Finance 29 449–70

Merton, RC 1976 Option pricing when underlying stock returns

are discontinuous. Journal of Financial Economics 3 125–44

Rasmussen, H & Wilmott, P 2002 Asymptotic analysis of

stochastic volatility models. In New Directions in Mathe-

matical Finance, Ed. Wilmott, P & Rasmussen, H, John Wiley

& Sons

Schönbucher, PJ 1999 A market model for stochastic implied

volatility. Phil. Trans. A 357 2071–2092

Schönbucher, PJ 2003 Credit Derivatives Pricing Models. John

Wiley & Sons

Vasicek, OA 1977 An equilibrium characterization of the term

structure. Journal of Financial Economics 5 177–188

Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second

edition. John Wiley & Sons

Chapter 6

The Black–Scholes

Formulæ and the

Greeks

300 Frequently Asked Questions In Quantitative Finance

I nthefollowingformulæ

x

N( x) =

1

√

e− 1 φ 2

2

dφ,

2 π −∞

ln( S/K) + ( r − D + 1 σ 2)( T − t) d

2

1 =

√

σ T − t

and

ln( S/K) + ( r − D − 1 σ 2)( T − t) d

2

2 =

√

.

σ T − t

The formulæ are also valid for time-dependent σ , D and

r, just use the relevant ‘average’ as explained in the

previous chapter.

Warning

The greeks which are ‘greyed out’ in the following can

sometimes be misleading. They are those greeks which

are partial derivatives with respect to a parameter ( σ , r

or D) as opposed to a variable ( S and t) and which are not single signed (i.e. always greater than zero or always

less than zero). Differentiating with respect a parameter,

which has been assumed to be constant so that we can

find a closed-form solution, is internally inconsistent.

For example, ∂V /∂σ is the sensitivity of the option price

to volatility, but if volatility is constant, as assumed in

the formula, why measure sensitivity to it? This may

not matter if the partial derivative with respect to the

parameter is of one sign, such as ∂V /∂σ for calls and

puts. But if the partial derivative changes sign then

there may be trouble. For example, the binary call has

a positive vega for low stock prices and negative vega

for high stock prices, in the middle vega is small, and

even zero at a point. However, this does not mean that

the binary call is insensitive to volatility in the middle.

It is precisely in the middle that the binary call value is

very sensitive to volatility, but not the level, rather the

volatility skew.

Chapter 6: Black–Scholes and Greeks

301

Table 6.1: Formulæ for European call.

Call

Payoff

max( S − K, 0)

Value V

Se− D( T − t) N ( d 1) − Ke− r( T− t) N( d 2) Black–Scholes value

Delta ∂V

∂

e− D( T − t) N ( d

S

1)

Sensitivity to underlying

e− D( T − t) N ( d

Gamma ∂ 2 V

1)

√

∂S 2

σS T− t

Sensitivity of delta to

underlying

Theta ∂V

∂

− σSe− D( T− t) N ( d 1)

√

+ DSN( d

t

2 T − t

1) e− D( T − t)

Sensitivity to time

− rKe− r( T− t) N( d 2)

√

Speed ∂ 3 V

− e− D( T− t) N ( d 1) × d

T − t

∂

1 + σ

S 3

σ 2 S 2( T− t)

Sensitivity of gamma to

underlying

Charm ∂ 2 V

∂

De− D( T − t) N ( d

S ∂t

1) + e− D( T − t) N ( d 1)

d

Sensitivity of delta to time

×

2

− r− D

√

2( T − t)

σ T− t

e− D( T − t) N ( d

Colour

∂ 3 V

1)

√

∂S 2 ∂t

σS T− t

Sensitivity of gamma to time

× D + 1− d 1 d 2 − d 1( r− D)

√

2( T − t)

σ T− t

√

Vega ∂V

∂σ

S T − te− D( T − t) N ( d 1)

Sensitivity to volatility

Rho ( r) ∂V

∂

K( T − t) e− r( T − t) N( d r

2)

Sensitivity to interest rate

Rho ( D) ∂V

∂

−( T − t) Se− D( T− t) N( d

D

1)

Sensitivity to dividend yield

d

Vanna

∂ 2 V

2

∂S ∂σ

− e− D( T− t) N ( d 1) σ

Sensitivity of delta to

volatility

√

d

Volga/Vomma ∂ 2 V

S T − te− D( T − t) N ( d

1 d 2

∂σ 2

1)

σ

Sensitivity of vega to volatility

302 Frequently Asked Questions In Quantitative Finance

Table 6.2: Formulæ for European put.

Put

Payoff

max( K − S, 0)

Value V

− Se− D( T− t) N(− d 1) + Ke− r( T− t) N(− d 2) Black–Scholes value

Delta ∂V

∂

e− D( T − t)( N ( d

S

1) − 1)

Sensitivity to underlying

e− D( T − t) N ( d

Gamma ∂ 2 V

1)

√

∂S 2

σS T− t

Sensitivity of delta to

underlying

Theta ∂V

∂

− σSe− D( T− t) N (− d 1)

√

− DSN(− d

t

2 T − t

1) e− D( T − t)

Sensitivity to time

+ rKe− r( T− t) N(− d 2)

√

Speed ∂ 3 V

− e− D( T− t) N ( d 1) × d

T − t

∂

1 + σ

S 3

σ 2 S 2( T− t)

Sensitivity of gamma to

underlying

Charm ∂ 2 V

∂

De− D( T − t)( N ( d

S ∂t

1) − 1) + e− D( T − t) N ( d 1)

d

Sensitivity of delta to time ×

2

− r− D

√

2( T − t)

σ T− t

e− D( T − t) N ( d

Colour

∂ 3 V

1)

√

∂S 2 ∂t

σS T− t

Sensitivity of gamma to

× D + 1− d 1 d 2 − d 1( r− D)

√

2( T − t)

σ T− t

time

√

Vega ∂V

∂σ

S T − te− D( T − t) N ( d 1)

Sensitivity to volatility

Rho ( r) ∂V

∂

− K( T − t) e− r( T− t) N(− d r

2)

Sensitivity to interest rate

Rho ( D) ∂V

∂

( T − t) Se− D( T − t) N(− d D

1)

Sensitivity to dividend yield

d

Vanna

∂ 2 V

2

∂S ∂σ

− e− D( T− t) N ( d 1) σ

Sensitivity of delta to

volatility

√

d

Volga/Vomma ∂ 2 V

S T − te− D( T − t) N ( d

1 d 2

∂σ 2

1)

σ

Sensitivity of vega to

volatility

Chapter 6: Black–Scholes and Greeks

303

Table 6.3: Formulæ for European binary call.

Binary Call

Payoff

1 if S > K otherwise 0

Value V

e− r( T − t) N ( d 2)

Black–Scholes value

e− r( T − t) N ( d

Delta ∂V

2)

∂

√

S

σS T− t

Sensitivity to underlying

Gamma ∂ 2 V

− e− r( T− t) d 1 N ( d 2)

∂S 2

σ 2 S 2( T− t)

Sensitivity of delta to

underlying

Theta ∂V

∂

re− r( T − t) N ( d

t

2) + e− r( T − t) N ( d 2)

d

Sensitivity to time

×

1

− r− D

√

2( T − t)

σ T− t

Speed ∂ 3 V

− e− r( T− t) N ( d 2) × −2 d

√

∂

1 + 1− d 1 d 2

S 3

σ 2 S 3( T− t)

σ T− t

Sensitivity of gamma to

underlying

e− r( T − t) N ( d

Charm ∂ 2 V

2)

∂

√

× r + 1− d 1 d 2 + d 2( r− D)

√

S ∂t

σS T− t

2( T − t)

σ T− t

Sensitivity of delta to time

Colour

∂ 3 V

− e− r( T− t) N ( d 2) × rd

− r− D

√

∂

1 + 2 d 1+ d 2

S 2 ∂t

σ 2 S 2( T− t)

2( T − t)

σ T− t

d

Sensitivity of gamma to

× d

1

1 d

√

2

− r− D

2( T − t)

σ T− t

time

d

Vega ∂V

1

∂σ

− e− r( T− t) N ( d 2) σ

Sensitivity to volatility

Rho ( r) ∂V

∂

−( T − t) e− r( T− t) N( d

r

2)

√

Sensitivity to interest rate + T − t

√ σ

e− r( T − t) N ( d 2)

Rho ( D) ∂V

T − t

∂

−

D

σ

e− r( T − t) N ( d 2)

Sensitivity to

dividend yield

Vanna

∂ 2 V

∂

√

N ( d

1 − d

S ∂σ

− e− r( T− t)

σ 2

2)

1 d 2

S T − t

Sensitivity of delta to

volatility

Volga/Vomma ∂ 2 V

e− r( T − t) N ( d

d 2 d

∂σ 2

σ 2

2)

1 2 − d 1 − d 2

Sensitivity of vega to

volatility

304 Frequently Asked Questions In Quantitative Finance

Table 6.4: Formulæ for European binary put.

Binary Put

Payoff

1 if S < K otherwise 0

Value V

e− r( T − t)(1 − N( d 2))

Black–Scholes value

Delta ∂V

∂

− e− r( T− t) N ( d 2)

√

S

σS T− t

Sensitivity to underlying

e− r( T − t) d

Gamma ∂ 2 V

1 N ( d 2)

∂S 2

σ 2 S 2( T− t)

Sensitivity of delta to

underlying

Theta ∂V

∂

re− r( T − t)(1 − N( d

t

2)) − e− r( T − t) N ( d 2)

d

Sensitivity to time

×

1

− r− D

√

2( T − t)

σ T− t

e− r( T − t) N ( d

Speed ∂ 3 V

2) × −2 d

√

∂

1 + 1− d 1 d 2

S 3

σ 2 S 3( T− t)

σ T− t

Sensitivity of gamma to

underlying

Charm ∂ 2 V

∂

− e− r( T− t) N ( d 2)

√

× r + 1− d 1 d 2 + d 2( r− D)

√

S ∂t

σS T− t

2( T − t)

σ T− t

Sensitivity of delta to

time

e− r( T − t) N ( d

Colour

∂ 3 V

2) × rd

− r− D

√

∂

1 + 2 d 1+ d 2

S 2 ∂t

σ 2 S 2( T− t)

2( T − t)

σ T− t

d

Sensitivity of gamma to

d

1

1 d

√

2

− r− D

2( T − t)

σ T− t

time

d

Vega ∂V

1

∂σ

e− r( T − t) N ( d 2) σ

Sensitivity to volatility

Rho ( r) ∂V

∂

−( T − t) e− r( T− t)(1 − N( d r

2))

√

Sensitivity to interest rate − T − t

√ σ

e− r( T − t) N ( d 2)

Rho ( D) ∂V

T − t

∂D

σ

e− r( T − t) N ( d 2)

Sensitivity to dividend

yield

Vanna

∂ 2 V

∂

√

N ( d

1 − d

S ∂σ

− e− r( T− t)

σ 2

2)

1 d 2

S T − t

Sensitivity of delta to

volatility

Volga/Vomma ∂ 2 V

e− r( T − t) N ( d

d 2 d

∂σ 2

σ 2

2)

1 2 − d 1 − d 2

Sensitivity of vega to

volatility

Chapter 7

Common Contracts

306 Frequently Asked Questions In Quantitative Finance

Things to Look Out for in Exotic

Contracts

There are six important features to look out for in exotic

contracts. Understanding these features will help you

price a contract. These features are as follows.

1. Time dependence

2. Cashflows

3. Path dependence

4. Dimensionality

5. Order

6. Embedded decisions

If you can classify an exotic contract according to these

characteristics you will be able to determine the following.

• What kind of pricing method should best be used

• Whether you can re-use some old code

• How long it will take you to code it up

• How fast it will eventually run

Time dependence

is when the terms of an exotic contract

specify special dates or periods on or during which

something happens, such as a cashflow, or early exercise,

or an event is triggered. Time dependence is first on our

list of features, since it is a very basic concept.

• Time dependence in an option contract means that

our numerical discretization may have to be lined up

to coincide with times at, or periods during which,

something happens.

• This means that our code will have to keep track of

time, dates, etc. This is not difficult, just annoying.

Chapter 7: Common Contracts

307

Cashflows

are when money changes hands during the life

of the contract (as opposed to an initial premium or a

final payoff). When there is a cashflow the value of the

contract will instantaneously jump by the amount of the

cashflow.

• When a contract has a discretely paid cashflow you

should expect to have to apply jump conditions. This

also means that the contract has time dependence,

see above.

• Continuously paid cashflows mean a modification,

although rather simple, to the governing equation.

Path dependence

is when an option has a payoff that

depends on the path taken by the underlying asset, and

not just the asset’s value at expiration. Path dependency

comes in two varieties, strong and weak.

Strong path dependent contracts have payoffs that

depend on some property of the asset price path in

addition to the value of the underlying at the present

moment in time; in the equity option language, we can-

not write the value as V ( S, t). The contract value is

a function of at least one more independent variable.

Strong path dependency comes in two forms, discretely

sampled and continuously sampled, depending on

whether a discrete subset of asset prices is used or

a continuous distribution of them.

• Strong path dependency means that we have to work

in higher dimensions. A consequence of this is that

our code may take longer to run.

Weak path dependence is when a contract does depend

on the history of the underlying but an extra state

variable is not required. The obvious example is a bar-

rier option.

308 Frequently Asked Questions In Quantitative Finance

• Weak path dependency means that we don’t have to

work in higher dimensions, so our code should be

pretty fast.

Dimensionality

refers to the number of underlying inde-

pendent variables. The vanilla option has two indepen-

dent variables, S and t, and is thus two dimensional. The

weakly path-dependent contracts have the same number

of dimensions as their non-path-dependent cousins.

We can have two types of three-dimensional problem.

The first type of problem that is three dimensional is

the strongly path-dependent contract. Typically, the

new independent variable is a measure of the path-

dependent quantity on which the option is contin-

gent. In this case, derivatives of the option value with

respect to this new variable are only of the first order.

Thus the new variable acts more like another time-like

variable.

The second type of three-dimensional problem occurs

when we have a second source of randomness, such as

a second underlying asset. In the governing equation

we see a second derivative of the option value with

respect to each asset. We say that there is diffusion in

two dimensions.

• Higher dimensions means longer computing time.

• The number of dimensions we have also tells us what

kind of numerical method to use. High dimensions

mean that we probably want to use Monte Carlo, low

means finite difference.

The order of an option

refers to options whose payoff,

and hence value, is contingent on the value of another

option. The obvious second-order options are compound

Chapter 7: Common Contracts

309

options, for example, a call option giving the holder the

right to buy a put option.

• When an option is second or higher order we have to

solve for the first-order option, first. We thus have a

layer cake, we must work on the lower levels and the

results of those feed into the higher levels.

• This means that computationally we have to solve

more than one problem to price our option.

Embedded decisions

are when the holder or the writer has

some control over the payoff. They may be able to

exercise early, as in American options, or the issuer may

be able to call the contract back for a specified price.

When a contract has embedded decisions you need an

algorithm for deciding how that decision will be made.

That algorithm amounts to assuming that the holder

of the contract acts to make the option value as high

as possible for the delta-hedging writer. The pricing algo-

rithm then amounts to searching across all possible

holder decision strategies for the one that maximizes

the option value. That sounds hard, but approached cor-

rectly is actually remarkably straightforward, especially

if you use the finite-difference method. The justification

for seeking the strategy that maximizes the value is that

the writer cannot afford to sell the option for anything

less, otherwise he would be exposed to ‘decision risk.’

When the option writer or issuer is the one with the

decision to make then the value is based on seeking the

strategy that minimizes the value.

• Decision features mean that we’d really like to price

via finite differences.

• The code will contain a line in which we seek the

best price, so watch out for ≥ or ≤ signs.

310 Frequently Asked Questions In Quantitative Finance

Examples

Accrual

is a generic term applied to contracts in which an

amount gradually builds up until it is paid off in a lump sum.

An example would be an accrual range note in which for

every day that some underlying is within a specified range

a specified amount is accrued, to eventually be paid off in a

lump sum on a specified day. As long as there are no decision

features in the contract then the accrual is easily dealt with

by Monte Carlo simulation. If one wants to take a partial

differential approach to modelling then an extra state variable

will often be required to keep track of how much money has

been accrued.

American option

is one where the holder has the right to

exercise at any time before expiration and receive the payoff.

Many contracts have such early exercise American features.

Mathematically, early exercise is the same as conversion of a

convertible bond. These contracts are priced assuming that

the holder exercises so as to give the contract its highest

value. Therefore a comparison must be made between the

value of the option assuming you don’t exercise and what

you would get if you immediately exercised. This makes finite

differences a much more natural numerical method for pricing

such contracts than Monte Carlo.

Asian option

is an option whose payoff depends on the aver-

age value of the underlying during some period of the option’s

life. The average can be defined in many ways, as an arith-

metic or geometric mean, for example, and can use a large

set of data points in a continuously sampled Asian or only

a smaller set, in the discretely sampled Asian. In an Asian

tail the averaging only occurs over a short period before

option expiration. There are closed-form formulæ for some of

the simpler Asian options based on geometric averages, and

approximations for others. Otherwise they can be priced using

Monte Carlo methods, or sometimes by finite differences.

Because the average of an asset price path is less volatile

than the asset path itself these options can be cheaper than

their equivalent vanillas, but this will obviously depend on the

nature of the payoff. These contracts are very common in the

commodity markets because users of commodities tend to be

Chapter 7: Common Contracts

311

exposed to prices over a long period of time, and hence their

exposure is to the average price.

Asset swap

is the exchange of one asset for interest payments

for a specified period.

Balloon option

is an option where the quantity of option

bought will increase if certain conditions are met, such as

barriers being triggered.

Barrier option

has a payoff that depends on whether or not a

specified level of the underlying is reached before expiration.

In an ‘out’ option if the level is reached (triggered) then

the option immediately becomes worthless. In an ‘in’ option

the contract is worthless unless the level is triggered before

expiration. An ‘up’ option is one where the trigger level is

above the initial price of the underlying and a ‘down’ option

is one where the trigger level is below the initial price of the

underlying. Thus one talks about contracts such as the ‘up-

and-in call’ which will have the same payoff as a call option

but only if the barrier is hit from below. In these contracts

one must specify the barrier level, whether it is in or out,

and the payoff at expiration. A double barrier option has both

an upper and a lower barrier. These contracts are bought

by those with very specific views on the direction of the

underlying, and its probability of triggering the barrier. These

contracts are weakly path dependent. There are formulæ

for many types of barrier option, assuming that volatility is

constant. For more complicated barrier contracts or when

volatility is not constant these contracts must be valued using

numerical methods. Both Monte Carlo and finite differences

can be used but the latter is often preferable.

Basis swap

is an exchange of floating interest payments of

one tenor for floating interest payments of another tenor,

a six-month rate for a two-year rate for example. Since the

two payments will generally move together if the yield curve

experiences parallel shifts the basis swap gives exposure to

non-parallel movements in the yield curve such as flatten-

ing or steepening. More generally basis swap refers to any

exchange in which the two floating rates are closely related,

and therefore highly correlated.

312 Frequently Asked Questions In Quantitative Finance

Basket option

has a payoff that depends on more than one

underlying. A modest example would be an option that gives

you at expiration the value of the higher performing out of

two stocks. Another example would be a contract that pays

the average of the values of 20 stocks at expiration provided

that value is above a specified strike level. These contracts

can be valued straightforwardly by Monte Carlo simulation as

long as there is no early exercise feature. You would not use

finite-difference methods because of the high dimensionality.

If the contract is European, non path dependent with all of

the underlyings following lognormal random walks with con-

stant parameters then there is a closed-form formula for the

value of the contract, and this can be calculated by numerical

integration (quadrature). Basket options are popular in for-

eign exchange for those with exposure to multiple exchange

rates. They can also be used as options on your own index.

Although pricing these contracts can be theoretically straight-

forward they depend crucially on the correlation between

the underlyings. These correlations can be very difficult to

estimate since they can be quite unstable.

Bermudan option

is one where the holder has the right to

exercise on certain dates or periods rather than only at

expiration (European exercise) or at any time (American

exercise). Bermudan options cannot be worth less than their

European equivalent and cannot be worth more than their

American equivalent.

Binary option

has a payoff that is discontinuous. For example

a binary call pays off a specified amount if the underlying ends

above the strike at expiration and is otherwise worthless. A

one-touch pays off the specified amount as soon as the strike

is reached, it can be thought of as an American version of the

European binary. These contracts are also called digitals.

Break/Cancellable forward

is a forward contract, usually FX,

where the holder can terminate the contract at certain times

if they so wish.

Coupe option

is a periodic option in which the strike gets reset

to the worst of the underlying and the previous strike. Similar

to a cliquet option, but cheaper.

Chapter 7: Common Contracts

313

Call option

is an option to buy the underlying asset for a

specified price, the strike or exercise price, at (European)

or before (American) a specified data, the expiry or expira-

tion. The underlying can be any security. They are bought to

benefit from upward moves in the underlying, or if volatility

is believed to be higher than implied. In the latter case the

buyer would delta hedge the option to eliminate exposure

to direction. Calls are written for the opposite reasons, of

course. Also a holder of the underlying stock might write a

call to gain some premium in a market where the stock is

not moving much. This is called covered call writing. Simulta-

neous buying of the stock and writing a call is a buy-write

strategy. For calls on lognormal underlyings in constant

or time-dependent volatility worlds there are closed-form

expressions for prices. With more complicated underlyings or

volatility models these contracts can be priced by Monte Carlo

or finite difference, the latter being more suitable if there is

early exercise.

Other contracts may have call features or an embedded call.

For example, a bond may have a call provision allowing the

issuer to buy it back under certain conditions at specified

times. If the issuer has this extra right then it may decrease

the value of the contract, so it might be less than an equiva-

lent contract without the call feature. Sometimes the addition

of a call feature does not affect the value of a contract,

this would happen when it is theoretically never optimal

to exercise the call option. The simplest example of this is

an American versus a European call on a stock without any

dividends. These both have the same theoretical value since

it is never optimal to exercise early.

Cap

is a fixed-income option in which the holder receives a

payment when the underlying interest rate exceeds a specified

level, the strike. This payment is the interest rate less the

strike. These payments happen regularly, monthly, or quarterly

etc., as specified in the contract, and the underlying interest

rate will usually be of the same tenor as this interval. The

life of the cap will be several years. They are bought for

protection against rises in interest rates. Market practice is to

quote prices for caps using the Black ’76 model. A contract

with a single payment as above is called a caplet.

314 Frequently Asked Questions In Quantitative Finance

Chooser option

is an option on an option, therefore a second-

order option. The holder has the right to decide between

getting a call or a put, for example, on a specified date. The

expiration of these underlying options is further in the future.

Other similar contracts can be readily imagined. The key to

valuing such contracts is the realization that the two (or more)

underlying options must first be valued, and then one values

the option on the option. This means that finite-difference

methods are the most natural solution method for this kind

of contract. There are some closed-form formulæ for simple

choosers when volatility is at most time dependent.

Cliquet option

is a path-dependent contract in which amounts

are locked in at intervals, usually linked to the return on some

underlying. These amounts are then accumulated and paid off

at expiration. There will be caps and/or floors on the locally

locked-in amounts and on the global payoff. Such contracts

might be referred to as locally capped, globally floored, for

example. These contracts are popular with investors because

they have the eternally appreciated upside participation and

the downside protection, via the exposure to the returns and

the locking in of returns and global floor. Because of the

locking in of returns and the global cap/floor on the sum of

returns, these contracts are strongly path dependent. Typically

there will be four dimensions, which may in special cases

be reduced to three via a similarity reduction. This puts

the numerical solution on the Monte Carlo, finite difference

border. Neither are ideal, but neither are really inefficient

either. Because these contracts have a gamma that changes

sign, the sensitivity is not easily represented by a simple vega

calculation. Therefore, to be on the safe side, these contracts

should be priced using a variety of volatility models so as to

see the true sensitivity to the model.

Constant Maturity Swap (CMS)

is a fixed-income swap. In the

vanilla swap the floating leg is a rate with the same maturity

as the period between payments. However, in the CMS the

floating leg is of longer maturity. This apparently trivial differ-

ence turns the swap from a simple instrument, one that can

be valued in terms of bonds without resort to any model, into

a model-dependent instrument.

Chapter 7: Common Contracts

315

Collateralized Debt Obligation (CDO)

is a pool of debt instru-

ments securitized into one financial instrument. The pool

may consist of hundreds of individual debt instruments.

They are exposed to credit risk, as well as interest risk,

of the underlying instruments. CDOs are issued in several

tranches which divide up the pool of debt into instruments

with varying degrees of exposure to credit risk. One can buy

different tranches so as to gain exposure to different levels of

loss.

The aggregate loss is the sum of all losses due to default. As

more and more companies default so the aggregate loss will

increase. The tranches are specified by levels, as percentages

of notional. For example, there may be the 0–3% tranche, and

the 3–7% tranche etc. As the aggregate loss increases past

each of the 3%, 7%, etc. hurdles so the owner of that tranche

will begin to receive compensation, at the same rate as the

losses are piling up. You will only be compensated once your

attachment point has been reached, and until the detachment

point. The pricing of these contracts requires a model for the

relationship between the defaults in each of the underlying

instruments. A common approach is to use copulas. However,

because of the potentially large number of parameters needed

to represent the relationship between underlyings, the corre-

lations, it is also common to make simplifying assumptions.

Such simplifications might be to assume a single common

random factor representing default, and a single parameter

representing all correlations.

Collateralized Debt Obligation Squared (CDO 2 )

is a CDO-like con-

tract in which the underlyings are other CDOs instead of being

the simpler risky bonds.

Collateralized Mortgage Obligation (CMO)

is a pool of mortgages

securitized into one financial instrument. As with CDOs there

are different tranches allowing investors to participate in dif-

ferent parts of the cashflows. The cashflows in a mortgage

are interest and principal, and the CMOs may participate in

either or both of these depending on the structure. The dif-

ferent tranches may correspond to different maturities of the

underlying mortgages, for example. The risk associated with

316 Frequently Asked Questions In Quantitative Finance

CMOs are interest rate risk and prepayment risk, therefore it

is important to have a model representing prepayment.

Compound option

is an option on an option, such as a call on a

put which would allow the holder the right to buy a specified

put at a later date for a specified amount. There is no element

of choice in the sense of which underlying option to buy (or

sell).

Contingent premium option

is paid for at expiration only if

the option expires in the money, not up front. If the option

expires below the strike, for a call, then nothing is paid, but

then nothing is lost. If the asset is just slightly in the money

then the agreed premium is paid, resulting in a loss for the

holder. If the underlying ends up significantly in the money

then the agreed premium will be small relative to the payoff

and so the holder makes a profit. This contract can be valued

as a European vanilla option and a European digital with the

same strike. This contract has negative gamma below the

strike (for a call) and then positive gamma at the strike and

above, so its dependence on volatility is subtle. The holder

clearly wants the stock to end up either below the strike (for

a call) or far in the money. A negative skew will lower the

price of this contract.

Convertible bond

is a bond issued by a company that can,

at the choosing of the holder, be converted into a speci-

fied amount of equity. When so converted the company will

issue new shares. These contracts are a hybrid instrument,

being part way between equity and debt. They are appealing

to the issuer since they can be issued with a lower coupon

than straight debt, yet do not dilute earnings per share. If

they are converted into stock that is because the company

is doing well. They are appealing to the purchaser because

of the upside potential with the downside protection. Of

course, that downside protection may be limited because

these instruments are exposed to credit risk. In the event of

default the convertible bond ranks alongside debt, and above

equity.

These instruments are best valued using finite-difference meth-

ods because that takes into account the optimal conversion

time quite easily. One must have a model for volatility and

Chapter 7: Common Contracts

317

also risk of default. It is common to make risk of default

depend on the asset value, so the lower the stock price the

greater the probability of default.

Credit Default Swap (CDS)

is a contract used as insurance

against a credit event. One party pays interest to another for

a prescribed time or until default of the underlying instrument.

In the event of default the counterparty then pays the princi-

pal in return. The CDS is the dominant credit derivative in the

structured credit market. The premium is usually paid period-

ically (quoted in basis points per notional). Premium can be

an up-front payment, for short-term protection. On the credit

event, settlement may be the delivery of the reference asset in

exchange for the contingent payment or settlement may be in

cash (that is, value of the instrument before default less value

after, recovery value). The mark-to-market value of the CDS

depends on changes in credit spreads. Therefore they can be

used to get exposure to or hedge against changes in credit

spreads. To price these contracts one needs a model for risk

of default. However, commonly, one backs out an implied risk

of default from the prices of traded CDSs.

Diff(erential) swap

is an interest rate swap of floating for fixed

or floating, where one of the floating legs is a foreign inter-

est rate. The exchange of payments are defined in terms of a

domestic notional. Thus there is a quanto aspect to this instru-

ment. One must model interest rates and the exchange rate,

and as with quantos generally, the correlation is important.

Digital option

is the same as a binary option.

Extendible option/swap

is a contract that can have its ex-

piration date extended. The decision to extend may be at

the control of the writer, the holder or both. If the holder has

the right to extend the expiration then it may add value to

the contract, but if the writer can extend the expiry it may

decrease the value. There may or may not be an additional

premium to pay when the expiration is extended. These con-

tracts are best valued by finite-difference means because the

contract contains a decision feature.

Floating Rate Note (FRN)

is a bond with coupons linked to a

variable interest rate issued by a company. The coupon will

318 Frequently Asked Questions In Quantitative Finance

typically have a spread in excess of a government interest

rate, and this spread allows for credit risk. The coupons

may also have a cap and/or a floor. The most common

measure of a floating interest rate is the London Interbank

Offer Rate or LIBOR. LIBOR comes in various maturities,

one month, three month, six month, etc., and is the rate

of interest offered between Eurocurrency banks for fixed-term

deposits.

Floor

is a fixed-income option in which the holder receives

a payment when the underlying interest rate falls below a

specified level, the strike. This payment is the strike less the

interest rate. These payments happen regularly, monthly, or

quarterly etc., as specified in the contract, and the underlying

interest rate will usually be of the same tenor as this interval.

The life of the floor will be several years. They are bought for

protection against falling interest rates. Market practice is to

quote prices for floors using the Black ’76 model. A contract

with a single payment as above is called a floorlet.

Forward

is an agreement to buy or sell an underlying, typi-

cally a commodity, at some specified time in the future. The

holder is obliged to trade at the future date. This is in con-

trast to an option where the holder has the right but not the

obligation. Forwards are OTC contracts. They are linear in the

underlying and so convexity is zero, meaning that the volatil-

ity of the commodity does not matter and a dynamic model is

not required. The forward price comes from a simple, static,

no-arbitrage argument.

Forward Rate Agreement (FRA)

is an agreement between two

parties that a specified interest rate will apply to a specified

principal over some specified period in the future. The value

of this exchange at the time the contract is entered into is

generally not zero and so there will be a transfer of cash from

one party to the other at the start date.

Forward-start option

is an option that starts some time in the

future. The strike of the option is then usually set to be the

value of the underlying on the start date, so that it starts

life as an at-the-money option. It is also possible to have

contracts that begin in or out of the money by a specified

Chapter 7: Common Contracts

319

amount. Although the option comes into being at a specified

date in the future it is usually paid for as soon as the contract

is entered into. In a Black–Scholes world, even with time-

dependent volatility, these contracts have simple closed-form

formulæ for their values. Provided the strike is set to be a

certain fraction of the underlying at the start date then the

value of a vanilla call or put at that start date is linear in the

price of the underlying, and so prior to the start date there is

no convexity. This means that forward-start options are a way

of locking in an exposure to the volatility from the option’s

start date to the expiration.

Future

is an agreement to buy or sell an underlying, typically

a commodity, at some specified time in the future. The holder

is obliged to trade at the future date. The difference between

a forward and a future is that forwards are OTC and futures

are exchange traded. Therefore futures have standardized

contract terms and are also marked to market on a daily basis.

Being exchange traded they also do not carry any credit risk

exposure.

Hawai’ian option

is a cross between Asian and American.

Himalayan option

is a multi-asset option in which the best

performing stock is thrown out of the basket at specified

sampling dates, leaving just one asset in at the end on which

the payoff is based. There are many other, similar, mountain

range options.

HYPER option

High Yielding Performance Enhancing Reversible

options are like American options but which you can exercise

over and over again. On each exercise the option flips from

call to put or vice versa. These can be priced by introducing

a price function when in the call state and another when in

the put state. The Black–Scholes partial differential equation

is solved for each of these, subject to certain optimality con-

straints.

Index amortizing rate swap

is just as a vanilla swap, an agree-

ment between two parties to exchange interest payments on

some principal, usually one payment is at a fixed rate and

the other at a floating rate. However, in the index amortizing

rate swap the size of the principal decreases, or amortizes,

320 Frequently Asked Questions In Quantitative Finance

according to the value of some financial quantity or index

over the life of the swap. The level of this principal may be

determined by the level of an interest rate on the payments

dates. Or the principal may be determined by a non-fixed

income index. In the first example we would only need a fixed-

income model, in the second we would also need a model

for this other quantity, and its correlation with interest rates.

In an index amortizing rate swap the principal typically can

amortize on each payment date. On later payment dates this

principal can then be amortized again, starting from its cur-

rent level at the previous payment date and not based on

its original level. This makes this contract very path depen-

dent. The contract can be priced in either a partial differential

equation framework based on a one- or two-factor spot-rate

based model, or using Monte Carlo simulations and a Libor

market-type model.

Interest rate swap

is a contract between two parties to ex-

change interest on a specified principal. The exchange may

be fixed for floating or floating of one tenor for floating of

another tenor. Fixed for floating is a particularly common form

of swap. These instruments are used to convert a fixed-rate

loan to floating, or vice versa. Usually the interval between the

exchanges is set to be the same as the tenor of the floating

leg. Furthermore, the floating leg is set at the payment date

before it is paid. This means that each floating leg is equiv-

alent to a deposit and a withdrawal of the principal with an

interval of the tenor between them. Therefore all the floating

legs can be summed up to give one deposit at the start of

the swap’s life and a withdrawal at maturity. This means that

swaps can be valued directly from the yield curve without

needing a dynamic model. When the contract is first entered

into the fixed leg is set so that the swap has zero value. The

fixed leg of the swap is then called the par swap rate and is

a commonly quoted rate. These contracts are so liquid that

they define the longer-maturity end of the yield curve rather

than vice versa.

Inverse floater

is a floating-rate interest-rate contract where

coupons go down as interest rates go up. The relationship is

linear (up to any cap or floor) and not an inverse one.

Chapter 7: Common Contracts

321

Knock-in/out option

are types of barrier option for which the

payoff is contingent on a barrier level being hit/missed before

expiration.

LIBOR-in-arrears swap

is an interest rate swap but one for

which the floating leg is paid at the same time as it is set,

rather than at the tenor later. This small difference means

that there is no exact relationship between the swap and

bond prices and so a dynamic model is needed. This amounts

to pricing the subtle convexity in this product.

Lookback option

is a path-dependent contract whose payoff

depends on the maximum or minimum value reached by the

underlying over some period of the option’s life. The maxi-

mum/minimum may be sampled continuously or discretely,

the latter using only a subset of asset prices over the option’s

life. These contracts can be quite expensive because of the

extreme nature of the payoff. There are formulæ for some

of the simpler lookbacks, under the assumption of a lognor-

mal random walk for the underlying and non-asset-dependent

volatility. Otherwise they can be valued via finite-difference

solution of a path-dependent partial differential equation in

two or three dimensions, or by Monte Carlo simulation.

Mortgage Backed Security (MBS)

is a pool of mortgages that

have been securitized. All of the cashflows are passed on to

investors, unlike in the more complex CMOs. The risks inher-

ent in MBSs are interest rate risk and prepayment risk, since

the holders of mortgages have the right to prepay. Because

of this risk the yield on MBSs should be higher than yields

without prepayment risk. Prepayment risk is usually modelled

statistically, perhaps with some interest rate effect. Holders

of mortgages have all kinds of reasons for prepaying, some

rational and easy to model, some irrational and harder to

model but which can nevertheless be interpreted statistically.

Outperformance option

is an option where the holder gets the

best performing out of two or more underlyings at expiration.

This option can be valued theoretically in a lognormal random

walk, constant parameter world, since it is not path dependent

and there is a closed-form solution in terms of a multiple inte-

322 Frequently Asked Questions In Quantitative Finance

gral (in the same number of dimensions as there are underly-

ings). This amounts to a numerical quadrature problem which

is easily achieved by Monte Carlo or quasi Monte Carlo meth-

ods. The theory may be straightforward but the practice is not

since the price will depend on the correlations between all of

the underlyings, and these parameters are usually quite fickle.

Parisian option

is a barrier option for which the barrier

feature (knock in or knock out) is only triggered after the

underlying has spent a certain prescribed time beyond the

barrier. The effect of this more rigorous triggering criterion

is to smooth the option value (and delta and gamma) near

the barrier to make hedging somewhat easier. It also makes

manipulation of the triggering, by manipulation of the underly-

ing asset, much harder. In the classical Parisian contract the

‘clock’ measuring the time outside the barrier is reset when

the asset returns to within the barrier. In the Parisian contract

the clock is not reset but continues ticking as long as the

underlying is beyond the barrier. These contracts are strongly

path dependent and can be valued either by Monte Carlo sim-

ulation or by finite-difference solution of a three-dimensional

partial differential equation.

Pass through

is a security which collects payments on various

underlying securities and then passes the amounts on to

investors. They are issued by Special Purpose Vehicles and

can be made to avoid appearing on balance sheets. This

achieves a variety of purposes, some rather nefarious.

Passport option

is a call option on the trading account of an

individual trader, giving the holder the amount in his account

at the end of the horizon if it is positive, or zero if it is

negative. For obvious reasons they are also called perfect

trader options. The terms of the contract will specify what the

underlying is that the trader is allowed to trade, his maximum

long and short position, how frequently he can trade and for

how long. To price these contracts requires a small knowledge

of stochastic control theory. The governing partial differential

equation is easily solved by finite differences. Monte Carlo

would be quite difficult to implement for pricing purposes.

Since the trader very quickly moves into or, more commonly,

Chapter 7: Common Contracts

323

out of the money, the option is usually hedged with vanilla

options after a while.

Put option

is the right to sell the underlying stock. See the

‘Call Option’ since comments about pricing methodology,

embedded features etc. are equally applicable. Deep out-of-

the-money puts are commonly bought for protection against

large downward moves in individual stocks or against market

crashes. These out-of-the-money puts therefore tend to be

quite expensive in volatility terms, although very cheap in

monetary terms.

Quanto

is any contract in which cashflows are calculated

from an underlying in one currency and then converted to

payment in another currency. They can be used to eliminate

any exposure to currency risk when speculating in a foreign

stock or index. For example, you may have a view on a UK

company but be based in Tokyo. If you buy the stock you

will be exposed to the sterling/yen exchange rate. In a quanto

the exchange rate would be fixed. The price of a quanto will

generally depend on the volatility of the underlying and the

exchange rate, and the correlation between the two.

Rainbow option

is any contract with multiple underlyings. The

most difficult part of pricing such an option is usually knowing

how to deal with correlations.

Range note

is a contract in which payments are conditional

upon an underlying staying within (or outside) a specified

range of values.

Ratchet

is a feature that periodically locks in profit.

Repo

is a repurchase agreement. It is an agreement to sell

some security to another party and buy it back at a fixed date

and for a fixed amount. The price at which the security is

bought back is greater than the selling price and the difference

implies an interest rate called the repo rate. Repos can be

used to lock in future interest rates.

Reverse repo

is the borrowing of a security for a short period

at an agreed interest rate.

324 Frequently Asked Questions In Quantitative Finance

Straddle

is a portfolio consisting of a long call and a long put

with the same strike and expiration. Such a portfolio is for

taking a view on the range of the underlying or volatility.

Strangle

is a portfolio of a call and a put, the call having

a higher strike than the put. It is a volatility play like the

straddle but is cheaper. At the same time it requires the

underlying to move further than for a straddle for the holder

to make a profit.

STRIPS

stands for Separate Trading of Registered Interest

and Principal of Securities. The coupons and principal of

normal bonds are split up, creating artificial zero-coupon

bonds of longer maturity than would otherwise be available.

Swap

is a general term for an over-the-counter contract in

which there are exchanges of cashflows between two parties.

Examples would be an exchange of a fixed interest rate for

a floating rate, or the exchange of equity returns and bond

returns, etc.

Swaption

is an option on a swap. It is the option to enter into

the swap at some expiration date, the swap having predefined

characteristics. Such contracts are very common in the fixed-

income world where a typical swaption would be on a swap of

fixed for floating. The contract may be European so that the

swap can only be entered into on a certain date, or American

in which the swap can be entered into before a certain date

or Bermudan in which there are specified dates on which the

option can be exercised.

Total Return Swap (TRS)

is the exchange of all the profit or

loss from a security for a fixed or floating interest payment.

Periodically, one party transfers the cashflows plus any posi-

tive value change of a reference asset to the other party,

this includes interest payments, appreciation, coupons, etc.,

while the other party pays a fixed or floating rate, probably

with some spread. The difference between a total return swap

and a default swap is that a default swap simply transfers

credit risk, by reference to some designated asset whereas

a total return swap transfers all the risks of owning the des-

ignated asset. Total return swaps were among the earliest

credit derivatives. TRSs existed before default swaps, but now

Chapter 7: Common Contracts

325

default swaps are the more commonly traded instruments. The

maturity is typically less than the maturity of the underlying

instrument. A TRS therefore provides a means of packaging

and transferring all of the risks associated with a reference

obligation, including credit risk. TRSs are more flexible than

transactions in the underlyings. For example, varying the terms

of the swap contract allows the creation of synthetic assets

that may not be otherwise available. The swap receiver never

has to make the outlay to buy the security. Even after posting

collateral and paying a high margin, the resulting leverage and

enhanced return on regulatory capital can be large.

Variance swap

is a swap in which one leg is the realized vari-

ance in the underlying over the life of the contract and the

other leg is fixed. This variance is typically measured using

regularly spaced data points according to whatever variance

formula is specified in the term sheet. The contract is popular

with both buyers and sellers. For buyers, the contract is a

simple way of gaining exposure to the variance of an asset

without having to go to all the trouble of dynamically delta

hedging vanilla options. And for sellers it is popular because it

is surprisingly easy to statically hedge with vanilla options to

almost eliminate model risk. The way that a variance swap is

hedged using vanillas is the famous ‘one over strike squared

rule.’ The variance swap is hedged with a continuum of vanilla

options with the quantity of options being inversely propor-

tional to the square of their strikes. In practice, there does

not exist a continuum of strikes, and also one does not go all

the way to zero strike (and an infinite quantity of them).

The volatility swap is similar in principle, except that the pay-

off is linear in the volatility, the square root of variance. This

contract is not so easily hedged with vanillas. The difference

in prices between a volatility swap and a variance swap can

be interpreted via Jensen’s Inequality as a convexity adjust-

ment because of volatility of volatility. The VIX volatility index

is a representation of SP500 30-day implied volatility inspired

by the one-over-strike-squared rule.

Chapter 8

Popular Quant Books

328 Frequently Asked Questions In Quantitative Finance

T he following are the dozen most popular quant

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Chapter 8: Popular Quant Books

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330 Frequently Asked Questions In Quantitative Finance

Advanced Modelling in Finance Using

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The book adopts a step-by-step approach to understanding

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Chapter 8: Popular Quant Books

331

Option Valuation under Stochastic

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Scholes model. These features include the ‘smile’ pattern and

the term structure of implied volatility. The book includes

Mathematica code for the most important formulæ and many

illustrations.

332 Frequently Asked Questions In Quantitative Finance

The Concepts and Practice of

Mathematical Finance by Mark Joshi

‘‘Mark Joshi’s work is one of the most thoughtful books in

applied finance I know. It is both intuitive and mathematically

correct and it deals with very deep concepts in derivatives

pricing while keeping the treatment simple and readily under-

standable.’’ Riccardo Rebonato

Publisher Cambridge University Press

Publication date 2003

Format Hardback

ISBN 0521823552

Uniquely, the book includes extensive discussion of the ideas

behind the models, and is even-handed in examining various

approaches to the subject. Thus, each pricing problem is

solved using several methods. Worked examples and exercises,

with answers, are provided in plenty, and computer projects

are given for many problems. The author brings to this book

a blend of practical experience and rigorous mathematical

background, and supplies here the working knowledge needed

to become a good quantitative analyst.

Chapter 8: Popular Quant Books

333

C++ Design Patterns and Derivatives

Pricing by Mark Joshi

‘‘This book is thought-provoking and rewarding. Even for

the less experienced programmer, the presentation is readily

accessible, and the coded examples can be directly used to

solve real-life problems.’’ Journal of the American Statistics

Association, Ana-Maria Matache

Publisher Cambridge University Press

Publication date 2004

Format Hardback

ISBN 0521832357

Design patterns are the cutting-edge paradigm for

programming in object-oriented languages. Here they are

discussed, for the first time in a book, in the context of

implementing financial models in C++.

Assuming only a basic knowledge of C++ and mathematical

finance, the reader is taught how to produce well-designed,

structured, re-usable code via concrete examples. Each ex-

ample is treated in depth, with the whys and wherefores of

the chosen method of solution critically examined.

334 Frequently Asked Questions In Quantitative Finance

Heard on the Street by Timothy Crack

Publisher Timothy Crack

Publication date 2004

Format Paperback

ISBN 0970055234

The book contains over 140 quantitative questions collected

from actual investment banking, investment management, and

options trading job interviews. The interviewers use the same

questions year after year and here they are! These questions

come from all types of interviews (corporate finance, sales and

trading, quantitative research, etc), but they are especially likely

in quantitative capital markets job interviews. The questions

come from all levels of interviews (undergraduate, MBA, PhD),

but they are especially likely if you have, or almost have,

an MBA. The questions cover pure quantitative/logic, financial

economics, derivatives, and statistics. Each quantitative question

in the book is accompanied by a very detailed solution and by

helpful advice.

The latest edition also includes about 120 non-quantitative

actual interview questions.

Chapter 8: Popular Quant Books

335

Monte Carlo Methods in Finance by

Peter Jäckel

‘‘Few expert practitioners also have the academic expertise to

match Peter Jäckel’s in this area, let alone take the trouble to

write a most accessible, comprehensive and yet self-contained

text.’’ Carol Alexander

Publisher John Wiley & Sons

Publication date 2002

Format Hardback

ISBN 047149741X

Monte Carlo Methods in Finance adopts a practical flavour

throughout, the emphasis being on financial modelling and

derivatives pricing. Numerous real-world examples help the

reader foster an intuitive grasp of the mathematical and

numerical techniques needed to solve particular financial

problems. At the same time, the book tries to give a detailed

explanation of the theoretical foundations of the various meth-

ods and algorithms presented.

336 Frequently Asked Questions In Quantitative Finance

Credit Derivatives Pricing Models by

Philipp Schönbucher

‘‘Philipp Schönbucher is one of the most talented researchers

of his generation. He has taken the credit derivatives world by

storm.’’ Paul Wilmott

Publisher John Wiley & Sons

Publication date 2003

Format Hardback

ISBN 0470842911

Credit Derivatives Pricing Models provides an extremely com-

prehensive overview of the most current areas in credit risk

modeling as applied to the pricing of credit derivatives. As one

of the first books to uniquely focus on pricing, this title is also

an excellent complement to other books on the application of

credit derivatives. Based on proven techniques that have been

tested time and again, this comprehensive resource provides

readers with the knowledge and guidance to effectively use

credit derivatives pricing models.

Chapter 8: Popular Quant Books

337

Principles of Financial Engineering by

Salih Neftci

‘‘This is the first comprehensive hands-on introduction to

financial engineering. Neftci is enjoyable to read, and finds a

natural balance between theory and practice.’’ Darrell Duffie

Publisher Academic Press

Publication date 2004

Format Hardback

ISBN 0125153945

On a topic where there is already a substantial body of

literature, Salih Neftci succeeds in presenting a fresh, orig-

inal, informative, and up-to-date introduction to financial

engineering. The book offers clear links between intuition

and underlying mathematics and an outstanding mixture of

market insights and mathematical materials. Also included are

end-of-chapter exercises and case studies.

338 Frequently Asked Questions In Quantitative Finance

Options, Futures, and Other

Derivatives by John Hull

Publisher Prentice Hall

Publication date 2005

Format Hardback

ISBN 0131499084

For advanced undergraduate or graduate business, economics,

and financial engineering courses in derivatives, options and

futures, or risk management. Designed to bridge the gap

between theory and practice, this successful book contin-

ues to impact the college market and is regarded as ‘the

bible’ in trading rooms throughout the world. This edition has

been completely reworked from beginning to end to improve

presentation, update material, and reflect recent market devel-

opments. Though nonessential mathematical material has been

either eliminated or moved to end-of-chapter appendices, the

concepts that are likely to be new to many readers have been

explained carefully, and are supported by numerical examples.

Chapter 8: Popular Quant Books

339

The Complete Guide to Option Pricing

Formulas by Espen Gaarder Haug

‘‘The truth of the matter is that if I am being so positive about

this book, it’s because I know for a fact that it has saved lives

more than once.’’ Alireza Javaheri

Publisher McGraw-Hill Professional

Publication date 1997

Format Hardback

ISBN 0786312408

When pricing options in today’s fast-action markets, expe-

rience and intuition are no longer enough. To protect your

carefully planned positions, you need precise facts and tested

information that has been proven time and again.

The Complete Guide to Option Pricing Formulas is the first

and only authoritative reference to contain every option tool

you need, all in one handy volume: Black–Scholes, two asset

binomial trees, implied trinomial trees, Vasicek, exotics.

Many important option pricing formulæ are accompanied by

computer code to assist in their use, understanding, and

implementation.

Chapter 9

The Most Popular

Search Words and

Phrases on

Wilmott.com

342 Frequently Asked Questions In Quantitative Finance

T hefollowingaresomeofthemostcommonsearch

words or phrases on wilmott.com, and a few com-

ments on each. If other people want to know about

these, then maybe you should too.

American option

An option which can be exercised at any time

of the holder’s choosing prior to expiration. See page 310.

Arbitrage

Arbitrage is the making of a totally riskless profit in

excess of the risk-free rate of return. See page 25.

Asian option

An option whose payoff depends on the average

value of the underlying asset over some time period prior to

expiration. See page 310.

Asset swap

The exchange of two investments, or the cashflows

to those investments, between two parties.

Barrier option

An option which either comes into being or

becomes worthless if a specified asset price is reached before

expiration. See page 311.

Base correlation

A correlation used in a CDO model to repre-

sent the relationship between all underlyings from zero up to

a given detachment point. For example, the 0–3% and a 3–6%

tranches are separate instruments but between them one can

price a 0–6% tranche and so back out an implied correlation

from 0–6%, that is the base correlation. See page 315.

Basket

A collection of financial instruments. In a basket

option the payoff depends on the behaviour of the many

underlyings. See page 312.

Bermudan swaption

An option to enter into a swap that may

be exercised on any of a specified number of dates.

C++

An enhanced version of the C programming language

developed by Bjarne Stroustrup in 1983. The enhancements

include classes, virtual functions, multiple inheritance, tem-

plates, etc.

Chapter 9: Popular Search Words

343

Calibration

Determining parameters (possibly state and time

dependent) such that one’s theoretical prices match traded

prices. Also called fitting. This is a static process using a

snapshot of prices. Calibration does not typically involve

looking at the dynamics or time series of the underlying. See

page 191.

Callable

A contract which the issuer or writer can buy back

(call). The amount he has to pay and the dates on which he

can exercise this right will be specified in the contract.

Cap

A fixed-income contract paying the holder when the

underlying interest rate exceeds a specified level. See page 313.

CDO

A Collateralized Debt Obligation is a pool of debt instru-

ments securitized into one financial instrument. See page 315.

CDS

A Credit Default Swap is a contract used as insurance

against a credit event. One party pays interest to another for

a prescribed time or until default of the underlying instrument.

See page 317.

CFA

Chartered Financial Analyst. A professional designation

offered by the CFA Institute for successfully completing three

examinations. The syllabus includes aspects of corporate and

quantitative finance, economics and ethics.

CMS

Constant Maturity Swap is a fixed-income swap in which

one leg is a floating rate of a constant maturity (from the date

it is paid). A convexity adjustment is required for the pricing

of these instruments. See page 314.

Convertible

An instrument that can be exchanged for another

of a different type. A convertible bond is a bond that can be

turned into stock at a time of the holder’s choosing. This gives

an otherwise simple instrument an element of optionality. See

page 316.

Convexity

Related to the curvature in the value of a derivative

(or its payoff) with respect to its underlying. A consequence

of Jensen’s Inequality for convex functions together with

randomness in an underlying is that convexity adds value to

a derivative. A positive convexity with respect to a random

344 Frequently Asked Questions In Quantitative Finance

underlying or parameters increases the derivative’s value,

a negative convexity decreases value. In equity derivatives

convexity is known as gamma.

Copula

A function used to combine many univariate distri-

butions to make a single multivariate distribution. Often used

to model relationships between many underlying in credit

derivatives. See page 212.

Correlation

Covariance between two random variables divided

by both of their standard deviations. It is a number between

(and including) minus one and plus one that measures the

amount of linear relationship between the two variables. Cor-

relation is a parameter in most option-pricing models in which

there are two or more random factors. However, the parame-

ter is often highly unstable.

CQF

Certificate in Quantitative Finance, a part-time qualifi-

cation offered by Wilmott and 7city Learning which teaches

the more practical aspects of quantitative finance, including

modelling, data analysis, implementation of the models and,

crucially, focuses on which models are good and which aren’t.

Default probability

The probability of an entity defaulting

or going bankrupt. A concept commonly used in credit risk

modelling where it is assumed that default is a probabilistic

concept, rather than a business decision. Pricing credit instru-

ments then becomes an exercise in modelling probability of

default, and recovery rates. See page 295.

Delta

The sensitivity of an option to the underlying asset.

See page 110.

Digital

An option with a discontinuous payoff. See page 312.

Dispersion

The amount by which asset, typically equity,

returns are independent. A dispersion trade involves a bas-

ket of options on single stocks versus the opposite position in

an option on a basket of stocks (an index).

Duration

The sensitivity of a bond to an interest rate or yield.

It can be related to the average life of the bond.

Chapter 9: Popular Search Words

345

Exotic

A contract that is made to measure, or bespoke, for a

client and which does not exist as an exchange-traded instru-

ment. Since it is not traded on an exchange it must be priced

using some mathematical model. See pages 305–325.

Expected loss

The average loss once a specified threshold

has been breached. Used as a measure of Value at Risk. See

page 48.

Finite difference

A numerical method for solving differential

equations wherein derivatives are approximated by differ-

ences. The differential equation thus becomes a difference

equation which can be solved numerically, usually by an

iterative process.

Gamma

The sensitivity of an option’s delta to the underlying.

Therefore it is the second derivative of an option price with

respect to the underlying. See page 111.

GARCH

Generalized Auto Regressive Conditional Hetero-

scedasticity, an econometric model for volatility in which the

current variance depends on the previous random increments.

Hedge

To reduce risk by exploiting correlations between finan-

cial instruments. See page 73.

Hybrid

An instrument that exhibits both equity and fixed-

income characteristics, and even credit risk. An example would

be a convertible bond. Pricing such instruments requires

knowledge of models from several different areas of quantita-

tive finance.

Implied

Used as an adjective about financial parameters

meaning that they have been deduced from traded prices.

For example, what volatility when put into the Black–Scholes

formula gives a theoretical price that is the same as the mar-

ket price? This is the implied volatility. Intimately related to

calibration.

Lévy

A probability distribution, also known as a stable

distribution. It has the property that sums of independent

identically distributed random variables from this distribution

346 Frequently Asked Questions In Quantitative Finance

have the same distribution. The normal distribution is a spe-

cial case. Of interest in finance because returns data matches

this distribution quite well. See page 231.

LIBOR

London Interbank Offered Rate. An interest rate at

which banks offer to lend funds to other banks in the London

wholesale money market. It is quoted at different maturities.

Being a standard reference rate it is often the underlying

interest rate in OTC fixed-income contracts.

Market maker

Someone who gives prices at which he will

buy or sell instruments, in the hope of making a profit on the

difference between the bid and offer prices. They are said to

add liquidity to the market.

MBS

A Mortgage Backed Security is a pool of mortgages that

have been securitized. See page 321.

Mean reversion

The returning of a quantity to an average level.

This is a feature of many popular interest rate and volatility

models, which may exhibit randomness but never stray too

far from some mean.

Monte Carlo

A name given to many methods for solving

mathematical problems using simulations. The link between

a probabilistic concept, such as an average, and simulations is

clear. There may also be links between a deterministic prob-

lem and a simulation. For example, you can estimate π by

throwing darts at a square, uniformly distributed, and count-

ing how many land inside the inscribed circle. It should be π/ 4

of the number thrown. To get six decimal places of accuracy

in π you would have to throw approximately 1012 darts, this

is the downside of Monte Carlo methods, they can be slow.

Normal distribution

A probability distribution commonly used

to model financial quantities. See page 201.

PDE

Partial differential equation, as its name suggest an

equation (there must be an ‘equals’ sign), involving derivatives

with respect to two or more variables. In finance almost all

PDEs are of a type known as parabolic, this includes the

famous heat or diffusion equation. See page 20.

Chapter 9: Popular Search Words

347

Quantlib

Definition taken from www.quantlib.org: ‘‘QuantLib

is a free/open-source library for modeling, trading, and risk

management in real-life.’’

Quanto

Any contract in which cashflows are calculated from

an underlying in one currency and then converted to payment

in another currency. See page 323.

Regression

Relating a dependent and one or more independent

variables by a relatively simple function.

Risk

The possibility of a monetary loss associated with

investments. See page 36.

Risk neutral

Indifferent to risk in the sense that a return in

excess of the risk-free rate is not required by a risk-neutral

investor who takes risks. To price derivatives one can imagine

oneself in a world in which investors are risk neutral. Options

are then priced to be consistent with the market prices of

the underlying and future states of the world. This is because

the risk premium on the stock is already incorporated into

its current price, and the price of risk for the option and its

underlying should be the same. Working in a risk-neutral world

is a shortcut to that result. See page 103.

SABR

An interest rate model, by Pat Hagan, Deep Kumar,

Andrew Lesniewski and Diane Woodward, that exploits asymp-

totic analysis to make an otherwise intractable problem

relatively easy to manage. See page 292.

Skew

The slope of the graph of implied volatility versus

strike. A negative skew, that is a downward slope going from

left to right, is common in equity options.

Smile

The upward curving shape of the graph of implied

volatility versus strike. A downward curving profile would be

a frown.

Sobol’

A Russian mathematician responsible for much of the

important breakthroughs in low-discrepancy sequences, now

commonly used for simulations in finance. See page 225 and

www.broda.co.uk.

348 Frequently Asked Questions In Quantitative Finance

Stochastic

Random. The branch of mathematics involving the

random evolution of a quantities usually in continuous time

commonly associated with models of the financial markets and

derivatives. To be contrasted with deterministic.

Structured products

Contracts designed to meet the specific

investment criteria of a client, in terms of market view, risk

and return.

Swap

A general term for an over-the-counter contract in

which there are exchanges of cashflows between two parties.

See page 324.

Swaptions

An option on a swap. They are commonly Bermu-

dan exercise. See page 324.

VaR

Value at Risk, an estimate of the potential downside

from one’s investments. See pages 40 and 48.

Variance swap

A contract in which there is an exchange of the

realized variance over a specified period and a fixed amount.

See page 325.

Volatility

The annualized standard deviation of returns of

an asset. The most important quantity in derivatives pricing.

Difficult to estimate and forecast, there are many competing

models for the behaviour of volatility. See page 151.

Yield curve

A graph of yields to maturity versus maturity (or

duration). Therefore a way of visualizing how interest rates

change with time. Each traded bond has its own point on the

curve.

Esoterica

And finally, some rather more exotic word or phrase

searches, without any descriptions:

Art of War; Atlas Shrugged; Background check; Bloodshed;

Bonus; Deal or no deal; Death; Depression; Drug test; Female;

Gay; How to impress; James Bond; Lawsuit; Lonely; Sex; Suit;

Test; The; Too old

From this final list one should be able to build up a personal-

ity profile of the typical quant.

Chapter 10

Brainteasers

350 Frequently Asked Questions In Quantitative Finance

T hefollowingBrainteasershaveallbeentakenfrom

wilmott.com. They are all the type of questions

you could easily face during a job interview. Some

of these questions are simple calculation exercises,

often probabilistic in nature reflecting the importance of

understanding probability concepts, some have a ‘trick’

element to them, if you can spot the trick you can solve

them, otherwise you will struggle. And some require

lateral, out of the box, thinking.

The Questions

Russian roulette

I have a revolver which holds up to six

bullets. There are two bullets in the gun, in adjacent

chambers. I am going to play Russian roulette (on my

own!), I spin the barrel so that I don’t know where the

bullets are and then pull the trigger. Assuming that I

don’t shoot myself with this first attempt, am I now bet-

ter off pulling the trigger a second time without spinning

or spin the barrel first?

(Thanks to pusher.)

Matching birthdays

You are in a room full of people,

and you ask them all when their birthday is. How

many people must there be for there to be a greater

than 50% chance that at least two will share the same

birthday?

(Thanks to baghead.)

Another one about birthdays

At a cinema the manager

announces that a free ticket will be given to the first per-

son in the queue whose birthday is the same as some-

one in line who has already bought a ticket. You have

the option of getting in line at any position. Assuming

Chapter 10: Brainteasers

351

that you don’t know anyone else’s birthday, and that

birthdays are uniformly distributed throughout a 365-

day year, what position in line gives you the best chance

of being the first duplicate birthday?

(Thanks to amit7ul.)

Biased coins

You have n biased coins with the k th coin

having probability 1 /(2 k + 1) of coming up heads. What

is the probability of getting an odd number of heads in

total?

(Thanks to FV.)

Two heads

When flipping an unbiased coin, how long do

you have to wait on average before you get two heads

in a row? And more generally, how long before n heads

in a row.

(Thanks to MikeM.)

Balls in a bag

Ten balls are put in a bag based on the

result of the tosses of an unbiased coin. If the coin

turns up heads, put in a black ball, if tails, put in a

white ball. When the bag contains ten balls hand it to

someone who hasn’t seen the colours selected. Ask

them to take out ten balls, one at a time with replace-

ment. If all ten examined balls turn out to be white,

what is the probability that all ten balls in the bag are

white?

(Thanks to mikebell.)

Sums of uniform random variables

The random variables x 1,

x 2, x 3, . . . are independent and uniformly distributed over zero to one. We add up n of them until the sum

exceeds one. What is the expected value of n?

(Thanks to balaji.)

352 Frequently Asked Questions In Quantitative Finance

Minimum and maximum correlation

If X, Y and Z are three

random variables such that X and Y have a correlation

of 0.9, and Y and Z have correlation of 0.8, what are

the minimum and maximum correlation that X and Z

can have?

(Thanks to jiantao.)

Airforce One

One hundred people are in line to board

Airforce One. There are exactly 100 seats on the plane.

Each passenger has a ticket. Each ticket assigns the

passenger to a specific seat. The passengers board the

aircraft one at a time. GW is the first to board the plane.

He cannot read, and does not know which seat is his,

so he picks a seat at random and pretends that it is his

proper seat.

The remaining passengers board the plane one at a

time. If one of them finds their assigned seat empty,

they will sit in it. If they find that their seat is already

taken, they will pick a seat at random. This contin-

ues until everyone has boarded the plane and taken

a seat.

What is the probability that the last person to board

the plane sits in their proper seat?

(Thanks to Wilbur.)

Hit-and-run taxi

There was a hit-and-run incident involv-

ing a taxi in a city in which 85% of the taxis are green

and the remaining 15% are blue. There was a witness to

the crime who says that the hit-and-run taxi was blue.

Unfortunately this witness is only correct 80% of the

time. What is the probability that it was indeed a blue

car that hit our victim?

(Thanks to orangeman44.)

Chapter 10: Brainteasers

353

Annual returns

Every day a trader either makes 50% with

probability 0.6 or loses 50% with probability 0.4. What is

the probability the trader will be ahead at the end of a

year, 260 trading days? Over what number of days does

the trader have the maximum probability of making

money?

(Thanks to Aaron.)

Dice game

You start with no money and play a game in

which you throw a dice over and over again. For each

throw, if 1 appears you win $1, if 2 appears you win $2,

etc. but if 6 appears you lose all your money and the

game ends. When is the optimal stopping time and what

are your expected winnings?

(Thanks to ckc226.)

100 kg of berries

You have 100 kg of berries. Ninety-nine

percent of the weight of berries is water. Time passes

and some amount of water evaporates, so our berries

are now 98% water. What is the weight of berries now?

Do this one in your head.

(Thanks to NoDoubts.)

Urban planning

There are four towns positioned on the

corners of a square. The towns are to be joined by

a system of roads such that the total road length is

minimized. What is the shape of the road?

(Thanks to quantie.)

Closer to the edge or the centre?

You have a square and a ran-

dom variable that picks a random point on the square

with a uniform distribution. What is the probability that

a randomly selected point is closer to the center than

to the edge?

(Thanks to OMD.)

354 Frequently Asked Questions In Quantitative Finance

Snowflake

Start with an equilateral triangle. Now stick on

to the middle of each side equilateral triangles with side

one third of the side of the original triangle. This gives

you a Star of David, with six points. Now add on to the

sides of the six triangles yet smaller triangles, with side

one third of the ‘parent’ triangle and so on ad infinitum.

What are the perimeter and area of the final snowflake?

(Thanks to Gerasimos.)

The doors

There are one hundred closed doors in a cor-

ridor. The first person who walks along the corridor

opens all of the doors. The second person changes the

current state of every second door starting from the

second door by opening closed doors and closing open

doors. The third person who comes along changes the

current state of every third door starting from the third

door. This continues until the 100th person. At the end

how many doors are closed and how many open?

(Thanks to zilch.)

Two thirds of the average

Everyone in a group pays $1

to enter the following competition. Each person has

to write down secretly on a piece of paper a number

from zero to 100 inclusive. Calculate the average of all

of these numbers and then take two thirds. The winner,

who gets all of the entrance fees, is the person who gets

closest to this final number. The players know the rule

for determining the winner, and they are not allowed to

communicate with each other. What number should you

submit?

(Thanks to knowtorious and the Financial Times.)

Ones and zeros

Show that any natural number has a mul-

tiple whose decimal representation only contains the

digits 0 and 1. For example, if the number is 13, we get

13 × 77 = 1001.

(Thanks to idgregorio.)

Chapter 10: Brainteasers

355

Bookworm

There is a two-volume book set on a shelf,

the volumes being side by side, first then second. The

pages of each volume are two centimeters thick and

each cover is two millimeters thick. A worm has nibbled

the set, perpendicularly to the pages, from the first page

of the first volume to the last page of the second one.

What is the length of the path he has nibbled?

(Thanks to Vito.)

Compensation

A number of quants are at dinner, and

start discussing compensation. They want to calculate

the average compensation among themselves, but are

too embarrassed to disclose their own salaries. How

can they determine the average compensation of their

group? They do not have pens or paper or any other

way of writing down their salaries.

(Thanks to Arroway.)

Einstein’s brainteaser

There are five houses of five differ-

ent colours. In each house lives a person of a different

nationality. Those five people drink different drinks,

smoke cigarettes of a different brand and have a differ-

ent pet. None of them has the same pet, smokes the

same cigarette or drinks the same drink.

We know:

• The Englishman lives in the red house.

• The Swede has a dog as a pet.

• The Dane drinks tea.

• The green house is on the left of the white one.

• The person who lives in the green house drinks

coffee.

• The person who smokes Pall Mall raises birds.

• The owner of the yellow house smokes Dunhill.

• The man who lives in the house that is in the middle

drinks milk.

356 Frequently Asked Questions In Quantitative Finance

• The Norwegian lives in the first house.

• The man who smokes Blends lives next to the one

who has cats.

• The man who raises horses lives next to the one who

smokes Dunhill.

• The man who smokes Bluemaster drinks beer.

• The German smokes Prince.

• The Norwegian lives next to the blue house.

• The man who smokes Blends is neighbour of the one

who drinks water.

Question: Who has the fish?

(Thanks to NoDoubts.)

Gender ratio

A country is preparing for a possible future

war. The country’s tradition is to send only males into

battle and so they want to increase the proportion of

males to females in the population through regulating

births. A law is passed that requires every married

couple to have children and they must continue to have

children until they have a male.

What effect do you expect this law to have on the

makeup of the population?

(Thanks to Wilbur.)

Aircraft armour

Where should you reinforce the armour

on bombers? You can’t put it everywhere because it will

make the aircraft too heavy. Suppose you have data for

every hit on planes returning from their missions, how

should you use this information in deciding where to

place the armour reinforcement?

(Thanks to Aaron.)

Ages of three children

A census taker goes to a house,

a woman answers the door and says she has three

children. The census taker asks their ages and she says

Chapter 10: Brainteasers

357

that if you multiply their ages, the result is 36. He says

he needs more info so she tells him that the total of

their ages is the address of the building next door. He

goes and looks, then comes back and says he still needs

more information. She tells him that she won’t answer

any more questions because her eldest child is sleeping

upstairs and she doesn’t want to wake him.

What are the children’s ages?

(Thanks to tristanreid.)

Ants on a circle

You have a circle with a number of ants

scattered around it at distinct points. Each ant starts

walking at the same speed but in possibly different

directions, either clockwise or anticlockwise. When two

ants meet they immediately change directions, and then

continue with the same speed as before. Will the ants

ever, simultaneously, be in the same positions as when

they started out?

(Thanks to OMD.)

Four switches and a lightbulb

Outside a room there are four

switches, and in the room there is a lightbulb. One of

the switches controls the light. Your task is to find out

which one. You cannot see the bulb or whether it is on

or off from outside the room. You may turn any number

of switches on or off, any number of times you want.

But you may only enter the room once.

(Thanks to Tomfr.)

Turnover

In a dark room there is a table, and on this

table there are 52 cards, 19 face up, 33 face down. Your

task is to divide the cards into two groups, such that in

each group there must be the same number of face up

cards. You can’t switch on a light, ask a friend for help,

all the usual disalloweds. Is this even possible?

(Thanks to golftango and Bruno Dupire.)

358 Frequently Asked Questions In Quantitative Finance

Muddy faces

A group of children are playing and some of

them get mud on their foreheads. A child cannot tell if

he has mud on his own forehead, although he can see

the mud on the foreheads of any other muddy children.

An adult comes to collect the children and announces

that at least one of the children has a dirty forehead,

and then asks the group to put up their hand if they

know that they have mud on their forehead. How can

each child determine whether or not their forehead is

muddy without communicating with anyone else?

(Thanks to weaves.)

Pirate puzzle

There are 10 pirates in a rowing boat. Their

ship has just sunk but they managed to save 1000 gold

doubloons. Being greedy bastards they each want all the

loot for themselves but they are also democratic and

want to make the allocation of gold as fair as possible.

But how?

They each pick a number, from one to 10, out of a hat.

Each person in turn starting with number one, decides

how to divvy up the loot among the pirates in the boat.

They then vote. If the majority of pirates approve of the

allocation then the loot is divided accordingly, other-

wise that particular pirate is thrown overboard into the

shark-infested sea. In the latter case, the next pirate in

line gets his chance at divvying up the loot. The same

rules apply, and either the division of the filthy lucre

gets the majority vote or the unfortunate soul ends up

in Davy Jones’s locker.

Question, how should the first pirate share out the

spoils so as to both guarantee his survival and get a

decent piece of the action?

Chapter 10: Brainteasers

359

The Answers

Russian roulette

I have a revolver which holds up to six bullets. There

are two bullets in the gun, in adjacent chambers. I am

going to play Russian roulette (on my own!), I spin the

barrel so that I don’t know where the bullets are and

then pull the trigger. Assuming that I don’t shoot myself

with this first attempt, am I now better off pulling the

trigger a second time without spinning or spin the barrel

first?

(Thanks to pusher.)

Solution

This is a very typical, simple probability Brainteaser.

It doesn’t require any sophisticated or lateral thought.

Just pure calculation.

Whenever you spin the barrel you clearly have a two

in six, or one in three chance of landing on a chamber

containing a bullet.

If you spin and pull the trigger on an empty chamber,

what are the chances of the next chamber containing

a bullet? You are equally likely to be at any one of the

four empty chambers but only the last of these is adja-

cent to a chamber containing a bullet. So there is now a

one in four chance of the next pull of the trigger being

fatal. Conclusion is that you should not spin the barrel.

After surviving two pulls of the trigger without spinning

the barrel the odds become one in three again, and

it doesn’t matter whether you spin or not (at least it

doesn’t matter in a probabilistic sense). After surviving

360 Frequently Asked Questions In Quantitative Finance

that ‘shot’ it becomes fifty-fifty and if you are successful

four times in a row then the next shot will definitely

be fatal.

Matching birthdays

You are in a room full of people, and you ask them all

when their birthday is. How many people must there be

for there to be a greater than 50% chance that at least

two will share the same birthday?

(Thanks to baghead.)

Solution

This is a classic, simple probability question that is

designed to show how poor is most people’s perception

of odds.

As with many of these type of questions it is easier to

ask what are the chances of two people not having the

same birthday. So suppose that there are just the two

people in the room, what are the chances of them not

having the same birthday? There are 364 days out of 365

days that the second person could have, so the proba-

bility is 364 / 365. If there are three people in the room

the second must have a birthday on one of 364 out of

365, and the third must have one of the remaining 363

out of 365. So the probability is then 364 × 363 / 3652.

And so on. If there are n people in the room the proba-

bility of no two sharing a birthday is

364!

.

(365 − n)!365 n−1

So the question becomes, what is the smallest n for

which this is less than one half? And the answer to this

is 23.

Chapter 10: Brainteasers

361

Another one about birthdays

At a cinema the manager announces that a free ticket

will be given to the first person in the queue whose

birthday is the same as someone in line who has already

bought a ticket. You have the option of getting in line

at any position. Assuming that you don’t know anyone

else’s birthday, and that birthdays are uniformly dis-

tributed throughout a 365-day year, what position in line

gives you the best chance of being the first duplicate

birthday?

(Thanks to amit7ul.)

Solution

This is solved by an application of Bayes’ theorem.

Prob ( A ∩ B) = Prob ( A| B) Prob ( B) .

You need to calculate two probabilities, first the proba-

bility of having the same birthday as someone ahead of

you in the queue given that none of them has a dupli-

cate birthday, and second the probability that none of

those ahead of you have duplicate birthdays. If there are

n people ahead of you then we know from the previous

birthday problem that the second probability is

364!

.

(365 − n)!365 n−1

The first probability is simply n/ 365. So you want to

maximize

n 364!

.

(365 − n)!365 n

This is shown as a function of n below. It is maximized

when n = 19 so you should stand in the 20th place.

This maximizes your chances, but they are still small at

only 3.23%.

362 Frequently Asked Questions In Quantitative Finance

0.035

Probability

0.03

0.025

0.02

0.015

0.01

0.005

0

0

5

10

15

20

25

30

35

Number ahead of you in queue

Biased coins

You have n biased coins with the k th coin having

probability 1 /(2 k + 1) of coming up heads. What is the

probability of getting an odd number of heads in total?

(Thanks to FV.)

Solution

I include this as a classic example of the induction

method. Use pn to denote the required probability.

After n − 1 tosses there is a probability pn−1 that there

have been an odd number of heads. And therefore a

probability of 1 − pn−1 of there having been an even

number of heads. To get the probability of an even

number of heads after another toss, n in total, you mul-

tiply the probability of an odd number so far by the

Chapter 10: Brainteasers

363

probability of the next coin being tails and add this to

the product of the probability of an even number and

the probability of getting a head next:

1

pn = pn−1 1 −

1

+ ( 1 − pn−1 )

.

2 n + 1

2 n + 1

This becomes

2 n − 1

pn = pn−1

+

1

.

2 n + 1

2 n + 1

Now we just have to solve this difference equation, with

the starting value that before any tossing we have zero

probability of an odd number, so p 0 = 0. If we write

pn = an/(2 n + 1) then the difference equation for an

becomes the very simple

an = an−1 + 1 .

The solution of this with a 0 = 0 is just n and so the

required probability is

pn =

n

.

2 n + 1

Two heads

When flipping an unbiased coin, how long do you have

to wait on average before you get two heads in a row?

And more generally, how long before n heads in a row.

(Thanks to MikeM.)

Solution

It turns out that you may as well solve the general prob-

lem for n in a row. Let Nn be the number of tosses

needed to get n heads in the row. It satisfies the recur-

sion relationship

Nn = 1 ( N

( N

2

n−1 + 1) + 12

n−1 + 1 + Nn) .

364 Frequently Asked Questions In Quantitative Finance

This is because with probability 1 we get the required

2

head, and with probability 1 we get a tail and will have

2

to start the whole thing anew. Therefore we obtain

Nn = 2 Nn−1 + 2 .

This has solution

Nn = 2 n+1 − 2 .

This means six tosses on average to get two heads in

a row.

Balls in a bag

Ten balls are put in a bag based on the result of the

tosses of an unbiased coin. If the coin turns up heads,

put in a black ball, if tails, put in a white ball. When the

bag contains ten balls hand it to someone who hasn’t

seen the colours selected. Ask them to take out ten

balls, one at a time with replacement. If all ten examined

balls turn out to be white, what is the probability that

all ten balls in the bag are white?

(Thanks to mikebell.)

Solution

This is a test of your understanding of conditional prob-

ability and Bayes’ theorem again. First a statement of

Bayes’ theorem.

Prob( A| B) = Prob( B| A)Prob( A) .

Prob( B)

Prob( A) is the probability of A, without any information

about B, this is unconditional probability. Prob( A| B)

means probability of A with the extra information

concerning B, this is a conditional probability.

Chapter 10: Brainteasers

365

In the balls example, A is the event that all of the balls

in the bag are white, B is the event that the balls

taken out of the bag are all white. We want to find

Prob( A| B).

Clearly, Prob( A) is just 1 10 = 0 . 000976563. Trivially

2

Prob( B| A) is 1. The probability that we take 10 white

balls out of the bag is a bit harder. We have to look at

the probability of having n white balls in the first place

and then picking, after replacement, 10 white. This is

then Prob( B). It is calculated as

10

10!

1 n 10 = 0 . 01391303 .

n!(10 − n)! 210 10

n=0

And so the required probability is 0 . 000976563 / 0 . 01391303

= 0 . 0701905. Just over 7%.

Sums of uniform random variables

The random variables x 1, x 2, x 3, . . . are independent and uniformly distributed over zero to one. We add up n of

them until the sum exceeds one. What is the expected

value of n?

(Thanks to balaji.)

Solution

There are two steps to finding the solution. First what

is the probability of the sum of n such random vari-

ables being less than one. Second, what is the required

expectation.

There are several ways to approach the first part. One

way is perhaps the most straightforward, simply calcu-

late the probability by integrating unit integrand over

366 Frequently Asked Questions In Quantitative Finance

the domain in the upper right ‘quadrant’ between the

point (0, 0, . . . , 0) and the plane x 1 + x 2 + . . . + xn = 1.

This is just

1

1− x 1

1− x 1− x 2

1− x 1− x 2− ... − xn−1

. . .

1 dxn. . . dx 3 dx 2 dx 1 .

0

0

0

0

After doing several of the inner integrals you will find

that the answer is simply 1 .

n!

From this it follows that the probability that the sum

goes over one for the first time on the n th random

variable is

n − 1

1 − 1

− 1 −

1

=

.

n!

( n − 1)!

n!

The required expectation is the sum of n( n − 1) /n! =

1 /( n − 2)! from two to infinity, or equivalently the sum

of 1 /n! for n zero to infinity. And this is our answer, e.

Minimum and maximum correlation

If X, Y and Z are three random variables such that

X and Y have a correlation of 0.9, and Y and Z have correlation of 0.8, what are the minimum and maximum

correlation that X and Z can have?

(Thanks to jiantao.)

Solution

The correlation matrix





1

ρXY ρXZ

 ρ



XY

1

ρYZ

ρXZ ρYZ

1

must be positive semi definite. A bit of fooling around

with that concept will result in the following

Chapter 10: Brainteasers

367

constraints

− (1 − ρ 2 )(1 − ρ 2 ) + ρ

XY

YZ

XY ρYZ

≤ ρXZ ≤ (1 − ρ 2 )(1 − ρ 2 ) + ρ

XY

YZ

XY ρYZ .

For this particular example we have 0 . 4585 ≤ ρXZ ≤

0 . 9815. It is interesting how small the correlation can be,

less than one half, considering how high the other two

correlations are. Of course, if one of the two correla-

tions is exactly one then this forces the third correlation

to be the same as the other.

Airforce One

One hundred people are in line to board Airforce One.

There are exactly 100 seats on the plane. Each passen-

ger has a ticket. Each ticket assigns the passenger to a

specific seat. The passengers board the aircraft one at a

time. GW is the first to board the plane. He cannot read,

and does not know which seat is his, so he picks a seat

at random and pretends that it is his proper seat.

The remaining passengers board the plane one at a

time. If one of them finds their assigned seat empty,

they will sit in it. If they find that their seat is already

taken, they will pick a seat at random. This continues

until everyone has boarded the plane and taken a seat.

What is the probability that the last person to board

the plane sits in their proper seat?

(Thanks to Wilbur.)

Solution

Sounds really complicated, because of all the people

who could have sat in the last person’s seat before

368 Frequently Asked Questions In Quantitative Finance

their turn. Start by considering just two people, GW and

you. If GW sits in his own seat, which he will do 50%

of the time, then you are certain to get your allocated

seat. But if he sits in your seat, again with 50% chance,

then you are certain to not get the right seat. So a priori

result, 50% chance. Now if there are three people, GW

either sits in his own seat or in your seat or in the

other person’s seat. The chances of him sitting in his

own seat or your seat are the same, and in the former

case you are certain to get your correct seat and in

the latter you are certain to not get it. So those two

balance out. If he sits in the other person’s seat then

it all hinges on whether the other person then sits in

GW’s seat or yours. Both equally likely, end result 50-50

again. You can build on this by induction to get to the

simple result that it is 50-50 whether or not you sit in

your allocated seat.

Hit-and-run taxi

There was a hit-and-run incident involving a taxi in a

city in which 85% of the taxis are green and the remain-

ing 15% are blue. There was a witness to the crime who

says that the hit-and-run taxi was blue. Unfortunately

this witness is only correct 80% of the time. What is the

probability that it was indeed a blue car that hit our

victim?

(Thanks to orangeman44.)

Solution

A classic probability question that has important conse-

quences for the legal and medical professions.

Suppose that we have 100 such incidents. In 85 of these

the taxi will have been green and 15 blue, just based

on random selection of taxi colour. In the cases where

Chapter 10: Brainteasers

369

the taxi was green the witness will mistakenly say that

the car is blue 20% of the time, i.e. 17 times. In the 15

blue cases the witness will correctly say blue 80% of

the time, i.e. 12 times. So although there were only 15

accidents involving a blue taxi there were 29 reports of

a blue taxi being to blame, and most of those (17 out of

29) were in error. These are the so-called false positives

one gets in medical tests.

Now, given that we were told it was a blue taxi, what is

the probability that it was a blue taxi? That is just 12/29

or 41.4%.

Annual returns

Every day a trader either makes 50% with probability 0.6

or loses 50% with probability 0.4. What is the probability

the trader will be ahead at the end of a year, 260 trading

days? Over what number of days does the trader have

the maximum probability of making money?

(Thanks to Aaron.)

Solution

This is nice one because it is extremely counterintuitive.

At first glance it looks like you are going to make money

in the long run, but this is not the case.

Let n be the number of days on which you make 50%.

After 260 days your initial wealth will be multiplied by

1 . 5 n 0 . 5260− n.

So the question can be recast in terms of finding n for

which this expression is equal to 1:

n =

260 ln 0 . 5

= 260 ln 2 = 164 . 04 .

ln 05 . − ln 1 . 5

ln 3

370 Frequently Asked Questions In Quantitative Finance

The answer to the first question is then what is the

probability of getting 165 or more ‘wins’ out of 260

when the probability of a ‘win’ is 0.6. The answer to this

standard probability question is just over 14%.

The average return per day is

1 − exp(0 . 6 ln 1 . 5 + 0 . 4 ln 0 . 5) = −3 . 34% .

The probability of the trader making money after one

day is 60%. After two days the trader has to win on

both days to be ahead, and therefore the probability is

36%. After three days the trader has to win at least two

out of three, this has a probability of 64.8%. After four

days, he has to win at least three out of four, probability

47.52%. And so on. With an horizon of N days he would

have to win at least N ln 2 / ln 3 (or rather the integer

greater than this) times. The answer to the second part

of the question is therefore three days.

As well as being counterintuitive, this question does

give a nice insight into money management and is

clearly related to the Kelly criterion. If you see a ques-

tion like this it is meant to trick you if the expected

profit, here 0 . 6 × 0 . 5 + 0 . 4 × (−0 . 5) = 0 . 1, is positive with the expected return, here −3 . 34%, negative.

Dice game

You start with no money and play a game in which you

throw a dice over and over again. For each throw, if 1

appears you win $1, if 2 appears you win $2, etc. but if

6 appears you lose all your money and the game ends.

When is the optimal stopping time and what are your

expected winnings?

(Thanks to ckc226.)

Chapter 10: Brainteasers

371

Solution

Suppose you have won an amount S so far and you

have to decide whether to continue. If you roll again

you have an expected winnings on the next throw of

1 × 1 + 1 × 2 + 1 × 3 + 1 × 4 + 1 × 5 − 1 × S = 15 − S .

6

6

6

6

6

6

6

So as long as you have less than 15 you would continue.

The expected winnings is harder.

You will stop at 15, 16, 17, 18 and 19. You can’t get to

20 because that would mean playing when you have 15,

and throwing a five. So we must calculate the probabili-

ties of reaching each of these numbers without throwing

a six. At this point we defer to our good friend Excel.

A simple simulation of the optimal strategy yields an

expected value for this game of $6.18.

100 kg of berries

You have 100 kg of berries. Ninety-nine percent of

the weight of berries is water. Time passes and some

amount of water evaporates, so our berries are now

98% water. What is the weight of berries now?

Do this one in your head.

(Thanks to NoDoubts.)

Solution

The unexpected, yet correct, answer is 50 kg. It seems

like a tiny amount of water has evaporated so how can

the weight have changed that much?

There is clearly 1 kg of solid matter in the berries. If

that makes up two percent (100 less 98%) then the total

weight must be 50 kg.

372 Frequently Asked Questions In Quantitative Finance

Urban planning

There are four towns positioned on the corners of a

square. The towns are to be joined by a system of roads

such that the total road length is minimized. What is the

shape of the road?

(Thanks to quantie.)

Solution

One is tempted to join the towns with a simple crossroad

shape but this is not optimal. Pythagoras and some basic

calculus will show you that the arrangement shown in the

figure is better, with the symmetrically placed crosspiece

√

in the middle of the ‘H’ shape having length 1 − 1 3.

Obviously there are two such solutions.

Closer to the edge or the centre?

You have a square and a random variable that picks a

random point on the square with a uniform distribution.

Chapter 10: Brainteasers

373

What is the probability that a randomly selected point

is closer to the center than to the edge?

(Thanks to OMD.)

Solution

Many people will think that the required probability is

the same as the probability of landing in the circle with

diameter half the side of the square. But this is not the

case. The line separating closer to centre from closer to

edge is a parabola. The answer is

√

2

√ 3 / 2

−1 + 2 + 2 3 − 2 2

.

3

Snowflake

Start with an equilateral triangle. Now stick on to the

middle of each side equilateral triangles with side one

third of the side of the original triangle. This gives you

a Star of David, with six points. Now add on to the sides

of the six triangles yet smaller triangles, with side one

third of the ‘parent’ triangle and so on ad infinitum.

What are the perimeter and area of the final snowflake?

(Thanks to Gerasimos.)

Solution

First count how many sides there are as a function of

number of iterations. Initially there are three sides, and

then 3 × 4. Every iteration one side turns into four. So

there will be 3 4 n after n iterations. The length of each

side is one third what it was originally. Therefore after

n iterations the perimeter will be

4 n

3

multiplied by the original perimeter. It is unbounded as

n tends to infinity.

374 Frequently Asked Questions In Quantitative Finance

The area increases by one third after the first iteration.

After the second iteration you add an area that is num-

ber of sides multiplied by area of a single small triangle

which is one ninth of the previously added triangle.

If we use An to be the area after n iterations (when

multiplied by the area of initial triangle) then

4 n−1

An = An−1 + 1

.

3

9

So

∞

4 i

An = 1 + 1

= 8 .

3

9

5

i=0

The final calculation exploits the binomial expansion.

This is the famous Koch snowflake, first described in

1904, and is an example of a fractal.

The doors

There are one hundred closed doors in a corridor. The

first person who walks along the corridor opens all

of the doors. The second person changes the current

state of every second door starting from the second

door by opening closed doors and closing open doors.

The third person who comes along changes the current

state of every third door starting from the third door.

This continues until the 100th person. At the end how

many doors are closed and how many open?

(Thanks to zilch.)

Solution

This is a question about how many divisors a number

has. For example the 15th door is divisible by one,

three, five and fifteen. So it will be opened, closed,

Chapter 10: Brainteasers

375

opened, closed. Ending up closed. What about door 37?

Thirty seven is only divisible by one and 37. But again

it will end up closed. Since only squares have an odd

number of divisors we have to count how many squares

there are from one to 100. Of course, there are only 10.

Two thirds of the average

Everyone in a group pays $1 to enter the following com-

petition. Each person has to write down secretly on a

piece of paper a number from zero to 100 inclusive.

Calculate the average of all of these numbers and then

take two thirds. The winner, who gets all of the entrance

fees, is the person who gets closest to this final number.

The players know the rule for determining the winner,

and they are not allowed to communicate with each

other. What number should you submit?

(Thanks to knowtorious and the Financial Times.)

Solution

This is a famous economics experiment, which examines

people’s rationality among other things.

If everyone submits the number 50, say, then the win-

ning number would be two thirds of 50, so 33. Perhaps

one should therefore submit 33. But if everyone does

that the winning number will be 22. Ok, so submit that

number. But if everyone does that . . . You can see where

this leads. The stable point is clearly zero because if

everyone submits the answer zero then two thirds of

that is still zero, and so zero is the winning number.

The winnings get divided among everyone and there

was no point in entering in the first place.

In discussions about this problem, people tend to carry

through the above argument and either quickly conclude

376 Frequently Asked Questions In Quantitative Finance

that zero is ‘correct’ or they stop the inductive process

after a couple of iterations and submit something around

20. It may be that the longer people have to think about

this, the lower the number they submit.

This is a nice problem because it does divide people

into the purely rational, game-theoretic types, who pick

zero, and never win, and the more relaxed types, who

just pick a number after a tiny bit of thought and do

stand a chance of winning.

Personal note from the author:

The Financial Times ran

this as a competition for their readers a while back.

(The prize was a flight in Concorde, so that dates it

a bit. And the cost of entry was just the stamp on a

postcard.)

I organized a group of students to enter this competi-

tion, all submitting the number 99 as their answer (it

wasn’t clear from the rules whether 100 was included).

A number which could obviously never win. The pur-

pose of this was twofold, a) to get a mention in the

paper when the answer was revealed (we succeeded)

and b) to move the market (we succeeded in that

as well).

There were not that many entries (about 1,500 if I

remember rightly) and so we were able to move the

market up by one point. The FT printed the distribution

of entries, a nice exponentially decaying curve with a

noticeable ‘blip’ at one end! The winner submitted the

number 13.

I didn’t tell my students this, but I can now reveal that I

secretly submitted my own answer, with the purpose of

winning . . . my submission was 12. Doh!

Chapter 10: Brainteasers

377

Ones and zeros

Show that any natural number has a multiple whose

decimal representation only contains the digits 0 and 1.

For example, if the number is 13, we get 13 = 1001.

(Thanks to idgregorio.)

Solution

Consider the n + 1 numbers 1, 11, 111, 1111, etc. Two of

them will be congruent modulo n. Subtract the smaller

one from the bigger one. You will get a number contain-

ing only 0s and 1s.

Bookworm

There is a two-volume book set on a shelf, the volumes

being side by side, first then second. The pages of each

volume are two centimeters thick and each cover is two

millimeters thick. A worm has nibbled the set, perpen-

dicularly to the pages, from the first page of the first

volume to the last page of the second one. What is the

length of the path he has nibbled?

(Thanks to Vito.)

Solution

Just four millimeters. Think about where the first page

of the first volume and the last page of the second

volume will be relative to each other.

Compensation

A number of quants are at dinner, and start dis-

cussing compensation. They want to calculate the

378 Frequently Asked Questions In Quantitative Finance

average compensation among themselves, but are too

embarrassed to disclose their own salaries. How can

they determine the average compensation of their

group? They do not have pens or paper or any other

way of writing down their salaries.

(Thanks to Arroway.)

Solution

One of the quants adds a random number to his salary.

The total he then whispers to his neighbour on the

right. This person adds his own salary to the number

he was given, and whispers it to the person on his right.

This continues all around the table until we get back

to the first quant who simply subtracts his random

number from the total and divides by the number of

quants at the table. That is the average compensation

of the group.

Einstein’s brainteaser

There are five houses of five different colours. In each

house lives a person of a different nationality. Those

five people drink different drinks, smoke cigarettes of a

different brand and have a different pet. None of them

has the same pet, smokes the same cigarette or drinks

the same drink.

We know:

• The Englishman lives in the red house.

• The Swede has a dog as a pet.

• The Dane drinks tea.

• The green house is on the left of the white one.

• The person who lives in the green house drinks

coffee.

• The person who smokes Pall Mall raises birds.

• The owner of the yellow house smokes Dunhill.

Chapter 10: Brainteasers

379

• The man who lives in the house that is in the middle

drinks milk.

• The Norwegian lives in the first house.

• The man who smokes Blends lives next to the one

who has cats.

• The man who raises horses lives next to the one who

smokes Dunhill.

• The man who smokes Bluemaster drinks beer.

• The German smokes Prince.

• The Norwegian lives next to the blue house.

• The man who smokes Blends is neighbour of the one

who drinks water.

Question: Who has the fish?

(Thanks to NoDoubts.)

Solution

This was a question posed by Einstein who said that

98% of people can’t solve it. More likely 98% of people

can’t be bothered. And in these days of Su Doku, the

percentage of people who can solve it will be higher.

Oh, and the answer is the German.

(Historical note: Smoking was something that the poor

and the uneducated used to do. For an explanation of

the process, see Newhart, R. ‘Introducing tobacco to civ-

ilization’. ‘‘What you got for us this time, Walt . . . you got

another winner for us? Tob-acco . . . er, what’s tob-acco,

Walt? It’s a kind of leaf, huh . . . and you bought eighty

tonnes of it?!! . . . Let me get this straight, Walt . . . you’ve bought eighty tonnes of leaves? . . . This may come as a

kind of a surprise to you Walt but . . . come fall in Eng-

land, we’re kinda upto our . . . It isn’t that kind of leaf,

huh? . . . Oh!, what kind is it then . . . some special kind

of food? . . . not exactly? . . . Oh, it has a lot of different uses . . . Like . . . what are some of the uses, Walt? . . . Are you saying ‘snuff’, Walt? . . . What’s snuff? . . . You take a

380 Frequently Asked Questions In Quantitative Finance

pinch of tobacco . . . (ha ha ha) . . . and you shove it up

your nose . . . (ha ha ha) . . . and it makes you sneeze? . . .

(ha ha ha) . . . Yeh, I imagine it would, Walt! Hey, Gold-

enrod seems to do it pretty well over here! It has other

uses though, huh? . . . you can chew it! . . . Or put it in a

pipe! . . . or you can shred it up . . . and put it in a piece

of paper . . . (ha ha ha) . . . and roll it up . . . (ha ha ha) . . .

don’t tell me, Walt, don’t tell me . . . (ha ha ha) . . . you

stick it in your ear, right? (ha ha ha) . . . Oh! . . . between

your lips! . . . Then what do you do, Walt? . . . (ha ha ha) . . .

you set fire to it! . . . (ha ha ha) Then what do you do,

Walt? . . . (ha ha ha) . . . You inhale the smoke, huh! . . .

(ha ha ha) You know, Walt . . . it seems you can stand

in front of your own fireplace and have the same thing

going for you!’’)

Gender ratio

A country is preparing for a possible future war. The

country’s tradition is to send only males into battle

and so they want to increase the proportion of males

to females in the population through regulating births.

A law is passed that requires every married couple to

have children and they must continue to have children

until they have a male.

What effect do you expect this law to have on the

makeup of the population?

(Thanks to Wilbur.)

Solution

A bit of a trick question, this, and open to plenty of

interesting discussion.

The obvious answer is that there is no effect on the

gender ratio. However, this would only be true under

Chapter 10: Brainteasers

381

certain assumptions about the distribution of the sexes

of offspring among couples. Consider a population in

which each couple can only ever have boys or only

ever have girls. Those who have boys could stop after

one child, whereas those who have girls can never stop

having children, with the end result being more girls

than boys. (Of course, this might not matter since

the goal is for there to be more males, there is no

requirement on the number of females.) And if there

is any autocorrelation between births this will also have

an impact. If autocorrelation is one, so that a male child

is always followed by a male, and a female by a female,

then the ratio of males to females decreases, but with a

negative correlation the ratio increases.

Aircraft armour

Where should you reinforce the armour on bombers?

You can’t put it everywhere because it will make the

aircraft too heavy. Suppose you have data for every hit

on planes returning from their missions, how should

you use this information in deciding where to place the

armour reinforcement?

(Thanks to Aaron.)

Solution

The trick here is that we only have data for aircraft that

survived. Since hits on aircraft are going to be fairly

uniformly distributed over all places that are accessible

by gunfire one should place the reinforcements at pre-

cisely those places which appeared to be unharmed in

the returning aircraft. They are the places where hits

would be ‘fatal.’ This is a true Second World War story

about the statistician Abraham Wald who was asked

precisely this.

382 Frequently Asked Questions In Quantitative Finance

Ages of three children

A census taker goes to a house, a woman answers the

door and says she has three children. The census taker

asks their ages and she says that if you multiply their

ages, the result is 36. He says he needs more info so she

tells him that the total of their ages is the address of the

building next door. He goes and looks, then comes back

and says he still needs more information. She tells him

that she won’t answer any more questions because her

eldest child is sleeping upstairs and she doesn’t want to

wake him.

What are the children’s ages?

(Thanks to tristanreid.)

Solution

First suitably factorize 36: (1,1,36), (1,4,9), (1,2,18),

(1,3,12), (1,6,6), (2,3,6), (2,2,9), (3,3,4).

When the census taker is unable to decide from the

information about nextdoor’s house number we know

that nextdoor must be number 13, because both (1,6,6)

and (2,2,9) add up to 13. All of the other combina-

tions give distinct sums. Finally the mother refers to

the ‘eldest child,’ and this rules out (1,6,6) because the

two older children have the same age. Conclusion the

ages must be two, two and nine.

Caveat: (1,6,6) is technically still possible because one

of the six-year olds could be nearing seven while the

other has only just turned six.

Ants on a circle

You have a circle with a number of ants scattered

around it at distinct points. Each ant starts walking

Chapter 10: Brainteasers

383

at the same speed but in possibly different directions,

either clockwise or anticlockwise. When two ants meet

they immediately change directions, and then continue

with the same speed as before. Will the ants ever,

simultaneously, be in the same positions as when they

started out?

(Thanks to OMD.)

Solution

What are the chances of that happening? Surely all that

bouncing around is going to shuffle them all up. Well,

the answer, which you’ve probably now guessed, is

that, yes, they do all end up at the starting point. And

the time at which this happens (although there may be

earlier times as well) is just the time it would take for

one ant to go around the entire circle unhindered. The

trick is to start by ignoring the collisions, just think of

the ants walking through each other. Clearly there will

then be a time at which the ants are in the starting pos-

itions. But are the ants in their own starting positions?

This is slightly harder to see, but you can easily con-

vince yourself, and furthermore at that time they will

also be moving in the same direction they were to start

with (this is not necessarily true of earlier times at

which they may all be in the starting positions).

Four switches and a lightbulb

Outside a room there are four switches, and in the

room there is a lightbulb. One of the switches controls

the light. Your task is to find out which one. You cannot

see the bulb or whether it is on or off from outside the

room. You may turn any number of switches on or off,

any number of times you want. But you may only enter

the room once.

(Thanks to Tomfr.)

384 Frequently Asked Questions In Quantitative Finance

Solution

The trick is to realize that there is more to the bulb

than light.

Step one: turn on lamps 1 and 2, and go and have

some coffee. Step two: turn off 1 and turn on 3, then

go quickly into the room and touch the lamp.

It is controlled by switch 1 if it is hot and dark, 2 if it is

hot and light, 3 if it cold and light, 4 if it cold and dark.

Turnover

In a dark room there is a table, and on this table there

are 52 cards, 19 face up, 33 face down. Your task is

to divide the cards into two groups, such that in each

group there must be the same number of face up cards.

You can’t switch on a light, ask a friend for help, all the

usual disalloweds. Is this even possible?

(Thanks to golftango and Bruno Dupire.)

Solution

An elegant lateral thinking puzzle, with a simple solution.

Move any 19 cards to one side and turn them all over.

Think about it!

The use of an odd number, 19 in this case, can be seen

as either a clue or as a red herring suggesting that the

task is impossible.

Muddy faces

A group of children are playing and some of them get

mud on their foreheads. A child cannot tell if he has

Chapter 10: Brainteasers

385

mud on his own forehead, although he can see the

mud on the foreheads of any other muddy children. An

adult comes to collect the children and announces that

at least one of the children has a dirty forehead, and

then asks the group to put up their hand if they know

that they have mud on their forehead. How can each

child determine whether or not their forehead is muddy

without communicating with anyone else?

(Thanks to weaves.)

Solution

If there is only one child with mud on their forehead

they will immediately know it because all of the other

children are clean. He will therefore immediately raise

his hand.

If there are two children with muddy foreheads they

will not immediately raise their hands because they will

each think that perhaps the adult is referring to the

other child. But when neither raises their hand both

will realize that the other is thinking the same as them

and therefore both will raise their hands.

Now if their are three muddy children they will follow

a similar line of thinking but now it will take longer for

them all to realize they are muddy. And so on for an

arbitrary number of muddy children.

To make this work we really need something to divide

time up into intervals, a bell perhaps, because no doubt

not all children will be thinking at quite the same speed!

Pirate puzzle

There are 10 pirates in a rowing boat. Their ship has

just sunk but they managed to save 1000 gold dou-

bloons. Being greedy bastards they each want all the

386 Frequently Asked Questions In Quantitative Finance

loot for themselves but they are also democratic and

want to make the allocation of gold as fair as possible.

But how?

They each pick a number, from one to 10, out of a hat.

Each person in turn starting with number one, decides

how to divvy up the loot among the pirates in the boat.

They then vote. If the majority of pirates approve of the

allocation then the loot is divided accordingly, other-

wise that particular pirate is thrown overboard into the

shark-infested sea. In the latter case, the next pirate in

line gets his chance at divvying up the loot. The same

rules apply, and either the division of the filthy lucre

gets the majority vote or the unfortunate soul ends up

in Davy Jones’s locker.

Question, how should the first pirate share out the

spoils so as to both guarantee his survival and get a

decent piece of the action?

Solution

This is obviously one of those questions where you

have to work backwards, inductively, to the solution for

10 pirates. Along the way we’ll see how it works for an

arbitrary number of pirates.

Let’s start with two pirates, with 1000 doubloons to

share. Pirate 2 gets to allocate the gold. Unless he gives

it all to Pirate 1 the vote will be 50:50 and insufficient to

save him. Splash! We are assuming here that an equal

split of votes isn’t quite enough to count as a majority.

So he gives Pirate 1 the entire hoard, and prays that he

lives. (Of course, Pirate 1 could say hard luck and dump

Pirate 2 overboard and still keep the money.)

Now on to the three-pirate case. In making his alloca-

tion Pirate 3 must consider what would happen if he

loses the vote and there are just two pirates left. In

Chapter 10: Brainteasers

387

other words, he should make his allocation so that it is

preferred to the next scenario by sufficient numbers of

pirates to ensure that he gets a favourable vote.

Pirate 3 allocates 1000 to himself and nothing to the

others. Obviously Pirate 3 will vote for this. And so will

Pirate 2, if he votes against in the hope of getting some

loot he will find himself in the two-pirate situation . . . in

which case he could easily end up over the side.

Pirate 3

Pirate 2

Pirate 1

0

1000

1000

0

0

Now to four pirates. Pirate number 3 is not going to vote

for anything number 4 says because he wants Pirate 4

in the deep. So there’s no point in giving him any share

at all. Pirates 2 and 1 will vote for anything better than

the zero they’d get from the three-pirate scenario, so he

gives them one each and 998 to himself.

Pirate 4

Pirate 3

Pirate 2

Pirate 1

1000

0

1000

0

0

998

0

1

1

With five pirates similar logic applies. Pirate 4 gets zero.

Then Pirate 5 needs to get two votes from the remaining

three pirates. What is the cheapest way of doing this?

He gives one to Pirate 3 and two to either of Pirates 2

and 1. Pirate 5 gets the remaining 997.

Pirate 5

Pirate 4

Pirate 3

Pirate 2

Pirate 1

1000

0

1000

0

0

998

0

1

1

997

0

1

2 / 0

0 / 2

Pirate 6 needs four votes to ensure survival, his own

plus three others. He’ll never get Pirate 5 so he needs

three votes from Pirates 4, 3, 2 and 1. Pirate 4 is cheap,

388 Frequently Asked Questions In Quantitative Finance

he only needs 1 doubloon. But how to get two votes

from the remaining Pirates 3, 2 and 1?

There are clearly several options here. And we are

going to have to think carefully about the actions of

the Pirates when faced with uncertainty.

Imagine being Pirate 2 when Pirate number 6 is allocat-

ing the gold. Suppose he gives you zero, what do you

do? You may as well vote against, because there is a

chance that on the next round you will get two doub-

loons. If Pirate 6 gives you two doubloons you should

vote in favour. Things cannot be improved on the next

round but may get worse. If given one doubloon now,

what should you do? Next round you will either get zero

or two. A tricky choice. And a possibly personal one.

But it is up to Pirate 6 to make sure you are not faced

with that tricky decision which may result in his expul-

sion from the boat.

The conclusion is that Pirate 6 should give two

doubloons to any of Pirates 3, 2 and 1. It doesn’t matter

which.

Pirate 6

Pirate 5

Pirate 4

Pirate 3

Pirate 2

Pirate 1

1000

0

1000

0

0

998

0

1

1

997

0

1

2 / 0

0 / 2

995

0

1

2 / 2 / 0

2 / 0 / 2

0 / 2 / 2

On Pirate 7’s turn he will give zero to Pirate 6, one to

Pirate 5 and two to any of Pirates 4 down to 1, again it

doesn’t matter which two, they will both now vote in

his favour.

Pirate 7

Pirate 6

Pirate 5

Pirate 4

Pirate 3

Pirate 2

Pirate 1

1000

0

1000

0

0

998

0

1

1

997

0

1

2 / 0

0 / 2

995

0

1

2 / 2 / 0

2 / 0 / 2

0 / 2 / 2

995

0

1 Two doubloons to any two of these four

Chapter 10: Brainteasers

389

Now we settle down into a rhythm. Here’s the entire

allocation table.

Pirate 10

Pirate 9

Pirate 8

Pirate 7

Pirate 6

Pirate 5

Pirate 4

Pirate 3

Pirate 2

Pirate 1

1000

0

1000

0

0

998

0

1

1

997

0

1

2 / 0

0 / 2

995

0

1

2 / 2 / 0

2 / 0 / 2

0 / 2 / 2

995

0

1 Two doubloons to any two of these four

993

0

1 Two doubloons to any three of these five

993

0

1 Two doubloons to any three of these six

991

0

1 Two doubloons to any four of these seven This Brainteaser is particularly relevant in quantitative

finance because of the backward induction nature of

the solution. This is highly reminiscent of the binomial

model in which you have to calculate today’s option

price by working backwards from expiration by consid-

ering option prices at different times.

Another of these backward induction types is the

famous Brainteaser, the unexpected hanging. In this

problem we have a prisoner who has been condemned

to be executed in ten days’ time and an unusually

considerate executioner. The executioner wants the

prisoner to suffer as little mental anguish as possible

during his last days and although the prisoner knows

that sometime in the next ten days he will be executed

he doesn’t know when. If the executioner can surprise

him then the prisoner will be able to enjoy his last few

days, at least relatively speaking. So, the executioner’s

task is to wake the prisoner up one morning and exe-

cute him but must choose the day so that his visit was

not expected by the prisoner.

Let’s see how to address this problem by induction

backwards from the last of the ten days. If the prisoner

has not been executed on one of the first nine days then

he goes to bed that night in no doubt that tomorrow he

will be woken by the executioner and hanged. So he

can’t be executed on the last day, because it clearly

wouldn’t be a surprise. Now, if he goes to bed on the

night of the eighth day, not having been executed during

390 Frequently Asked Questions In Quantitative Finance

the first eight days then he knows he can’t be executed

on the last day because of the above, and so he knows

that he must be executed tomorrow, day nine. Therefore

it won’t be a surprise and therefore the execution can’t

happen on the ninth day either. We have ruled out the

last two days, and by working backwards we can rule

out every one of the ten days.

On day four the prisoner is awoken by the executioner,

and hanged. Boy, was he surprised!

Where did our backward induction argument go wrong?

Ok, now I can tell you that this brainteaser is called the

unexpected hanging paradox. There have been many

explanations for why the prisoner’s logic fails. For

example, because the prisoner has concluded that he

can’t be hanged then to surprise him is rather simple.

Chapter 11

Paul & Dominic’s Guide

to Getting a Quant Job

392 Frequently Asked Questions In Quantitative Finance

I fyouenjoyedthisbook,andarelookingforajobin

quantitative finance, you might be interested in Paul

& Dominic’s Guide to Getting a Quant Job. To whet your

appetite there follows the opening sections of version

1.0 of this famous guide. For details on how to get the

full guide email paul@wilmott.com.

Introduction

This guide is for people looking for their first or second

job in Quant Finance, the theory being that after a few

years you ought to know most of this stuff.

Making a difference

If the hiring process is working well,

the people seen by the bank will be roughly the same

quality and from comparable backgrounds. Thus you

need to stand out in order to win. We speak to a lot

of recruiting managers, and we find that the difference

between the one who got the job, and the person who

came second is often very small for the employer, but

obviously rather more important for you.

You have to walk a line between standing out, and not

seeming too much for them to handle.

Understand the process

Interviewing people is a major indus-

try all by itself, multiply the number of applicants by the

number of interviews they attend and you sometimes

wonder how any useful work ever gets done. Certainly

this thought occurs to interviewers on a regular basis.

They want it to end, soon, and although it is impor-

tant to get the right people almost no one enjoys the

process, and this is made worse by the fact that > 80%

of the work is wasted on those you never hire. Thus

a core objective must be to make life as easy for the

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

393

interviewer as possible. This means turning up on time,

but not too early, being flexible on interview times, and

trying to be pleasant.

What you need to prove

• You are smart

• You can work with people

• You can get things done

• You can manage yourself and your time

• You are committed to this line of work

Kissing frogs

Like trying to find a prince by kissing frogs,

you have to accept that it is rare for your first attempt

to succeed, so you must be prepared for a long haul,

and to pursue multiple options at the same time. This

means applying to several banks, and not being deterred

by failure to get into a particular firm.

Writing a CV

A CV is not some passive instrument that simply tells a

recruiter why he should interview you, it also to some

extent sets the agenda for the questions you will get

when they meet you. Thus it is important to choose

what you disclose as a balance between what you think

they want and the areas in which you are confident

answering questions.

Read the job specification

You should think how you can

present your skills and experience so as to be as close

a match as possible. At one level this might sound

obvious, but you should be aware that in many banks

your CV will not be read by the hiring manager at all.

Although at P&D we’ve actually done this stuff, it is

often the case that CVs are filtered by people with little

394 Frequently Asked Questions In Quantitative Finance

or no skills in finance. Often they resort to looking for

keywords. Thus you should not rely upon them working

out that if you have done subject X, you must have skills

Y and Z. If you believe those skills are critical for this

job, then make sure this can easily be spotted. Read the

specification carefully, and if they specifically ask for a

skill or experience, then include whatever you can to

illustrate your strengths. If you believe particular skills

to be critical mention them in your covering letter as

well (or if you believe the headhunter is especially dim).

Make sure you can be contacted

Make sure your contact

details are reliable and that you regularly monitor the

relevant email account(s) and telephones. It is sad when

someone’s CV shows great promise, but they don’t

respond in time to be interviewed. If you are at uni-

versity, be aware that your current email address may

stop working soon after you complete your course.

GMail is to be preferred over Yahoo for personal email

address.

Get it checked

Have your CV and covering letter proof

read by a native English speaker. This is important

because people really do judge your ability by how

you express yourself. Quant Finance is an international

sport, with speakers of every language, and the ability

to communicate difficult ideas is important, and if you

can’t get the name of your university correct, it makes

one wonder if you can explain your views on jump dif-

fusion. Also CVs use a particular style of English, which

is subtly different from the one you learned in school.

As there are a lot more applicants than jobs, the early

stages are optimized to filter out those who stand no

chance of getting in. Thus you must take consider-

able care to make sure you don’t fail at an early stage

because of trivial errors.

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

395

Covering letter

In your covering email, mention where

you saw the advertisement, and importantly which job

you are applying for. If you don’t say which job you are

applying for, you are relying upon the person receiving

it to guess correctly. That does not always happen, and

the larger the firm, the lower the probability, and at the

very least it makes their lives harder which is not the

way to start the relationship.

A good format for a covering letter is to respond to the

job specification point by point. State your ability to

handle each item, together with why you think you can

do it. This makes your CV more digestible, and shows

that you are serious about the application.

Opinion is divided about whether you should have some

‘‘statement of intent.’’ If you can think of something

useful to say here, by all means put it, but be aware

that a lot of new entrants to the market ‘‘want to pursue

a career in finance with a leading firm.’’

Above we emphasize getting your CV checked, and this

applies especially to the covering letter. Some managers

discard the CV unread if the letter is sloppy.

Fonts and layout

Some things oN YOUR cv are

important, and you may want tO draw their attention

to them. Do not do this excessively. It is really irritat-

ing. The only time breaking THIS rule has worked to

our knowledge was a hardcore programmer who

learned the POstscript language that PCs use to

talk directly to printers and he developed A pro-

gram that printed his CV as concentric spirals of text

in varying size. Viewed on screen it would slowly

spin. YES, Dominic hired him.

396 Frequently Asked Questions In Quantitative Finance

If you’re not prepared to spend at least a month

learning reverse Polish notation, use a standard

template. (Stick to two main font families, a sanserif,

such as Arial, for large headings, and a serif font, such

as Times, for main body text.)

PDF

Make a PDF if possible. These have a more pro-

fessional feel than Word documents, they do not have

virus problems (yet) and they retain original fonts and

layout. Whatever software you use, print it out to make

sure that what you see is really what you get. Perhaps

view on and print from another PC to double check.

Name

Give your document a name that will be meaning-

ful to the recruiter. Call it YourNameHere.pdf and not

CV.pdf in the spirit of making it easier for the recruiter.

It’s not nice to have a large number of files with the

same name, and it’s actually quite easy to get your CV

written over by the CV someone else who also called

it CV.

Dates

Make sure your dates ‘‘join up’’ as much as

possible. Some people in the recruitment process worry

about gaps.

Be honest

If you claim skills in some area, it’s a good

bet that you will be asked questions about it. The CV

should be a fair and positive statement of what you

have to offer. No one expects you to share your history

of skin diseases, but you’re going to be expected to

back the talk with action.

Show that you can do things

By this point in your life you’ve

soaked up a lot of information, and acquired skills,

which is, of course, good. But a big question in the

inquisitor’s mind is whether you can translate this into

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

397

real actions that are finished, complete and correct. One

can pass most exams by getting answers wrong, but by

showing good working, and an understanding of the

principles. However, banks aren’t really all that happy

if you lose a pile of money by getting ‘‘nearly’’ the right

answer, where you put a minus where it should have

been a plus. They like to see projects where you’ve

started, worked to a goal and completed without having

to have your hand held. This is a big reason why they

like PhDs since it’s a powerful argument that you can

complete things. However, if you’re going for a PhD-level

job, you still have to beat others who have reached that

level.

Projects completed are good, and you should be pre-

pared to answer questions on them. The people who

interview you will often have your sort of academic

background, so these questions may be deep.

You may have changed direction in your career, and

you should be prepared to answer why you made any

given choice. It is important to be able to show that you

didn’t just ‘‘give up’’ on an area when it got tough.

Interests and hobbies

Several of the people you meet will

want to understand what sort of personality you have,

or indeed whether you actually have one.

In finance you spend more of your waking hours with

your colleagues than the person you marry, so it is

good to present yourself as interesting as well as smart.

They all want to feel you can work with others, so the

cliché of ‘‘reading, walking and listening to music,’’ don’t

really cut it. Certainly you shouldn’t fake an interest in

something, but do try to find something with which

you can speak with a little passion. One candidate had

somehow acquired a formal qualification in stage com-

bat. Although it’s relatively rare for quants to need to

398 Frequently Asked Questions In Quantitative Finance

fight each other with swords, it’s the sort of thing that

catches people’s eyes, and can make a crucial differ-

ence. It also gives the non-specialist people who you

will meet something they can talk to you about.

Last job first

In your CV your most recent/current

employment should stand out and be relevant to the

job for which you are applying. Someone reading your

CV may never get beyond this one piece of information.

Make sure your dates are correct. As part of the pre-

employment screen at most banks, they check your

past employment, and people have had offers withdrawn

because of mistakes on this.

Paul & Dominic

When applying to P&D, we also like to

see a simple list of the various skills you have acquired,

together with some estimate of how good you are. If

you’re new to QF then it won’t be obvious which are

most important, that’s our job, so include as many as

possible.

Multiple CVs

Finally, there is no reason why you should

have only one CV. Presumably your entire life doesn’t

fit on two pages, so you can turn out a variety that each

emphasize different aspects of your experience and

education. You may take this as an exercise to work

out the optimal number of variants, and you will quickly

find out that it is not one. This is made more acute by

the fact that failed CVs get little if any feedback. Think

of it as shooting in the dark. If you don’t hear a scream

when you fire in one direction, you aim somewhere else.

Finding banks

In this document, we use the term ‘‘bank’’

for the firm you want to work for. It is of course the

case that quants work for many types of outfit, including

brokers, the government, hedge funds, insurers, thrifts,

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

399

consultancies, building societies, and of course in the

case of P&D, for a headhunting firm. The wilmott.com

website mentions any number of firms, and before you

approach anyone it’s good to do a few searches so that

you know the nature of the target.

If you’re still linked with your college then it has many

resources to help you. Most have a careers office with

directories of banks, and they will have some con-

tacts with banks in that country. The library will have

directories, and of course there is Google and Yahoo for

getting a list of targets. All large firms have entry-level

programmes of some form, and you can relatively easily

find a good number to apply for. At this stage numbers

are important, since the ratio of new entrants to the

market to jobs is quite high

Interviews

Be Prepared

Before you go for the interview, find out the

names of the people you are seeing, and do a Google on

their name, as well as the bank/business unit you are

joining. Try to avoid the error made by one candidate

who could not understand why the interviewer was

so interested in one part of her thesis. The candidate

had quoted papers by the interviewer, but somehow

managed to fail connecting the interviewer’s name with

the paper.

Be Confident

Almost no one at banks actually enjoys

interviewing people, some even see it as a form of pun-

ishment. That means they only interview you if there’s

a good chance they will want to hire you. Most people

who are considered for any job never even get a first

interview.

400 Frequently Asked Questions In Quantitative Finance

Be punctual

This shouldn’t need saying. If you can’t be

on time for your interview how can they expect you to

put in 12-hour days? If you are going to be late (and

assuming it was unavoidable) telephone ahead with an

accurate ETA. The best strategy is to schedule having

a coffee before the interview, a little caffeine and sugar

may well help, and this is a useful time buffer. Probably

the worst bit about being late is not what it does to

the interviewer, but what it does to you. The idea is to

present yourself as cool, smart and in control. If you’ve

been stressed out dealing with transport you knock a

few points off your performance.

Set traps

Although some questions are set in advance,

most interviewers like to drill down based upon your

answers. Thus you should try to mention areas where

you feel confident in answering hard questions. This

is best done subtly, by phrases like ‘‘this is quite like

X, but the answer is Y,’’ where X is a bastion of your

competence; or by saying thoughtfully ‘‘this isn’t like

X at all,’’ if you feel you are being drawn into an area

where you will sink.

Show you can do things

We mention this in the CV section,

and here’s a chance to ‘‘casually’’ drop in things you’ve

done that show you can dig in and finish the job. It’s OK

to mention problems you overcame, and the lessons you

learned from initial difficulties. Good managers are scep-

tical of people who claim to glide effortlessly through

life, and don’t want to be there when such a person

hits a rock. Practical ability is therefore something that

you will need to demonstrate a little more than theory.

You wouldn’t have reached this point if you didn’t have

a respectable record for absorbing theory, so the next

step is to see if you can apply what you’ve learned.

When asked your motivation for moving into finance,

it’s worth asking yourself if this is a reason.

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

401

Questions for the interviewer

It is a good idea to have a

question thought out in advance—it makes you look

interested in the position. You have two objectives

when they ask if you have questions for them.

Getting the message across

A question can be a good way

of bringing in things you want them to know, or to

emphasize a point you want them to remember. You

can ask the importance of your experience in MC, C++

or PDEs to the work you’d be doing. This gets the

message across, either as a reminder or to bring it to

their notice.

Find out more about the job

Good questions are on the

direction for the team over the next year, and how your

work would help them get where they want to be. It

shows interest, and may give a better insight into what

you really will be doing. Although they are interviewing

you, it is also the case that they are selling the job

to you, since they want you to accept if they offer.

So it’s up to you to work out whether it’s a good job

or not.

Remember, do not ask things that you should already

know. You should discuss the job and the bank as much

as you can with your recruiting consultant ahead of the

interview and consult websites and any recruitment

brochures. You don’t want to give the interviewer the

impression that you aren’t interested enough in their

bank to find out about it before the interview. Interview-

ers often say that this is the thing that really irritates

them most at interviews. Instead, it is good to preface

a question with a statement about some achievement

that the bank is proud of (i.e., talks about at length on

their website or in recruitment materials) e.g., ‘‘I know

your office won the International Finance Press Award

for Being a Bank last year, but could you tell me . . . ’’

402 Frequently Asked Questions In Quantitative Finance

Appearance

Good clothes

It is entirely possible that in your interview

process that every person you meet is not wearing a

suit, some may not have shaved. That doesn’t make

it wise for you to turn up in ‘‘smart casual.’’ How you

look is not a big deal for quants, you’re being paid to

think. However, some people do get remembered for the

wrong reason, and it can undermine your application a

little. You should feel comfortable, and if that means

a bit of perfume or good cufflinks then that’s fine, but

see below

Neatness is good

More important than colour of cloth or

design of tie, is the general impression of being in con-

trol of how you look. This means wearing it well, and

being ordered in your appearance. It is worth checking

this before you go into the bank.

Colours

Black is the new black. White is nice for shirts

and for no other visible item of clothing. Shoes should

be clean and preferably black for men, and muted tones

for women. A particular issue for women is the poor

workmanship in most of their shoes. Do not attempt to

walk long distances in new shoes that hurt your feet

so badly they bleed (we know one person who stained

the carpet with her blood). Make sure your clothes

fit—badly fitting clothes do not look presentable and if

your trousers are too tight you (and everyone else) will

find this distracts from the matter at hand. There are

some complexions that are generally complemented by

certain colours, and apparently in some circles ‘‘brown’’

is seen as a colour for your clothing. It is not; it merely

says things about you that are never said to your face.

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

403

Dark blue is good as well.

Ties are best boring, novelty is bad.

Another reason for white shirts is that they don’t show

sweat, some colours do this terribly and it’s not the

image you want to project. A good shirt doesn’t crease

badly in wear.

Jewellery

This will never help you get a job, no matter

how expensive or fashionable. Thus if you have any

doubt at all, don’t wear it. If you’re female and you

have some brooch or bracelet, that’s fine, but there’s

no upside for a man at all in bling. Cufflinks of course

are fine, as long as they are not ‘‘novelty’’—you have

no idea as to the sense of humour your interviewer may

have; he may not have one at all. Some banking people

spend quite appalling amounts on their watches, so

don’t even try to compete.

Perfume and aftershave

Feel free to smell nice, but make

sure that it’s not too strong.

Make-up

The following is for women. If you’re a male

reader, you really should not be reading this paragraph

and we are rather concerned that you are. Unless you

really never wear make-up, a small amount is a good

idea. Again, this gives the impression that you are mak-

ing an effort and will possibly counter the deadening

effect of all the monochrome clothing you are wear-

ing. It should be discreet (i.e. no bright colours) and

presentable rather than intending to make you look

prettier. There are jobs that you can obtain by being

attractive, but they are rarely fun and never intellec-

tually rewarding. Any make-up should always be well

404 Frequently Asked Questions In Quantitative Finance

applied—if you can’t get eyeliner on straight, don’t put

it on, and never wear nail polish if there is any chance

it will chip before the interview.

What People Get Wrong

Zeroth law of holes

When you find yourself in a hole, stop

digging. You will be asked questions for which you can’t

think of any answer at all. Some interviewers make the

questions harder until that point is reached. The trick

is to cut your losses. With any luck they will just move

on, unless it’s a critical topic. Of course if it’s critical

then it’s game over anyway. What you must avoid is

wasting time wandering like the lost spirit of ignorance

over a vast formless expanse of your incompetence. A

good response is to look them in the eye after a little

thought, then simply say ‘‘don’t know, sorry.’’

The exception to this are the ‘‘all the tea in China’’ ques-

tions where you are asked to estimate some quantity

like the number of bull testicles consumed by McDon-

alds customers per year. You aren’t expected to know

the answer to these, indeed knowing it would seem

rather strange. They want to see how well you can

estimate an unknown quantity and how you think.

But the biggest hole catches people who get very ner-

vous when things go wrong. This is about the most

negative personality defect you might have in a bank.

When you realize you’ve said something dumb, stop,

say something like ‘‘let me think about that for a sec-

ond,’’ and correct yourself. Make the pause work for

you. Think the answer through, and show that you are

capable of recovering. Remember that no one can talk

about things at the edge of their competence for 4–5

hours without saying something silly. You don’t have

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

405

to be defect free, but self knowledge and recovery will

score you vital points.

Sleep regularly, sleep often

Probably the most common

error we’ve seen is not getting enough sleep the night

before. As we said earlier, the difference between you

and your competitors is tiny, and losing a small percent-

age of your thinking ability through being tired has an

exponential effect on your probability of getting a job.

Hours in a bank can be quite hard, so it’s really not a

good idea to mention feeling tired. Not only will they not

be impressed, but if you get drawn into a conversation

about how it degrades your performance it won’t end

well. Conversely, a cup of coffee doesn’t do any harm,

but we have seen people who clearly had drunk rather

too much, and it didn’t work well for them.

Make eye contact

You need to make sure you look at your

interrogators, they can smell fear. No need to stare at

them, just remind yourself to look at them when they or

you are speaking.

Apply for the right job

You may feel you are unique indi-

vidual, and an obvious match for the job. Sadly, that

turns out not to be the case. If you are applying for

a job called ‘‘Henchman to Assistant Quant’s Minion–

PD0701067,’’ then do try to include that in your applica-

tion, prominently. If you don’t include this, then you are

critically dependant upon whoever opens your applica-

tion guessing

Don’t send a blue CV

Just don’t, OK?

Barbarians

The word barbarian comes from the ancient

Greeks who took anyone who didn’t speak Greek as

making ‘‘bar bub bar’’ noises, like a drunk Homer Simp-

son, not Barbarian as in the icy commanding tones of

406 Frequently Asked Questions In Quantitative Finance

Governor Schwarzenegger. Although Dr Simpson has

enjoyed careers as an astronaut, rock star and nuclear

engineer, few of us would hire him as a quant. It’s

important to get the right balance between gushing

at people so fast that they have trouble following you,

or being too quiet. You should try to practise looking

at the reaction of people talking to you, and if the inter-

viewer is clearly trying to move on, you usually should

let them. If you think of the conversation style used

when first meeting someone you find attractive, you

won’t go far wrong. (Just remember it’s a first date.)

It is also the case that no one wants to discriminate

against those who aren’t English speakers. This is good,

but means that if you aren’t understood they may just

skip over what you say, rather than pass comment on

your accent. This is especially true when having a tele-

phone interview where you will not get visual feedback,

and the sound quality is degraded.

Read your CV

Make sure that your CV is correct. A

surprisingly large number have dates that are clearly

wrong, or that by accident give the wrong impression.

These worry interviewers a lot, and if your dates don’t

match, this can lose you an offer when they do the

basic background check on all employees. Also read it

to work out which questions it might provoke them to

ask, ‘‘Why did you pick X?’’ ‘‘I see you’ve done a lot of

Y, here’s a hard question about it.’’

Mobile phone interviews

We’re old people ( > 35), and thus

sometimes use quaint ’phone technology which involves

long wires physically connecting us to a huge ancient

Unix computer miles away (yes, we still use miles). A

typical quant has done enough physics to know that you

can actually talk down copper wires rather than a 1 mm

thick cell phone that has more processing capacity than

its owner.

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407

Sadly, the quality of cell phone speech is hideously

degraded, and on many systems you can’t both talk at

the same time. This is occasionally awkward when both

speakers have the same first language, but if both have

English as second language neither comes out of the

conversation impressed with the other.

Do not attempt to do a phone interview on a cell phone.

Focus

Forging a rapport with the interviewer is a good

thing, but some interviews drift off topic as the people

involved chat. However, there is a time budget for each

interview, and most managers have specific objectives

in checking your ability. If they don’t get covered it can

hurt your progress to the next stage. Although it is the

interviewer’s responsibility to get things done, it’s your

problem if he doesn’t. This is where the politeness we

mention elsewhere is important. When you feel that time

is moving against you, ask to make sure that everything

they need to know is covered.

Asking questions

Actually there are stupid questions. Bad

questions are ones which embarrass the interviewer, or

force them into corners, that’s their job. Do not try to

score points off the interviewer, either you fail and look

silly, or worse still, you succeed. It’s a bad idea to bring

up any screw-ups that the bank has been involved in,

or where the manager has to admit that he hasn’t read

your CV.

Buzzwords

Your interrogator will often come from a sim-

ilar background to you, but even within maths and

physics there are many specializations that are mutually

incomprehensible. You’re just emerging from a disci-

pline where you think in terms of these names and

equations and it’s easy to emit a stream of noises that

your interviewer can barely understand. It’s actually

408 Frequently Asked Questions In Quantitative Finance

worse if they are from a similar background, since they

may feel embarrassed to ask what you actually mean.

You lose points here. But it is generally polite to enquire

about the background of your audience when asked to

explain some part of your work. This both shows con-

sideration, and prevents you making this error.

Show some market insight

This doesn’t mean you have to

know the ticker symbols of all SP500 stocks, but it does

mean you should be able to comment on the reliabil-

ity of models, what are their common pitfalls and how

the quant and the trader might communicate about

this. If you can quantify some practical phenomenon

that is rarely discussed in academic literature then you

will impress. (Tip: Because banks are often delta hedg-

ing, everything usually boils down to gamma and/or

volatility.)

It is also worth reading The Economist before the inter-

view. Some interviewers are keen to see if you have

awareness of the world in general. The Economist may

disturb some people since it covers other countries and

has no astrology column or coverage of golf.

Brainteasers

There are several different types of brain-

teasers you might get asked, all designed to test how

your mind works under pressure, and to try and gauge

how smart you are, rather than how much you have

learned.

• Straightforward calculation. Example: How many

trailing zeros are there in 100 factorial?

• Lateral thinking. Example: Several coworkers would

like to know their average salary. How can they

calculate it, without disclosing their own salaries?

• Open to discussion. Example: What’s the probability

that a quadratic function has two real roots?

Chapter 11: Paul & Dominic’s Guide to Getting a Quant Job

409

• Off the wall. Example: How many manhole covers are

there within the M25?

Work through the Brainteaser Forum on wilmott.com. You

can practice for IQ tests, and the more you do, the better

your score. Brainteasers are no different. And you’d be

surprised how often the same questions crop up.

It’s worth having a few numbers at your fingertips for

the ‘‘manhole covers.’’ One manager recently told us

in rather despairing tones of the stream of candidates

who didn’t have even a rough approximation to the

population of the country they were born and educated

in. Several put the population of Britain between 3 and

5 million (it’s around 60 million). A good trick when

‘‘estimating’’ is to pick numbers with which it is easy to

do mental arithmetic. Sure you can multiply by 57, but

why expose yourself to silly arithmetic errors.

In many types of question, they want to hear your

train of thought, and have simply no interest in the

actual answer. Thus you need to share your thoughts

about how you get to each stage. You also should ‘‘san-

ity check’’ your answers at each step, and make sure

they’re aware you’re doing it. This is a soft skill that’s

very important in financial markets where the money

numbers you are manipulating are rather larger than

your credit card bill.

At entry level we also see people being asked what we

call ‘‘teenage maths.’’ You’ve probably been focusing

on one area of maths for some years now, and to get

this far you’ve probably been good at it. However some

banks will ask you to do things like prove Pythagoras’

theorem, calculate π to a few decimal places, or prove

that the sum of N numbers is N( N + 1) / 2. That last fact being surprisingly useful in brainteasers.

410 Frequently Asked Questions In Quantitative Finance

Be polite

Your mother told you this would be important

one day, this is the day. ‘‘Please,’’ ‘‘thank you,’’ and

actually looking as if you are listening are good things.

Fidgeting, playing with your tie, or looking like you’d

rather be somewhere else aren’t polite. Standing when

people come into the room is good. Occasionally you

will find it appropriate to disagree, this is good, but get

in the habit of using phrases like ‘‘I’m not sure if that’s

the case, perhaps it is . . . ’’

You can’t just wake up one day and be polite on a whim.

(Hint: ‘‘Pretty Woman’’ is fiction, we know this for a fact.)

Without practice, it may even come over as sarcasm. In

some languages ‘‘please’’ and ‘‘thank you’’ are implied in

the context of the sentence, and that habit can spill over

into English. Break that habit, break it now.

Practise sounding positive about things.

Of the things you can change between now and your

interview, this one may have the biggest payback. If

you’ve been doing calculus for a decade, you aren’t

going to improve much in a week. However, you become

better at presenting yourself as someone who’s easy to

work with.

This is so important because your team will spend more

waking hours together than most married couples, and

senior people want to know you will ‘‘fit in.’’ Like much

of this whole process it’s a game. No one really cares if

you have a deep respect for your fellow man, but if you

can emulate it well under pressure it’s a difference that

makes no difference.

Be True to yourself

You are selling yourself, so obviously

you will be putting a positive spin on things. However,

this is a career, not a job. If you feel the job may really

not be what you want, then it’s important that you think

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411

that through. If in the interview you hear something

that sounds bad, ask about it. This does not have to

be confrontational; you can use phrases like ‘‘How does

that work out in practice?’’ and ‘‘What sort of flexibility

is there to choose the work?’’ when told you’re going to

be counting buttons for the first six months.

Do not sound as if you work for Accenture

Even if you do

work for Accenture or Arthur Andersen, you don’t want

to sound like you do. Avoid the sort of management

speak that resembles Dilbert cartoons. A common type

of interview question is of the form: ‘‘You find that

something has gone terribly wrong, what would you do

about it.’’ An Accenture answer is ‘‘I would see it as a

challenge that would allow me to work as a good team

player, as part of the global synergy’’; or perhaps you

might respond ‘‘I will grasp the opportunity to show

excellent leadership in integrity’’ which is interview

suicide.

This part may sound quite silly, but there is a grow-

ing trend for some universities to have formal coaching

in interview technique. In theory this should be very

useful. In theory. The real practice is rather scary. It

frustrates interviewers a lot to be faced with an obvi-

ously bright candidate who parrots clichés that some

consultant has fed into him. We say at the beginning

that you need to stand out, and given that the people

you are competing with may well include people from

your institution, it does you very little good.

By all means listen to these people, but take it with a

pinch of salt. When you know little about the process,

it’s easy to give too much weight to the few things you

get told.

Interview overlap

It is tempting to schedule lots of inter-

views as close together as possible, because travel does

412 Frequently Asked Questions In Quantitative Finance

eat into your budget. You should be very conservative

about the amount of time you allow for each interview.

It’s not easy to get a manager to speed up his process

because you want to get across town to talk to one of

his competitors. The worry about time, just like late-

ness, can reduce your effectiveness, so make sure this

doesn’t come up.

To be continued . . .

More

To find out more about this quant-job guide

please send either of us an email (Dominic

Connor, dominic@pauldominic.com, or Paul Wilmott,

paul@wilmott.com) or visit www.quantguides.com.

And if you are looking for a quant job, visit www.

pauldominic.com and send us your CV.

Document Outline

- Frequently Asked Questions In Quantitative Finance

- Contents

- Preface

- 1 Quantitative Finance Timeline

- 2 FAQs

- 3 The Most Popular Probability Distributions and Their Uses in Finance

- 4 Ten Different Ways to Derive Black–Scholes

- 5 Models and Equations

- 6 The Black–Scholes Formulæ and the Greeks

- 7 Common Contracts

- 8 Popular Quant Books

- 9 The Most Popular Search Words and Phrases on Wilmott.com

- 10 Brainteasers

- 11 Paul & Dominic’s Guide to Getting a Quant Job

Table of Contents

Frequently Asked Questions In Quantitative Finance

Contents

Preface

1 Quantitative Finance Timeline

2 FAQs

3 The Most Popular Probability Distributions and Their Uses in Finance

4 Ten Different Ways to Derive Black–Scholes

5 Models and Equations

6 The Black–Scholes Formulæ and the Greeks

7 Common Contracts

8 Popular Quant Books

9 The Most Popular Search Words and Phrases on Wilmott.com

10 Brainteasers

11 Paul & Dominic’s Guide to Getting a Quant Job