Paul Wilmott On

Quantitative Finance

Paul Wilmott On

Quantitative Finance

Second Edition

www.wilmott.com

Copyright 2006 Paul Wilmott

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Library of Congress Cataloging-in-Publication Data

Wilmott, Paul.

Paul Wilmott on quantitative ﬁnance.—2nd ed.

p. cm.

Includes bibliographical references and index.

ISBN 13 978-0-470-01870-5 (cloth/cd : alk. paper)

ISBN 10 0-470-01870-4 (cloth/cd : alk. paper)

1. Derivative securities—Mathematical models. 2. Options (Finance)—

Mathematical models. 3. Options (Finance)—Prices—Mathematical models. I. Title.

HG6024.A3W555 2006

332.64 53—dc22

2005028317

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN-13: 978-0-470-01870-5 (HB)

ISBN-10: 0-470-01870-4 (HB)

Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India

Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

This book is printed on acid-free paper responsibly manufactured from sustainable forestry

in which at least two trees are planted for each one used for paper production.

In memory of Detlev Vogel

contents of volume one

Visual Basic Code

Prolog to the Second Edition

xxv

xxvii

PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASIC

THEORY OF DERIVATIVES; RISK AND RETURN

1

1

Products and Markets

5

2

Derivatives

25

3

The Random Behavior of Assets

55

4

Elementary Stochastic Calculus

71

5

The Black–Scholes Model

91

6

Partial Differential Equations

101

7

The Black–Scholes Formulae and the ‘Greeks’

109

8

Simple Generalizations of the Black–Scholes World

139

9

Early Exercise and American Options

151

10 Probability Density Functions and First-exit Times

169

11 Multi-asset Options

183

12 How to Delta Hedge

197

13 Fixed-income Products and Analysis: Yield, Duration and Convexity

225

14 Swaps

251

viii

contents

15 The Binomial Model

261

16 How Accurate is the Normal Approximation?

295

17 Investment Lessons from Blackjack and Gambling

301

18 Portfolio Management

317

19 Value at Risk

331

20 Forecasting the Markets?

343

21 A Trading Game

359

contents

contents of volume two

PART TWO EXOTIC CONTRACTS AND PATH DEPENDENCY

365

22 An Introduction to Exotic and Path-dependent Derivatives

367

23 Barrier Options

385

24 Strongly Path-dependent Derivatives

417

25 Asian Options

427

26 Lookback Options

445

27 Derivatives and Stochastic Control

453

28 Miscellaneous Exotics

461

29 Equity and FX Term Sheets

481

PART THREE FIXED-INCOME MODELING AND DERIVATIVES

507

30 One-factor Interest Rate Modeling

509

31 Yield Curve Fitting

525

32 Interest Rate Derivatives

533

33 Convertible Bonds

553

34 Mortgage-backed Securities

571

35 Multi-factor Interest Rate Modeling

581

36 Empirical Behavior of the Spot Interest Rate

595

37 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models

609

38 Fixed-income Term Sheets

627

PART FOUR CREDIT RISK

637

39 Value of the Firm and the Risk of Default

639

40 Credit Risk

649

ix

x

contents

41 Credit Derivatives

675

42 RiskMetrics and CreditMetrics

701

43 CrashMetrics

709

44 Derivatives **** Ups

731

contents

contents of volume three

PART FIVE ADVANCED TOPICS

745

45 Financial Modeling

749

46 Defects in the Black–Scholes Model

755

47 Discrete Hedging

763

48 Transaction Costs

783

49 Overview of Volatility Modeling

813

50 Deterministic Volatility Surfaces

833

51 Stochastic Volatility

853

52 Uncertain Parameters

869

53 Empirical Analysis of Volatility

881

54 Stochastic Volatility and Mean-variance Analysis

889

55 Asymptotic Analysis of Volatility

901

56 Volatility Case Study: The Cliquet Option

915

57 Jump Diffusion

927

58 Crash Modeling

939

59 Speculating with Options

953

60 Static Hedging

969

61 The Feedback Effect of Hedging in Illiquid Markets

989

62 Utility Theory

1005

63 More About American Options and Related Matters

1013

64 Advanced Dividend Modeling

1035

65 Serial Autocorrelation in Returns

1045

66 Asset Allocation in Continuous Time

1051

xix

xx

contents

67 Asset Allocation Under Threat of a Crash

1061

68 Interest-rate Modeling Without Probabilities

1077

69 Pricing and Optimal Hedging of Derivatives, the Non-probabilistic

Model Cont’d

1099

70 Extensions to the Non-probabilistic Interest-rate Model

1117

71 Modeling Inﬂation

1129

72 Energy Derivatives

1141

73 Real Options

1151

74 Life Settlements and Viaticals

1161

75 Bonus Time

1175

PART SIX NUMERICAL METHODS AND PROGRAMS

1189

76 Overview of Numerical Methods

1191

77 Finite-difference Methods for One-factor Models

1199

78 Further Finite-difference Methods for One-factor Models

1227

79 Finite-difference Methods for Two-factor Models

1253

80 Monte Carlo Simulation

1263

81 Numerical Integration

1285

82 Finite-difference Programs

1295

83 Monte Carlo Programs

1311

Appendix A All the Math You Need. . . and No More (An Executive Summary)

1317

Bibliography

1329

Index

1351

visual basic code

Implied volatility, Newton–Raphson

Cumulative distribution for Normal variable

The binomial method, European option

The binomial method, American option

Double knock-out barrier option, ﬁnite difference

Instalment knock-out barrier option, ﬁnite difference

Range Note, ﬁnite difference

Lookback, ﬁnite difference

Index Amortizing Rate Swap, ﬁnite difference

Cliquet option, uncertain volatility, ﬁnite difference

Optimization subroutine

Setting up ﬁnal condition, ﬁnite difference

Finite difference time loop, ﬁrst example

European option, ﬁnite difference, three dimensions

American option, ﬁnite difference, three dimensions

European or American option, ﬁnite difference, two dimensions

Upwind differencing, interest rate

LU decomposition

Matrix solution

Successive over relaxation

Successive over relaxation, early exercise

Jump condition for discrete dividends

Jump condition for path-dependent quantities

Two-factor explicit ﬁnite difference

Convertible bond constraint

Box–Muller

Cholesky factorization

Numerical integration, Monte Carlo

Halton number generation

Kolmogorov equation, explicit ﬁnite difference

Convertible bond time stepping fragment, explicit ﬁnite difference

American option, implicit ﬁnite difference

Parisian option, explicit ﬁnite difference

Passport option, explicit ﬁnite difference

130

131

286

290

490

493

497

501

634

923

983

1212

1213

1215

1219

1221

1225

1234

1235

1238

1246

1248

1249

1257

1257

1269

1276

1287

1290

1295

1297

1298

1299

1300

xxii

visual basic code

Chooser Passport option, explicit ﬁnite difference

Stochastic volatility, explicit ﬁnite difference

Uncertain volatility, gamma rule

Crash model, ﬁnite difference code fragment

Epstein–Wilmott model, ﬁnite difference

Risky bond, explicit ﬁnite difference

Basket option, Monte Carlo

Basket option, quasi Monte Carlo

American option, Monte Carlo

1301

1303

1304

1305

1305

1307

1311

1313

1314

prolog to the second edition

This book is a greatly updated and expanded version of the ﬁrst edition. The content continues

to reﬂect my own interests and prejudices, based on my skills, such as they are. In the period

between the ﬁrst and second editions, the ﬁnancial markets have expanded, the tools available

to the modeler have expanded, and my girth has expanded. On a personal basis I have spent as

much time being a practitioner in a hedge fund as being an independent researcher. Much of the

new material therefore represents both my desire as a scientist to build the best, most accurate

models, and my need as a practitioner to have models that are fast and robust and simple to

understand. As I said, this book is a very personal account of my areas of expertise. Since the

subject of quant ﬁnance has been galloping apace of late, I advise that you supplement this

book with the specialized books that I recommend throughout, and in particular those in the

quant library at the end.

I would like to re-thank those people I mentioned in the prolog to the ﬁrst edition: Areﬁn Huq,

Asli Oztukel, Bafkam Bim, Buddy Holly, Chris McCoy, Colin Atkinson, Daniel Bruno, Dave

Thomson, David Bakstein, David Epstein, David Herring, David Wilson, Edna Hepburn-Ruston,

Einar Holstad, Eli Lilly, Elisabeth Keck, Elsa Cortina, Eric Cartman, Fouad Khennach, Glen

Matlock, Henrik Rassmussen, Hyungsok Ahn, Ingrid Blauer, Jean Laidlaw, Jeff Dewynne, John

Lydon, John Ockendon, Karen Mason, Keesup Choe, Malcolm McLaren, Mauricio Bouabci,

Patricia Sadro, Paul Cook, Peter J¨ackel, Philip Hua, Philipp Sch¨onbucher, Phoebus Theologites,

Quentin Crisp, Rich Haber, Richard Arkell, Richard Sherry, Sam Ehrlichman, Sandra Maler,

Sara Statman, Simon Gould, Simon Ritchie, Stephen Jefferies, Steve Jones, Truman Capote,

Varqa Khadem, and Veronika Guggenbichler.

I would also like to thank the following people. My partners in various projects: Paul and

Jonathan Shaw at 7city, unequaled in their dedication to training and their imagination for new

projects. Also Riaz Ahmad and Seb Lleo who have helped make the Certiﬁcate in Quantitative

Finance so successful, and for taking some of the pressure off me; Everyone involved in the

magazine, especially Aaron Brown, Alan Lewis, Bill Ziemba, Caitlin Cornish, Dan Tudball, Ed

Lound, Ed Thorp, Elie Ayache, Espen Gaarder Haug, Graham Russel, Henriette Pr¨ast, Jenny

McCall, Kent Osband, Liam Larkin, Mike Staunton, Paula Soutinho and Rudi Bogni. I am

particularly fortunate and grateful that John Wiley & Sons have been so supportive in what

must sometimes seem to them rather wacky schemes; Thanks to Ron Henley, the best hedge

fund partner a quant could wish for, ‘It’s just a jump to the left. And then a step to the

right.’ And to John Morris of Fulcrum, interesting times; and to Nassim Nicholas Taleb for

interesting chats.

xxiv

prolog to the second edition

Thanks to, John, Grace, Sel and Stephen, for instilling in me their values: values which have

invariably served me well. And to Oscar and Zachary who kept me sane throughout many a

series of unfortunate events!

Finally, thanks to my number one fan, Andrea Estrella, from her number one fan, me.

ABOUT THE AUTHOR

Paul Wilmott’s professional career spans almost every aspect of mathematics and ﬁnance, in

both academia and in the real world. He has lectured at all levels, founded a magazine, the

leading website for the quant community, and a quant certiﬁcate program. He has managed

money as a partner in a very successful hedge fund. He lives in London, is married, and has

two sons. His only remaining dream is to get some sleep.

prolog to the second edition

More info about the particular meaning of an icon is contained in its ‘speech box.’

You will see this icon whenever a method is implemented on the CD.

xxv

PART ONE

mathematical and ﬁnancial

foundations; basic theory

of derivatives; risk and

return

The ﬁrst part of the book contains the fundamentals of derivatives theory and practice. We look

at both equity and ﬁxed income instruments. I introduce the important concepts of hedging and

no arbitrage, on which most sophisticated ﬁnance theory is based. We draw some insight from

ideas ﬁrst seen in gambling, and we develop those into an analysis of risk and return.

The assumptions, key concepts and results in Part One make up what is loosely known as the

‘Black–Scholes world,’ named for Fischer Black and Myron Scholes who, together with Robert

Merton, ﬁrst conceived them. Their original work was published in 1973, after some resistance

(the famous equation was ﬁrst written down in 1969). In October 1997 Myron Scholes and

Robert Merton were awarded the Nobel Prize for Economics for their work, Fischer Black

having died in August 1995. The New York Times of Wednesday, 15th October 1997 wrote:

‘Two North American scholars won the Nobel Memorial Prize in Economic Science yesterday

for work that enables investors to price accurately their bets on the future, a breakthrough

that has helped power the explosive growth in ﬁnancial markets since the 1970’s and plays a

profound role in the economics of everyday life.’1

Part One is self contained, requiring little knowledge of ﬁnance or any more than elementary

calculus.

Chapter 1: Products and Markets An overview of the workings of the ﬁnancial markets and

their products. A chapter such as this is obligatory. However, my readers will fall into one of

two groups. Either they will know everything in this chapter and much, much more besides.

Or they will know little, in which case what I write will not be enough.

1

We’ll be hearing more about these two in Chapter 44 on ‘Derivatives **** Ups.’

2

Part One mathematical and ﬁnancial foundations

Chapter 2: Derivatives An introduction to options, options markets, market conventions.

Deﬁnitions of the common terms, simple no arbitrage, put-call parity and elementary trading

strategies.

Chapter 3: The Random Behavior of Assets An examination of data for various ﬁnancial

quantities, leading to a model for the random behavior of prices. Almost all of sophisticated

ﬁnance theory assumes that prices are random, the question is how to model that randomness.

Chapter 4: Elementary Stochastic Calculus We’ll need a little bit of theory for manipulating

our random variables. I keep the requirements down to the bare minimum. The key concept is

Itˆo’s lemma which I will try to introduce in as accessible a manner as possible.

Chapter 5: The Black–Scholes Model I present the classical model for the fair value of options

on stocks, currencies and commodities. This is the chapter in which I describe delta hedging

and no arbitrage and show how they lead to a unique price for an option. This is the foundation

for most quantitative ﬁnance theory and I will be building on this foundation for much, but by

no means all, of the book.

Chapter 6: Partial Differential Equations Partial differential equations play an important role

in most physical applied mathematics. They also play a role in ﬁnance. Most of my readers

trained in the physical sciences, engineering and applied mathematics will be comfortable with

the idea that a partial differential equation is almost the same as ‘the answer,’ the two being

separated by at most some computer code. If you are not sure of this connection I hope that

you will persevere with the book. This requires some faith on your part; you may have to read

the book through twice: I have necessarily had to relegate the numerics, the real ‘answer,’ to

the last few chapters.

Chapter 7: The Black–Scholes Formulae and the ‘Greeks’ From the Black–Scholes partial

differential equation we can ﬁnd formulae for the prices of some options. Derivatives of option

prices with respect to variables or parameters are important for hedging. I will explain some

of the most important such derivatives and how they are used.

Chapter 8: Simple Generalizations of the Black–Scholes World Some of the assumptions

of the Black–Scholes world can be dropped or stretched with ease. I will describe several of

these. Later chapters are devoted to more extensive generalizations.

Chapter 9: Early Exercise and American Options Early exercise is of particular importance

ﬁnancially. It is also of great mathematical interest. I will explain both of these aspects.

Chapter 10: Probability Density Functions and First-exit Times The random nature of ﬁnancial quantities means that we cannot say with certainty what the future holds in store. For that

reason we need to be able to describe that future in a probabilistic sense.

Chapter 11: Multi-asset Options Another conceptually simple generalization of the basic

Black–Scholes world is to options on more than one underlying asset. Theoretically simple,

this extension has its own particular problems in practice.

Chapter 12: How to Delta Hedge Not everyone believes in no arbitrage, the absence of free

lunches. In this chapter we see how to proﬁt if you have a better forecast for future volatility

than the market.

mathematical and ﬁnancial foundations Part One

Chapter 13: Fixed-income Products and Analysis: Yield, Duration and Convexity This

chapter is an introduction to the simpler techniques and analyses commonly used in the market.

In particular I explain the concepts of yield, duration and convexity. In this and the next chapter

I assume that interest rates are known, deterministic quantities.

Chapter 14: Swaps Interest-rate swaps are very common and very liquid. I explain the basics

and relate the pricing of swaps to the pricing of ﬁxed-rate bonds.

Chapter 15: The Binomial Model One of the reasons that option theory has been so successful

is that the ideas can be explained and implemented very easily with no complicated mathematics.

The binomial model is the simplest way to explain the basic ideas behind option theory using

only basic arithmetic. That’s a good thing, right? Yes, but only if you bear in mind that the

model is for demonstration purposes only, it is not the real thing. As a model of the ﬁnancial

world it is too simplistic, as a concept for pricing it lacks the elegance that makes other methods

preferable, and as a numerical scheme it is prehistoric. Use once and then throw away, that’s

my recommendation.

Chapter 16: How Accurate is the Normal Approximation? One of the major assumptions of

ﬁnance theory is that returns are Normally distributed. In this chapter we take a look at why

we make this assumption, and how good it really is.

Chapter 17: Investment Lessons from Blackjack and Gambling We draw insights and inspiration from the not-unrelated world of gambling to help us in the treatment of risk, return, and

money/risk management.

Chapter 18: Portfolio Management If you are willing to accept some risk how should you

invest? I explain the classical ideas of Modern Portfolio Theory and the Capital Asset Pricing

Model.

Chapter 19: Value at Risk How risky is your portfolio? How much might you conceivably

lose if there is an adverse market move? These are the topics of this chapter.

Chapter 20: Forecasting the Markets? Although almost all sophisticated ﬁnance theory

assumes that assets move randomly, many traders rely on technical indicators to predict the

future direction of assets. These indicators may be simple geometrical constructs of the asset

price path or quite complex algorithms. The hypothesis is that information about short-term

future asset price movements are contained within the past history of prices. All traders use

technical indicators at some time. In this chapter I describe some of the more common techniques.

Chapter 21: A Trading Game Many readers of this book will never have traded anything more

sophisticated than baseball cards. To get them into the swing of the subject from a practical

point of view I include some suggestions on how to organize your own trading game based on

the buying and selling of derivatives. I had a lot of help with this chapter from David Epstein

who has been running such games for several years.

3

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