Paul wilmott on quantitative finance vol 1 3, 2nd ed

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