Published 2003 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
Telephone (+44) 1243 779777
Email (for orders and customer service enquiries): cs-books@wiley.co.uk
Visit our Home Page on www.wileyeurope.com or www.wiley.com
Copyright
c
2003 Peter James
All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system
or transmitted in any form or by any means, electronic, mechanical, photocopying, recording,
scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988
or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham
Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher.
Requests to the Publisher should be addressed to the Permissions Department, John Wiley &
Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed
to permreq@wiley.co.uk, or faxed to (+44) 1243 770620.
This publication is designed to provide accurate and authoritative information in regard to the
subject matter covered. It is sold on the understanding that the Publisher is not engaged in
rendering professional services. If professional advice or other expert assistance is required, the
services of a competent professional should be sought.
Other Wiley Editorial Offices
John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA
Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA
Wiley-VCH Verlag GmbH, Boschstr. 12, D-69469 Weinheim, Germany
John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia
John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809
John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1
Wiley also publishes its book in a variety of electronic formats. Some content that appears in print
may not be available in electronic books.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 0-471-49289-2
Typeset in 10/12pt Times by TechBooks, New Delhi, India
Printed and bound in Great Britain by TJ International Ltd, Padstow, Cornwall, UK
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.
To Vivien
Contents
Preface xiii
PART 1 ELEMENTS OF OPTION THEORY 1
1 Fundamentals 3
1.1 Conventions 3
1.2 Arbitrage 7
1.3 Forward contracts 8
1.4 Futures contracts 11
2 Option Basics 15
2.1 Payoffs 15
2.2 Option prices before maturity 16
2.3 American options 18
2.4 Put–call parity for american options 20
2.5 Combinations of options 22
2.6 Combinations before maturity 26
3 Stock Price Distribution 29
3.1 Stock price movements 29
3.2 Properties of stock price distribution 30
3.3 Infinitesimal price movements 33
3.4 Ito’s lemma 34
4 Principles of Option Pricing 35
4.1 Simple example 35
4.2 Continuous time analysis 38
4.3 Dynamic hedging 44
4.4 Examples of dynamic hedging 46
4.5 Greeks 48
Contents
5 The Black Scholes Model 51
5.1 Introduction 51
5.2 Derivation of model from expected values 51
5.3 Solutions of the Black Scholes equation 52
5.4 Greeks for the Black Scholes model 53
5.5 Adaptation to different markets 56
5.6 Options on forwards and futures 58
6 American Options 63
6.1 Black Scholes equation revisited 63
6.2 Barone-Adesi and Whaley approximation 65
6.3 Perpetual puts 68
6.4 American options on futures and forwards 69
PART 2 NUMERICAL METHODS 73
7 The Binomial Model 75
7.1 Random walk and the binomial model 75
7.2 The binomial network 77
7.3 Applications 80
8 Numerical Solutions of the Black Scholes Equation 87
8.1 Finite difference approximations 87
8.2 Conditions for satisfactory solutions 89
8.3 Explicit finite difference method 91
8.4 Implicit finite difference methods 93
8.5 A worked example 97
8.6 Comparison of methods 100
9 Variable Volatility 105
9.1 Introduction 105
9.2 Local volatility and the Fokker Planck equation 109
9.3 Forward induction 113
9.4 Trinomial trees 115
9.5 Derman Kani implied trees 118
9.6 Volatility surfaces 123
10 Monte Carlo 125
10.1 Approaches to option pricing 125
10.2 Basic Monte Carlo method 127
10.3 Random numbers 130
10.4 Practical applications 133
10.5 Quasi-random numbers 135
10.6 Examples 139
viii
Contents
PART 3 APPLICATIONS: EXOTIC OPTIONS 143
11 Simple Exotics 145
11.1 Forward start options 145
11.2 Choosers 147
11.3 Shout options 148
11.4 Binary (digital) options 149
11.5 Power options 151
12 Two Asset Options 153
12.1 Exchange options (Margrabe) 153
12.2 Maximum of two assets 155
12.3 Maximum of three assets 156
12.4 Rainbow options 158
12.5 Black Scholes equation for two assets 158
12.6 Binomial model for two asset options 160
13 Currency Translated Options 163
13.1 Introduction 163
13.2 Domestic currency strike (compo) 163
13.3 Foreign currency strike: fixed exchange rate (quanto) 165
13.4 Some practical considerations 167
14 Options on One Asset at Two Points in Time 169
14.1 Options on options (compound options) 169
14.2 Complex choosers 173
14.3 Extendible options 173
15 Barriers: Simple European Options 177
15.1 Single barrier calls and puts 177
15.2 General expressions for single barrier options 180
15.3 Solutions of the Black Scholes equation 181
15.4 Transition probabilities and rebates 182
15.5 Binary (digital) options with barriers 183
15.6 Common applications 184
15.7 Greeks 186
15.8 Static hedging 187
16 Barriers: Advanced Options 189
16.1 Two barrier options 189
16.2 Outside barrier options 190
16.3 Partial barrier options 192
16.4 Lookback options 193
16.5 Barrier options and trees 195
ix
Contents
17 Asian Options 201
17.1 Introduction 201
17.2 Geometric average price options 203
17.3 Geometric average strike options 206
17.4 Arithmetic average options: lognormal solutions 206
17.5 Arithmetic average options: Edgeworth expansion 209
17.6 Arithmetic average options: geometric conditioning 211
17.7 Comparison of methods 215
18 Passport Options 217
18.1 Option on an investment strategy (trading option) 217
18.2 Option on an optimal investment strategy (passport option) 220
18.3 Pricing a passport option 222
PART 4 STOCHASTIC THEORY 225
19 Arbitrage 227
19.1 Simplest model 227
19.2 The arbitrage theorem 229
19.3 Arbitrage in the simple model 230
20 Discrete Time Models 233
20.1 Essential jargon 233
20.2 Expectations 234
20.3 Conditional expectations applied to the one-step model 235
20.4 Multistep model 237
20.5 Portfolios 238
20.6 First approach to continuous time 240
21 Brownian Motion 243
21.1 Basic properties 243
21.2 First and second variation of analytical functions 245
21.3 First and second variation of Brownian motion 246
22 Transition to Continuous Time 249
22.1 Towards a new calculus 249
22.2 Ito integrals 252
22.3 Discrete model extended to continuous time 255
23 Stochastic Calculus 259
23.1 Introduction 259
23.2 Ito’s transformation formula (Ito’s lemma) 260
23.3 Stochastic integration 261
23.4 Stochastic differential equations 262
23.5 Partial differential equations 265
23.6 Local time 266
x
Contents
23.7 Results for two dimensions 269
23.8 Stochastic control 271
24 Equivalent Measures 275
24.1 Change of measure in discrete time 275
24.2 Change of measure in continuous time: Girsanov’s theorem 277
24.3 Black Scholes analysis 280
25 Axiomatic Option Theory 283
25.1 Classical vs. axiomatic option theory 283
25.2 American options 284
25.3 The stop–go option paradox 287
25.4 Barrier options 290
25.5 Foreign currencies 293
25.6 Passport options 297
Mathematical Appendix 299
A.1 Distributions and integrals 299
A.2 Random walk 309
A.3 The Kolmogorov equations 314
A.4 Partial differential equations 318
A.5 Fourier methods for solving the heat equation 322
A.6 Specific solutions of the heat equation (Fourier methods) 325
A.7 Green’s functions 329
A.8 Fokker Planck equations with absorbing barriers 336
A.9 Numerical solutions of the heat equation 344
A.10 Solution of finite difference equations by LU decomposition 347
A.11 Cubic spline 349
A.12 Algebraic results 351
A.13 Moments of the arithmetic mean 353
A.14 Edgeworth expansions 356
Bibliography and References 361
Commentary 361
Books 363
Papers 364
Index 367
xi
Preface
Options are financial instruments which are bought and sold in a market place. The people
who do it well pocket large bonuses; companies that do it badly can suffer staggering losses.
These are intensely practical activities and this is a technical book for practical people working
in the industry. While writing it I have tried to keep a number of issues and principles to the
forefront:
r
The emphasis is on developing the theory to the point where it is capable of yielding a
numerical answer to a pricing question, either through a formula or through a numerical
procedure. In those places where the theory is fairly abstract, as in the sections explaining
stochastic calculus, the path back to reality is clearly marked.
r
An objective of the book is to demystify option theory. An essential part of this is giving
explanations and derivations in full. I have (almost) completely avoided the “it can be shown
that . . . ” syndrome, except for the most routine algebraic steps, since this can be very time-
wasting and frustrating for the reader. No quant who values his future is going to just lift a
formula or set of procedures from a textbook and apply them without understanding where
they came from and what assumptions went into them.
r
It is a sad fact that readers do not start at the beginning of a textbook and read every page until
they get to the end – at least not the people I meet in the derivatives market. Practitioners are
usually looking for something specific and want it quickly. I have therefore tried to make
the book reasonably easy to dip in and out of. This inevitably means a little duplication and
a lot of signposts to parts of the book where underlying principles are explained.
r
Option theory can be approached from several different directions, using different mathe-
matical techniques. An option price can be worked out by solving a differential equation or
by taking a risk-neutral expectation; results can be obtained by using formulas or trees or
by integrating numerically or by using finite difference methods; and the theoretical under-
pinnings of option theory can be explained either by using conventional, classical statistical
methods or by using axiomatic probability theory and stochastic calculus. This book demon-
strates that these are all saying the same thing in different languages; there is only one option
theory, although several branches of mathematics can be used to describe it. I have taken
pains to be unpartisan in describing techniques; the best technique is the one that produces
the best answer, and this is not the same for all options.
The reader of this book might have no previous knowledge of option theory at all, or he
might be an accomplished quant checking an obscure point. He might be a student looking
Không có nhận xét nào:
Đăng nhận xét