Paul Wilmott On Quantitative Finance

Paul Wilmott On Quantitative Finance

Paul Wilmott On Quantitative Finance Second Edition www.wilmott.com

Copyright 2006 Paul Wilmott Published by 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.wiley.com 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. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. 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, 42 McDougall Street, 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 books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. 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 ix 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

x contents 41 Credit Derivatives 675 42 RiskMetrics and CreditMetrics 701 43 CrashMetrics 709 44 Derivatives **** Ups 731

contents xix 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

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 130 Cumulative distribution for Normal variable 131 The binomial method, European option 286 The binomial method, American option 290 Double knock-out barrier option, finite difference 490 Instalment knock-out barrier option, finite difference 493 Range Note, finite difference 497 Lookback, finite difference 501 Index Amortizing Rate Swap, finite difference 634 Cliquet option, uncertain volatility, finite difference 923 Optimization subroutine 983 Setting up final condition, finite difference 1212 Finite difference time loop, first example 1213 European option, finite difference, three dimensions 1215 American option, finite difference, three dimensions 1219 European or American option, finite difference, two dimensions 1221 Upwind differencing, interest rate 1225 LU decomposition 1234 Matrix solution 1235 Successive over relaxation 1238 Successive over relaxation, early exercise 1246 Jump condition for discrete dividends 1248 Jump condition for path-dependent quantities 1249 Two-factor explicit finite difference 1257 Convertible bond constraint 1257 Box Muller 1269 Cholesky factorization 1276 Numerical integration, Monte Carlo 1287 Halton number generation 1290 Kolmogorov equation, explicit finite difference 1295 Convertible bond time stepping fragment, explicit finite difference 1297 American option, implicit finite difference 1298 Parisian option, explicit finite difference 1299 Passport option, explicit finite difference 1300