Mathematical Modeling and Methods of Option Pricing

Mathematical Modeling and Methods of Option Pricing

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Mathematical Modeling and Methods of Option Pricing Lishang Jiang Tongji University, China Translated by Canguo Li \Hp World Scientific

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. First published in Chinese in 2003 by Higher Education Press. MATHEMATICAL MODELING AND METHODS OF OPTION PRICING Copyright 2005 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN 981-256-369-5 Printed in Singapore by World Scientific Printers (S) Pte Ltd

Preface This text is based on the lecture notes of a graduate-level course "Mathematical Theory of Financial Derivatives", which I gave first at Department of Mathematics, University of Iowa, U.S.A., and later at Department of Applied Mathematics, Tongji University, Shanghai, China. In this course, I intended to present a systematic and in-depth introduction to the Black- Scholes-Merton's option pricing theory from the perspective of partial differential equation theory. It is the author's hope that this text may contribute to filling a gap in the existing literature. Option is a financial derivative. Therefore its price depends on the underlying asset's price movement. In the case of the continuous time model, the movement of the underlying asset's price can be described by a stochastic differential equation. Consequently, according to the idea of Black and Scholes, the option price can be modeled as a terminal-boundary problem for a partial differential equation (PDE). Therefore it is reasonable to adopt the existing theory and methods of PDE as a fundamental approach to the study of the option pricing theory. This includes establishing the PDE models for various types of options, deriving the pricing formulas as solutions of the corresponding PDE problems, making qualitative and in-depth analysis of the structure of the option price, and designing efficient algorithms for solving option pricing problems from the viewpoint of numerical solutions of PDE problems. As a textbook for graduate students in applied mathematics, the depth and scope of this book must be appropriate. In order to limit the prerequisites, we tried our best to make this text self-contained when topics of modern mathematics are involved. In fact, we only assume a basic knowledge of calculus, linear algebra, elementary probability theory, and mathematical physics equations. When topics of stochastic analysis, numerical V

vi Mathematical Modeling and Methods of Option Pricing methods of PDE and free boundary problems are encountered in the text, only a brief presentation of the basic concepts and results is given. That is, the conclusion is stated, the basic idea of the proof is explained, but the details are not presented, and references are provided to guide the reader for further study. Furthermore, we restrict our discussion to those financial topics whose option pricing can be formulated as a PDE problem via the A-hedging technique, to illustrate the basic idea of the PDE approach. The book is organized as follows: Fundamental concepts of financial derivatives are introduced in Chapter 1, and basics of stochastic analysis are covered in Chapter 4. Chapters 2, 3, and Chapter 5 form the core of this book. In these three chapters, in addition to presenting the mathematical models, algorithms and formulas of option pricing, we expound the basic ideas behind the Black-Scholes-Merton option pricing theory from several perspectives and levels: starting from the arbitrage-free assumption, via the A-hedging technique, put the investors in a risk-neutral world where all risky assets have the same expected return the risk-free interest rate, then option as a contingent claim is given a fair market price independent of the risk preference of each individual investor. In the case of the continuous time model, the pricing formula for European vanilla option is the well-known Black-Scholes formula. In Chapter 6 and 7.7, we study American option pricing problems. Since American option offers early exercise, the holder needs to select the optimal exercise strategy to get optimal returns. Mathematically, this is modeled as a free boundary problem, where the free boundary is the optimal exercise boundary of an American option. Since it is a nonlinear problem, explicit closed form solution does not exist in general, hence qualitative analysis and quantitative numerical solution play an important role. Naturally, American option pricing as free boundary problem becomes the central topic and apex of the book, where the power of the theory and methods of PDE are fully demonstrated. In Chapters 7-9, we study the models and solution methods for multi-asset options and various types of path-dependent options. New multi-dimensional PDE pricing models are introduced in those chapters, which include not only the multi-dimensional Black-Scholes equation, but also various types of terminal-boundary problems for hyperparabolic equations. In addition to studying various methods of numerical solution, we are particularly interested in the possibility of reducing a multi-dimensional problem to a one-dimensional problem. Finally, in Chapter 10, we study the inverse problem of option pricing, that is, how to recover the volatility of the underlying asset from the information of its option market. It is called the

Preface vii implied volatility problem. We first derive the Dupire's method from the PDE viewpoint, and then proceed to work in the optimal control framework, thus obtain a system of partial differential equations and propose a well-posed algorithm for recovering the implied volatility. I would like to thank my colleagues and students at Financial Mathematics Group in Tongji University, who have read earlier versions of the manuscript and made helpful suggestions. My special thanks go to Mrs. Xiaoping Zhang, my editor of the original Chinese edition of this text at the Higher Education Press(Beijing), for her expertise and enthusiastic work, and to Dr. Canguo Li for his elegant and painstaking translation work. The publication of the English edition of this text would not be possible without their efforts. Lishang Jiang Tongji University, 2004

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Contents Preface 1. Risk Management and Financial Derivatives 1 1.1 Risk and Risk Management 1 1.2 Forward Contracts and Futures 2 1.3 Options 3 1.4 Option Pricing 5 1.5 Types of Traders 6 2. Arbitrage-Free Principle 9 2.1 Financial Market and Arbitrage-Free Principle 9 2.2 European Option Pricing and Call-Put Parity 13 2.3 American Option Pricing and Early Exercise 15 2.4 Dependence of Option Pricing on the Strike Price 19 3. Binomial Tree Methods Discrete Models of Option Pricing 25 3.1 An Example 25 3.2 One-Period and Two-State Model 26 3.3 Binomial Tree Method of European Option Pricing (I) Non-Dividend-Paying 32 3.4 Binomial Tree Method of European Options (II) Dividend-Paying 39 3.5 Binomial Tree Method of American Option Pricing 42 3.6 Call-Put Symmetry 48 v 4. Brownian Motion and I to Formula 55 ix

x Mathematical Modeling and Methods of Option Pricing 4.1 Random Walk and Brownian Motion 55 4.2 Continuous Models of Asset Price Movement 58 4.3 Quadratic Variation Theorem 61 4.4 Ito Integral 64 4.5 Ito Formula 66 5. European Option Pricing Black-Scholes Formula 73 5.1 History 73 5.2 Black-Scholes Equation 74 5.3 Black-Scholes Formula 79 5.4 Generalized Black-Scholes Model (I) Dividend-Paying Options 82 5.5 Generalized Black-Scholes Model (II) Binary Options and Compound Options 88 5.6 Numerical Methods (I) Finite Difference Method... 93 5.7 Numerical Methods (II) Binomial Tree Method and Finite Difference Method 100 5.8 Properties of European Option Price 104 5.9 Risk Management 107 6. American Option Pricing and Optimal Exercise Strategy 113 6.1 Perpetual American Option 113 6.2 Models of American Options 124 6.3 Decomposition of American Options 127 6.4 Properties of American Option Price 134 6.5 Optimal Exercise Boundary 146 6.6 Numerical Method (I) Finite Difference Method... 165 6.7 Numerical Methods(II) Line Method 178 6.8 Other Types of American Options 189 7. Multi-Asset Option Pricing 201 7.1 Stochastic Models of Multi-Assets Pricing 201 7.2 Black-Scholes Equation 203 7.3 Black-Scholes Formula 204 7.4 Rainbow Options 210 7.5 Basket Options 216 7.6 Quanto Options 218 7.7 American Multi-Asset Options 222

Contents xi 8. Path-Dependent Options (I) Weakly Path-Dependent Options 247 8.1 Barrier Options 247 8.2 Time-Dependent Barrier Options 255 8.3 Reset Options 260 8.4 Modified Barrier Options 263 9. Path-Dependent Options (II) Strongly Path-Dependent Options 275 9.1 Asian Options 275 9.2 Model and Simplification 277 9.3 Valuation Formula for European-Style Geometric Average Asian Option 284 9.4 Call-Put Parities for Asian Options 288 9.5 Lookback Option 292 9.6 Numerical Methods 301 10. Implied Volatility 311 10.1 Preliminaries 311 10.2 Dupire Method 313 10.3 Optimal Control Method 315 10.4 Numerical Method 320 Bibliography 323 Index 327