Introduction to the Mathematics of Finance: Arbitrage and Option Pricing PDF
Preview Introduction to the Mathematics of Finance: Arbitrage and Option Pricing
Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University Kenneth Ribet University of California, Berkeley Advisory Board: Colin C. Adams, Williams College Alejandro Adem, University of British Columbia Ruth Charney, Brandeis University Irene M. Gamba, The University of Texas at Austin Roger E. Howe, Yale University David Jerison, Massachusetts Institute of Technology Jeffrey C. Lagarias, University of Michigan Jill Pipher, Brown University Fadil Santosa, University of Minnesota Amie Wilkinson, University of Chicago Undergraduate Texts in Mathematics are generally aimed at third- and fourth- year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding. Forfurthervolumes: h ttp://www.springer.com/series/666 Steven Roman Introduction to the Mathematics of Finance Arbitrage and Option Pricing Second Edition Steven Roman Irvine, CA USA ISSN 0172-6056 ISBN 978-1-4614-3581-5 ISBN 978-1-4614-3582-2 (eBook) DOI 10.1007/978-1-4614-3582-2 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2012936125 MathematicsSubjectClassification(2010):91-01, 91B25 © Steven Roman 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To Donna Preface This book has one specific goal in mind, namely to determine a fair price for a financial derivative, such as a stock option. The problem can be put in a very simple context as follows. Imagine that you are an investor in precious metals, such as gold or silver. Consider a one-ounce nugget of gold whose current value is $")!!. The owner of this gold is willing to enter into a contract with you that gives you the right to buy the gold from him for $"(&! at any time during the next month. Obviously, the owner is not going to enter into such a contract for free, since he would lose $&! if you were to exercise your right immediately. But the owner will probably want more than $&!, since there is a definite possibility that the price of gold will exceed $")!! over the next month. On the other hand, there are limits to what you should be willing to pay for the right to buy the gold nugget. For instance, you would probably not pay $#&! for this right. Assuming that both parties are eager to speculate (that is, gamble) on the future price of gold, there may be a price that both you and the owner of the gold will accept in order to enter into this contract. The purpose of this book is to build mathematical models that determine a fair price for such a contract. In technical terms, the contract to buy the gold is a call option on gold, the buying price $"(&! is the strike price and the date one month from today is the expiration date of the call option. Since the value of the contract at any given moment depends solely on the value of gold, the option is called a derivative and the gold is the underlying asset for the derivative. Our goal is to determine a fair price for this and other derivative financial instruments. The intended audience of the book is upper division undergraduate or beginning graduate students in mathematics, finance or economics. Accordingly, no measure theory is used in this book. It is my hope that this book will be read by people with rather diverse backgrounds, some mathematical and some financial. Students of mathematics vii viii Preface may be well prepared in the ways of mathematical thinking but not so well prepared when it comes to matters related to finance (portfolios, stock options, forward contracts and so on). For these readers, I have included the necessary background in financial matters. On the other hand, students of finance and economics may be well versed in financial topics but not as mathematically minded as students of mathematics. Nevertheless, since the subject of this book is the mathematics of finance, I have not watered down the mathematics in any way (appropriate to the level of the book, of course). That is, I have endeavored to be mathematically rigorous at the appropriate level. However, for the benefit of those with less mathematical background, I have made the book as mathematically self-contained as possible. Probability theory is ever present in the area of mathematical finance and in this respect the book is completely self-contained. The Second Edition This second edition is a complete rewriting of the first edition and has been influenced greatly by my having taught a class based on the first edition for the last five years running. In particular, the topic organization has been changed significantly, making the book flow much more smoothly. Most proofs have been rewritten and many have been improved significantly. The material on probability has been condensed into fewer chapters. The discussion of options has been expanded, including some information about the history of options and the reason why option pricing has become so important. The discussion of pricing nonattainable alternatives has been expanded significantly. In particular, a new appendix has been added that contains proofs that the minimum dominating price of any nonattainable alternative is actually achieved by some dominating attainable alternative; that the maximum extension price is achieved by some nonnegative extension and that the minimum dominating price is equal to the maximum extension price. Finally, the material on the capital asset pricing model has been removed. Organization of the Book The book is organized as follows. The first chapter is devoted to the basics of stock options. In Chapter 2, we illustrate the technique of derivative asset pricing through the assumption of no arbitrage by pricing plain-vanilla forward contracts and discussing some simple issues related to option pricing, such as the put-call option parity formula. Chapters 3 and 4 provide a thorough introduction to the topics of discrete probability that are needed for the subject at hand. Chapter 3 is an elementary and quite standard introduction to discrete probability and will probably be familiar to those who have had a course in basic probability. On the other hand, Chapter 4 covers topics that are generally not covered in basic probability Preface ix classes, such as information structures, state trees, stochastic processes and martingales. This material is discussed only for discrete sample spaces and always keeping in mind that it is probably being seen by the reader for the first time. Chapter 5 is devoted to the theory of discrete-time pricing models, where we discuss portfolios, arbitrage trading strategies, martingale measures and the first and second fundamental theorems of asset pricing. This prepares the way for the discussion in Chapter 6 on the binomial pricing model. This chapter introduces the important topics of drift, volatility and random walks. In Chapter 7, we discuss the problem of pricing nonattainable alternatives in an incomplete discrete model. This chapter may be omitted if desired. Chapter 8 is devoted to optimal stopping times and American options. This chapter is perhaps a bit more mathematically challenging than the previous chapters and may also be omitted if desired. Chapter 9 introduces the very basics of continuous probability. We need the notions of convergence in distribution and the Central Limit Theorem so that we can take the limit of the binomial model as the length of the time periods goes to !. We perform this limiting process in Chapter 10 to get the famous Black– Scholes option pricing formula. In Appendix A, we give optional background information on convexity that is used in Chapter 6. As mentioned earlier, Appendix B supplies some proofs related to pricing nonattainable alternatives. A Word on Definitions Unlike many areas of mathematics, the subject of this book, namely, the mathematics of finance, does not have an extensive literature at the undergraduate level. Put more simply, there are very few undergraduate textbooks on the mathematics of finance. Accordingly, there has not been a lot of precedent with respect to setting down the basic theory at the undergraduate level, where pedagogy and use of intuition are (or should be) at a premium. One area in which this seems to manifest itself is the lack of terminology to cover certain situations. Therefore, on rare occasions I have felt it necessary to invent new terminology to cover a specific concept. Let me assure the reader that I have not done this lightly. It is not my desire to invent terminology for any other reason than as an aid to pedagogy. In any case, the reader will encounter a few definitions that I have labeled as nonstandard. This label is intended to convey the fact that the definition is not
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