Risk neutral pricing and financial mathematics a primer

RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS A Primer Peter M Knopf John L Teall AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier Academic Press is an imprint of Elsevier 525 B Street, Suite 1800, San Diego, CA 92101-4495, USA 225 Wyman Street, Waltham, MA 02451, USA The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK Copyright r 2015 Elsevier Inc All rights reserved No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the Publisher Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein) Notices Knowledge and best practice in this field are constantly changing As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary Practitioners and researchers may always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein ISBN 978-0-12-801534-6 Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library For information on all Academic Press publications visit our website at http://store.elsevier.com/ Publisher: Nikki Levy Acquisition Editor: J Scott Bentley Editorial Project Manager: Susan Ikeda Production Project Manager: Melissa Read Designer: Mark Rogers Printed and bound in the United States of America Dedication to the lovely and supportive ladies in our lives, Anne and Arli, and the many others who shall remain nameless here About the Authors New York University, Cornell University, Pace University, Dublin City University, University of Melbourne as well as other institutions in the United States, Europe and Asia His primary areas of research and publication have been related to corporate finance and financial institutions He is the author/co-author of books Dr Teall obtained his Ph.D from the Stern School of Business at New York University and is a former member of the American Stock Exchange Dr Teall has consulted with numerous financial institutions including Deutsche Bank, Goldman Sachs, National Westminster Bank and Citicorp Peter Knopf obtained his Ph.D from Cornell University and subsequently taught at Texas A&M University and Rutgers University He is currently Professor of Mathematics at Pace University He has numerous research publications in both pure and applied mathematics His recent research interests have been in the areas of difference equations and stochastic delay equation models for pricing securities John L Teall is a visiting professor of finance at the LUISS Business School, LUISS Guido Carli University He was Jackson Tai ’72 Professor of Practice at Rensselaer Polytechnic Institute, and also served on the faculties of ix Preface It’s no easy task to write a book that makes quantitative finance seem easy Writing such a book with two bull-headed authors can be a real battle, at least at times But, now that we’re finished, or at least ready to begin pondering the book’s eventual second edition, we both are surprised and quite pleased with the result Our various pre-publication readers claim to like the chapters, and seem to believe that the book well suits our intended audience (we’ll get to this shortly) Competition, skepticism, and stubbornness seem to have a place in a joint writing endeavor Somewhere in the middle of an early draft of Chapter 7, a passage appeared that sparked the following exchange between the coauthors of this book, altered somewhat to soften the language: groups of readers: ambitious financial mathematics students with relatively modest mathematics backgrounds, say with one or two calculus courses along with a course or two in linear mathematics and another in statistics While a stronger math background would only help the reader, we have provided a brief review (Chapter and much of Chapter 2) for readers who are a bit rusty or have gaps in their undergraduate preparation We believe that readers with technical backgrounds such as in mathematics and engineering as well as readers with nontechnical backgrounds such as studies in undergraduate finance and economics will be able to follow and benefit from the presentations of this book After all, the book is still focused on finance, and not even most mathematicians will have seen at least some of the mathematics used here and in the financial industry more generally We also believe that the large number of solved end-ofchapter exercises and online materials will boost reader comfort with the material and deepen the learning experience In order to keep the price of the textbook down, certain materials such as additional readings, proofs and verifications have been placed on the authors’ website at http://www jteall.com/books.html (click the “Resources” link), and are also available online at the Elsevier textbook site (see overleaf for address) These readings, proofs and verifications offer a wealth of insight into many of the processes used to model financial instruments and to the mathematics underlying valuation and hedging For example, in the This paragraph doesn’t mean anything! That’s because you’re dense (exasperated) “You’re such a pinhead Why did I ever agree to write this book?” Actually, we knew from the start exactly why we needed to write this book We’ve seen a nice assortment of books in financial mathematics, written by mathematicians, that are nicely suited to readers with solid technical backgrounds in engineering, the physical sciences and math There are a smaller number of financial math books geared towards undergraduates and MBA students who really aren’t interested in the stochastic processes underlying models for pricing derivatives and fixed income instruments Your coauthors sought to fill a rapidly growing gap between these two xi xii PREFACE additional readings for Chapter 6, we derive the solution to the Black-Scholes differential equation Of similar importance to Chapter 8, the text website contains a derivation of the Vasicek single factor model for pricing bonds These website readings are presented to improve the balance between rigor and pedagogy in order for the student to gain as much intuition and understanding of the most important derivations in quantitative finance without losing the ability to apply the important results Introduction to Risk-Neutral Pricing and Financial Mathematics: A Primer seeks to introduce financial mathematics to students in quantitative finance, financial engineering, actuarial science and computational finance The central theme of the book is risk-neutral (martingale) pricing, though it does venture into a number of other areas The text endeavors to provide a foundation in financial mathematics for use in an introductory financial engineering, financial modeling, or financial mathematics course, primarily to students whose math backgrounds not extend much beyond two semesters of calculus, linear math and statistics Students with stronger mathematics preparations may find this book somewhat easier to follow, but most will not find significant amounts of material repeating what they have seen elsewhere The book will present and apply topics such as stochastic processes, equivalent martingales, RadonÀ Nikodym derivatives, and stochastic calculus, but it will start from the perspective that reader mathematical training does not extend beyond basic calculus, linear math and statistics Writing a textbook is never a solo effort, or even an effort undertaken by only two coauthors As have most authors, we benefitted immensely from the help from many people, including friends, family, students, colleagues and practitioners For example, we owe special thanks to some of our favorite students, including Xiao (Sean) Tang, T.J Wu, Alban Leung, Haoyu Li, Victor Shen, and Matthew Spector A number of our colleagues were also most helpful, including Professors Steven Kalikow and Maury Bramson for valuable input concerning stochastic processes and Professor Matthew Hyatt for programming and use of software tools We are particularly grateful to Hong Tu Yan for his careful reading of the text and competent coauthoring of supplemental materials We thank Ryan Cummings for her delightful cartoon appearing on the text companion website, and Ying Sue Huang and Imon Palit for their thoughtful comments And then there’s Chris Stone, always there when we need help, and always going above and beyond the call of duty, such as for her 11th hour sacrifice of vacation time to ensure that we met our manuscript submission deadline Of course, we are also grateful to Scott Bentley, Melissa Read, Susan Ikeda, and Mckenna Bailey at Academic Press/Elsevier And finally, we enjoyed the unfailing support and encouragement from Arli Epton and the unique culinary skills of Anne Teall Emily Teall’s unbounded interest in everything related to finance boosted our morale as well Unfortunately, a number of errors will surely persist even after the book goes into production We apologize sincerely for these errors, and we blame our parents for them Additional readings, proofs and verifications can be found on the companion website Go online to access it at: http://booksite.elsevier.com/9780128015346 C H A P T E R Preliminaries and Review 1.1 FINANCIAL MODELS A model can be characterized as an artificial structure describing the relationships among variables or factors Practically all of the methodology in this book is geared toward the development and implementation of financial models to solve financial problems For example, valuation models provide a foundation for investment decision-making and models describing stochastic processes provide an important tool to account for risk in decision-making The use of models is important in finance because “real world” conditions that underlie financial decisions are frequently extraordinarily complicated Financial decision-makers frequently use existing models or construct new ones that relate to the types of decisions they wish to make Models proposing decisions that ought to be made are called normative models.1 The purpose of models is to simulate or behave like real financial situations When constructing financial models, analysts exclude the “real world” conditions that seem to have little or no effect on the outcomes of their decisions, concentrating on those factors that are most relevant to their situations In some instances, analysts may have to make unrealistic assumptions in order to simplify their models and make them easier to analyze After simple models have been constructed with what may be unrealistic assumptions, they can be modified to match more closely “real world” situations A good financial model is one that accounts for the major factors that will affect the financial decision (a good model is complete and accurate), is simple enough for its use to be practical (inexpensive to construct and easy to understand), and can be used to predict actual outcomes A model is not likely to be useful if it is not able to project an outcome with an acceptable degree of accuracy Completeness and simplicity may directly conflict with one another The financial analyst must determine the appropriate trade-off between completeness and simplicity in the model he wishes to construct In finance, mathematical models are usually the easiest to develop, manipulate, and modify These models are usually adaptable to computers and electronic spreadsheets Mathematical models are obviously most useful for those comfortable with math; the primary purpose of this book is to provide a foundation for improving the quantitative preparation of the less mathematically oriented analyst Other models used in finance include those based on graphs and those involving simulations However, these models are often based on or closely related to mathematical models P.M Knopf & J.L Teall: Risk Neutral Pricing and Financial Mathematics: A Primer DOI: http://dx.doi.org/10.1016/B978-0-12-801534-6.00001-1 © 2015 Elsevier Inc All rights reserved PRELIMINARIES AND REVIEW The concepts of market efficiency and arbitrage are essential to the development of many financial models Market efficiency is the condition in which security prices fully reflect all available information Such efficiency is more likely to exist when wealth-maximizing market participants can instantaneously and costlessly execute transactions as information is revealed Transactions costs, irrationality, and poor execution systems reduce efficiency Arbitrage, in its simplest scenario, is the simultaneous purchase and sale of the same asset, or more generally, the nearly simultaneous purchase and sale of assets generating nearly identical cash flow structures In either case, the arbitrageur seeks to produce a profit by purchasing at a price that is less than the selling price Proceeds of the sales are used to finance purchases such that the portfolio of transactions is self-financing, and that over time, no additional capital is devoted to or lost from the portfolio Thus, the portfolio is assured a non-negative profit at each time period The arbitrage process is riskless if purchase and sale prices are known at the times they are initiated Arbitrageurs frequently seek to profit from market inefficiencies The existence of arbitrage profits is inconsistent with market efficiency 1.2 FINANCIAL SECURITIES AND INSTRUMENTS A security is a tradable claim on assets Real assets contribute to the productive capacity of the economy; securities are financial assets that represent claims on real assets or other securities Most securities are marketable to the general public, meaning that they can be sold or assigned to other investors in the open marketplace Some of the more common types of securities and tradable instruments are briefly introduced in the following: Debt securities: Denote creditorship of an individual, firm or other institution They typically involve payments of a fixed series of interest (often known as coupon payments) or amounts towards principal along with principal repayment (often known as face value) Examples include: • Bonds: Long-term debt securities issued by corporations, governments, or other institutions Bonds are normally of the coupon variety (they make periodic interest payments on the principal) or pure discount (they are zero coupon instruments that are sold at a discount from face value, the bond’s final maturity value) • Treasury securities: Debt securities issued by the Treasury of the United States federal government They are often considered to be practically free of default risk Equity securities (stock): Denote ownership in a business or corporation They typically permit for dividend payments if the firm’s debt obligations have been satisfied Derivative securities: Have payoff functions derived from the values of other securities, rates, or indices Some of the more common derivative securities are: • Options: Securities that grant their owners rights to buy (call) or sell (put) an underlying asset or security at a specific price (exercise price) on or before its expiration date • Forward and futures contracts: Instruments that oblige their participants to either purchase or sell a given asset or security at a specified price (settlement price) on the future settlement date of that contract A long position obligates the investor to purchase the given asset on the settlement date of the contract and a short position obligates the investor to sell the given asset on the settlement date of the contract • Swaps: Provide for the exchange of cash flows associated with one asset, rate, or index for the cash flows associated with another asset, rate, or index RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS: A PRIMER 1.3 REVIEW OF MATRICES AND MATRIX ARITHMETIC Commodities: Contracts, including futures and options on physical commodities such as oil, metals, corn, etc Commodities are traded in spot markets, where the exchange of assets and money occurs at the time of the transaction or in forward and futures markets Currencies: Exchange rates denote the number of units of one currency that must be given up for one unit of a second currency Exchange transactions can occur in either spot or forward markets As with commodities, in the spot market, the exchange of one currency for another occurs when the agreement is made In a forward market transaction, the actual exchange of one currency for another actually occurs at a date later than that of the agreement Spot and forward contract participants take one position in each of two currencies: • Long: An investor has a “long” position in that currency that he will accept at the later date • Short: An investor has a “short” position in that currency that he must deliver in the transaction Indices: Contracts pegged to measures of market performance such as the Dow Jones Industrials Average or the S&P 500 Index These are frequently futures contracts on portfolios structured to perform exactly as the indices for which they are named Index traders also trade options on these futures contracts This list of security types is far from comprehensive; it only reflects some of those instruments that will be emphasized in this book In addition, most of the instrument types will have many different variations 1.3 REVIEW OF MATRICES AND MATRIX ARITHMETIC A matrix is simply an ordered rectangular array of numbers A matrix is an entity that enables one to represent a series of numbers as a single object, thereby providing for convenient systematic methods for completing large numbers of repetitive computations Such objects are essential for the management of large data structures Rules of matrix arithmetic and other matrix operations are often similar to rules of ordinary arithmetic and other operations, but they are not always identical In this text, matrices will usually be denoted with bold uppercase letters When the matrix has only one row or one column, bold lowercase letters will be used for identification The following are examples of matrices: 3 " # 23 7 7 A56 c56 43 45B5 5 d ½ 4Š 3=4 21=2 25 The dimensions of a matrix are given by the ordered pair m n, where m is the number of rows and n is the number of columns in the matrix The matrix is said to be of order m n where, by convention, the number of rows is listed first Thus, A is 3 3, B is 2, c is 3 1, and d is Each number in a matrix is referred to as an element The symbol ai,j denotes the element in Row i and Column j of Matrix A, bi,j denotes the element in Row i and Column j of Matrix B, and so on Thus, a3,2 is 25 and c2,1 5 There are specific terms denoting various types of matrices Each of these particular types of matrices has useful applications and unique properties for working with For example, a vector is RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS: A PRIMER ... to apply the important results Introduction to Risk- Neutral Pricing and Financial Mathematics: A Primer seeks to introduce financial mathematics to students in quantitative finance, financial. .. ^ ^ cam,1 cam,2 cam,n RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS: A PRIMER PRELIMINARIES AND REVIEW Matrix Arithmetic Illustration: Consider the following matrices A and B below: A5 22... m,n D E F RISK NEUTRAL PRICING AND FINANCIAL MATHEMATICS: A PRIMER 1.3 REVIEW OF MATRICES AND MATRIX ARITHMETIC Now consider a third matrix operation The transpose AT of Matrix A is obtained by