An Elementary introduction to Mathematical Finance.pdf

This page intentionally left blankAn Elementary Introduction toMathematical Finance,Third EditionThis textbook on the basics of option pricing is accessible to readers withlimited mathematical training.It is for both professional traders and un-dergraduates studying the basics of finance.Assuming no prior knowledgeof probability,Sheldon M.Ross offers clear,simple explanations of arbi-trage,the BlackScholes option pricing formula,and other topics such asutility functions,optimal portfolio selections,and the capital assets pricingmodel.Among the many new features of this third edition are new chap-ters on Brownian motion and geometric Brownian motion,stochastic orderrelations,and stochastic dynamic programming,along with expanded setsof exercises and references for all the chapters.Sheldon M.Ross is the Epstein Chair Professor in the Department ofIndustrial and Systems Engineering,University of Southern California.Hereceived his Ph.D.in statistics from Stanford University in 1968 and was aProfessor at the University of California,Berkeley,from 1976 until 2004.He has published more than 100 articles and a variety of textbooks in theareas of statistics and applied probability,includingTopics in Finite andDiscrete Mathematics(2000),Introduction to Probability and Statis-tics for Engineers and Scientists,Fourth Edition(2009),A First Coursein Probability,Eighth Edition(2009),andIntroduction to ProbabilityModels,Tenth Edition(2009).Dr.Ross serves as the editor forProbabil-ity in the Engineering and Informational Sciences.An Elementary Introductionto Mathematical FinanceThird EditionSHELDON M.ROSSUniversity of Southern CaliforniaCAMBRIDGE UNIVERSITY PRESSCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,So Paulo,Delhi,Tokyo,Mexico CityCambridge University Press32 Avenue of the Americas,New York,NY 10013-2473,USAwww.cambridge.orgInformation on this title:www.cambridge.org/9780521192538 Cambridge University Press 1999,2003,2011This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.First published 1999Second edition published 2003Third edition published 2011Printed in the United States of AmericaA catalog record for this publication is available from the British Library.Library of Congress Cataloging in Publication dataRoss,Sheldon M.(Sheldon Mark),1943An elementary introduction to mathematical finance/Sheldon M.Ross.Third edition.p.cm.Includes index.ISBN 978-0-521-19253-81.Investments Mathematics.2.Stochastic analysis.3.Options(Finance)Mathematical models.4.Securities Prices Mathematical models.I.Title.HG4515.3.R67 2011332.601?51dc222010049863ISBN 978-0-521-19253-8 HardbackCambridge University Press has no responsibility for the persistence or accuracy of URLs forexternal or third-party internet websites referred to in this publication and does not guarantee thatany content on such websites is,or will remain,accurate or appropriate.To my parents,Ethel and Louis RossContentsIntroduction and Prefacepage xi1 Probability11.1 Probabilities and Events11.2 Conditional Probability51.3 Random Variables and Expected Values91.4 Covariance and Correlation141.5 Conditional Expectation161.6 Exercises172 Normal Random Variables222.1 Continuous Random Variables222.2 Normal Random Variables222.3 Properties of Normal Random Variables262.4 The Central Limit Theorem292.5 Exercises313 Brownian Motion and Geometric Brownian Motion343.1 Brownian Motion343.2 Brownian Motion as a Limit of Simpler Models353.3 Geometric Brownian Motion383.3.1 Geometric Brownian Motion as a Limitof Simpler Models403.4The Maximum Variable403.5 The Cameron-Martin Theorem453.6 Exercises464 Interest Rates and Present Value Analysis484.1 Interest Rates484.2 Present Value Analysis524.3 Rate of Return624.4 Continuously Varying Interest Rates654.5 Exercises67viiiContents5 Pricing Contracts via Arbitrage735.1 An Example in Options Pricing735.2 Other Examples of Pricing via Arbitrage775.3 Exercises866 The Arbitrage Theorem926.1 The Arbitrage Theorem926.2 The Multiperiod Binomial Model966.3 Proof of the Arbitrage Theorem986.4 Exercises1027 The BlackScholes Formula1067.1 Introduction1067.2 The BlackScholes Formula1067.3 Properties of the BlackScholes Option Cost1107.4 The Delta Hedging Arbitrage Strategy1137.5 Some Derivations1187.5.1 The BlackScholes Formula1197.5.2 The Partial Derivatives1217.6 European Put Options1267.7 Exercises1278 Additional Results on Options1318.1 Introduction1318.2 Call Options on Dividend-Paying Securities1318.2.1 The Dividend for Each Share of the SecurityIs Paid Continuously in Time at a Rate Equalto a Fixed Fraction f of the Price of theSecurity1328.2.2 For Each Share Owned,a Single Payment offS(td)Is Made at Time td1338.2.3 For Each Share Owned,a Fixed Amount D Isto Be Paid at Time td1348.3 Pricing American Put Options1368.4 Adding Jumps to Geometric Brownian Motion1428.4.1 When the Jump Distribution Is Lognormal1448.4.2 When the Jump Distribution Is General1468.5 Estimating the Volatility Parameter1488.5.1 Estimating a Population Mean and Variance1498.5.2 The Standard Estimator of Volatility150Contentsix8.5.3 Using Opening and Closing Data1528.5.4 Using Opening,Closing,and HighLow Data1538.6 Some Comments1558.6.1 When the Option Cost Differs from theBlackScholes Formula1558.6.2 When the Interest Rate Changes1568.6.3 Final Comments1568.7 Appendix1588.8 Exercises1599 Valuing by Expected Utility1659.1 Limitations of Arbitrage Pricing1659.2 Valuing Investments by Expected Utility1669.3 The Portfolio Selection Problem1749.3.1 Estimating Covariances1849.4 Value at Risk and Conditional Value at Risk1849.5 The Capital Assets Pricing Model1879.6 Rates of Return:Single-Period and GeometricBrownian Motion1889.7 Exercises19010 Stochastic Order Relations19310.1 First-Order Stochastic Dominance19310.2 Using Coupling to Show Stochastic Dominance19610.3 Likelihood Ratio Ordering19810.4 A Single-Period Investment Problem19910.5 Second-Order Dominance20310.5.1 Normal Random Variables20410.5.2 More on Second-Order Dominance20710.6 Exercises21011 Optimization Models21211.1 Introduction21211.2 A Deterministic Optimization Model21211.2.1 A General Solution Technique Based onDynamic Programming21311.2.2 A Solution Technique for ConcaveReturn Functions21511.2.3 The Knapsack Problem21911.3 Probabilistic Optimization Problems221xContents11.3.1 A Gambling Model with Unknown WinProbabilities22111.3.2 An Investment Allocation Model22211.4 Exercises22512 Stochastic Dynamic Programming22812.1 The Stochastic Dynamic Programming Problem22812.2 Infinite Time Models23412.3 Optimal Stopping Problems23912.4 Exercises24413 Exotic Options24713.1Introduction24713.2Barrier Options24713.3Asian and Lookback Options24813.4Monte Carlo Simulation24913.5Pricing Exotic Options by Simulation25013.6More Efficient Simulation Estimators25213.6.1 Control and Antithetic Variables in theSimulation of Asian and LookbackOption Valuations25313.6.2 Combining Conditional Expectation andImportance Sampling in the Simulation ofBarrier Option Valuations25713.7Options with Nonlinear Payoffs25813.8Pricing Approximations via Multiperiod BinomialModels25913.9Continuous Time Approximations of Barrierand Lookback Options26113.10 Exercises26214 Beyond Geometric Brownian Motion Models26514.1 Introduction26514.2 Crude Oil Data26614.3 Models for the Crude Oil Data27214.4 Final Comments27415 Autoregressive Models and Mean Reversion28515.1 The Autoregressive Model28515.2 Valuing Options by Their Expected Return28615.3 Mean Reversion28915.4 Exercises291Index303Introduction and PrefaceAn option gives one the right,but not the obligation,to buy or sell asecurity under specified terms.A call option is one that gives the rightto buy,and a put option is one that gives the right to sell the security.Both types of options will have an exercise price and an exercise time.Inaddition,therearetwostandardconditionsunderwhichoptionsoper-ate:European options can be utilized only at the exercise time,whereasAmerican options can be utilized at any time up to exercise time.Thus,for instance,a European call option with exercise price K and exercisetime t gives its holder the right to purchase at time t one share of theunderlying security for the price K,whereas an American call optiongives its holder the right to make the purchase at any time before or attime t.A prerequisite for a strong market in options is a computationally effi-cient way of evaluating,at least approximately,their worth;this wasaccomplished for call options(of either American or European type)bythe famous BlackScholes formula.The formula assumes that pricesof the underlying security follow a geometric Brownian motion.Thismeans that if S(y)is the price of the security at time y then,for anyprice history up to time y,the ratio of the price at a specified future timet+y to the price at time y has a lognormal distribution with mean andvariance parameters t and t2,respectively.That is,log?S(t+y)S(y)?will be a normal random variable with mean t and variance t2.Blackand Scholes showed,under the assumption that the prices follow a geo-metric Brownian motion,that there is a single price for a call option thatdoes not allow an idealized trader one who can instantaneously maketrades without any transaction costs to follow a strategy that will re-sult in a sure profit in all cases.That is,there will be no certain profit(i.e.,no arbitrage)if and only if the price of the option is as given bythe BlackScholes formula.In addition,this price depends only on thexiiIntroduction and Prefacevariance parameter of the geometric Brownian motion(as well as onthe prevailing interest rate,the underlying price of the security,and theconditions of the option)and not on the parameter.Because the pa-rameter is a measure of the volatility of the security,it is often calledthe volatility parameter.Arisk-neutral investorisonewhovaluesaninvestmentsolelythroughtheexpectedpresentvalueofitsreturn.Ifsuchaninvestormodelsasecu-ritybyageometricBrownianmotionthatturnsallinvestmentsinvolvingbuying and selling the security into fair bets,then this investors valu-ation of a call option on this security will be precisely as given by theBlackScholes formula.For this reason,the BlackScholes valuation isoften called a risk-neutral valuation.Our first objective in this book is to derive and explain the BlackScholes formula.Its derivation,however,requires some knowledge ofprobability,and this is what the first three chapters are concerned with.Chapter 1 introduces probability and the probability experiment.Ran-dom variables numerical quantities whose values are determined bythe outcome of the probability experiment are discussed,as are theconcepts of the expected value and variance of a random variable.InChapter2weintroducenormalrandomvariables;thesearerandomvari-ables whose probabilities are determined by a bell-shaped curve.Thecentral limit theorem is presented in this chapter.This theorem,prob-ably the most important theoretical result in probability,states that thesum of a large number of random variables will approximately be a nor-malrandomvariable.InChapter3weintroducethegeometricBrownianmotion process;we define it,show how it can be obtained as the limit ofsimpler processes,and discuss the justification for its use in modelingsecurity prices.With the probability necessities behind us,the second part of the textbegins in Chapter 4 with an introduction to the concept of interest ratesand present values.A key concept underlying the BlackScholes for-mulaisthatofarbitrage,whichisthesubjectofChapter5.Inthischapterwe show how arbitrage can be used to determine prices in a variety ofsituations,including the single-period binomial option model.In Chap-ter6wepresentthearbitragetheoremanduseittofindanexpressionforthe unique nonarbitrage option cost in the multiperiod binomial model.In Chapter 7 we use the results of Chapter 6,along with the approxima-tions of geometric Brownian motion presented in Chapter 4,to obtain aIntroduction and PrefacexiiisimplederivationoftheBlackScholesequationforpricingcalloptions.Properties of the resultant option cost as a function of its parameters arederived,as is the delta hedging replication strategy.Additional resultson options are presented in Chapter 8,where we derive option pricesfor dividend-paying securities;show how to utilize a multiperiod bino-mialmodeltodetermineanapproximationoftherisk-neutralpriceofanAmerican put option;determine no-arbitrage costs when the securitysprice follows a model that superimposes random jumps on a geomet-ric Brownian motion;and present different estimators of the volatilityparameter.In Chapter 9 we note that,in many situations,arbitrage considerationsdo not result in a unique cost.We show the importance in such casesof the investors utility function as well as his or her estimates of theprobabilities of the possible outcomes of the investment.The conceptsof mean variance analysis,value and conditional value at risk,and thecapital assets pricing model are introduced.In Chapter 10 we introduce stochastic order relations.These relationscanbeusefulindeterminingwhichofaclassofinvestmentsisbestwith-out completely specifying the investors utility function.For instance,if the return from one investment is greater than the return from anotherinvestmentinthesenseoffirst-orderstochasticdominance,thenthefirstinvestmentistobepreferredforanyincreasingutilityfunction;whereasif the first return is greater in the sense of second-order stochastic dom-inance,then the first investment is to be preferred as long as the utilityfunction is concave and increasing.In Chapters 11 and 12 we study some optimization models in finance.In Chapter 13 we introduce some nonstandard,or“exotic,”optionssuch as barrier,Asian,and lookback options.We explain how to useMonte Carlo simulation,implementing variance reduction techniques,to efficiently determine their geometric Brownian motion risk-neutralvaluations.The BlackScholes formula is useful even if one has doubts about thevalidity of the underlying geometric Brownian model.For as long asone accepts that this model is at least approximately valid,its use givesone an idea about the appropriate price of the option.Thus,if the ac-tual trading option price is below the formula price then it would seemthat the option is underpriced in relation to the security itself,thus lead-ing one to consider a strategy of buying options and selling the securityxivIntroduction and Preface(with the reverse being suggestedwhen the trading optionprice is abovethe formula price).In Chapter 14 we show that real data cannot awaysbe fit by a geometric Brownian motion model,and that more generalmodels may need to be considered.In the case of commodity prices,there is a strong belief by many traders in the concept of mean price re-version:that the market prices of certain commodities ha