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Probability Theory in Finance A Mathematical Guide to the Black-Scholes Formula Second ediTion Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Probability Theory in Finance A Mathematical Guide to the Black-Scholes Formula Second ediTion Seán dineen Graduate Studies in Mathematics Volume 70 American Mathematical Society Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. EDITORIAL COMMITTEE David Cox (Chair) Daniel S. Freed Rafe Mazzeo Gigliola Staffilani 2010 Mathematics Subject Classification. Primary 60-01, 91Bxx. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-70 Library of Congress Cataloging-in-Publication Data Dineen,Seań,1944- Probability theory in finance : a mathematical guide to the Black-Scholes formula / Seań Dineen.–Secondedition. pagescm. —(Graduatestudiesinmathematics;v.70) Includesbibliographicalreferencesandindex. ISBN978-0-8218-9490-3(alk.paper) 1.Businessmathematics. I.Title. HF5691.D57 2013 332.01(cid:2)519—dc23 2013003088 Copying and reprinting. Individual readers of this publication, and nonprofit libraries actingforthem,arepermittedtomakefairuseofthematerial,suchastocopyachapterforuse in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication is permitted only under license from the American Mathematical Society. Requests for such permissionshouldbeaddressedtotheAcquisitionsDepartment,AmericanMathematicalSociety, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by [email protected]. (cid:2)c 2013bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttp://www.ams.org/ 10987654321 181716151413 Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Contents Preface ix Chapter 1. Money and Markets 1 Summary 1 §1.1. Introduction 1 §1.2. Money 2 §1.3. Interest Rates 3 §1.4. The Market 13 §1.5. Exercises 15 Chapter 2. Fair Games 17 Summary 17 §2.1. Fair Games 17 §2.2. Hedging and Arbitrage 21 §2.3. Exercises 26 Chapter 3. Set Theory 29 Summary 29 §3.1. Approaching Abstract Mathematics 29 §3.2. Infinity 33 §3.3. σ–Fields 40 §3.4. Partitions 48 §3.5. Filtrations and Information 52 §3.6. Exercises 55 v Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. vi Contents Chapter 4. Measurable Functions 59 Summary 59 §4.1. Measurable Functions 59 §4.2. Convergence 69 §4.3. Exercises 74 Chapter 5. Probability Spaces 77 Summary 77 §5.1. Probability Spaces 77 §5.2. Call Options 1 83 §5.3. Independence 91 §5.4. Random Variables 100 §5.5. Stochastic Processes 103 §5.6. Exercises 104 Chapter 6. Expected Values 107 Summary 107 §6.1. Simple Random Variables 107 §6.2. Positive Bounded Random Variables 118 §6.3. Positive Random Variables 125 §6.4. Integrable Random Variables 133 §6.5. Summation of Series 139 §6.6. Exercises 142 Chapter 7. Continuity and Integrability 143 Summary 143 §7.1. Continuous Functions 143 §7.2. Convex Functions 146 §7.3. The Riemann Integral 151 §7.4. Independent Random Variables 156 §7.5. The Central Limit Theorem 161 §7.6. Exercises 163 Chapter 8. Conditional Expectation 165 Summary 165 §8.1. Call Options 2 165 §8.2. Conditional Expectation 171 Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Contents vii §8.3. Hedging 182 §8.4. Exercises 186 Chapter 9. Lebesgue Measure 189 Summary 189 §9.1. Product Measures 189 §9.2. Lebesgue Measure 197 §9.3. Density Functions 203 §9.4. Exercises 208 Chapter 10. Martingales 209 Summary 209 §10.1. Discrete-Time Martingales 209 §10.2. Martingale Convergence 214 §10.3. Continuous-Time Martingales 220 §10.4. Exercises 225 Chapter 11. The Black-Scholes Formula 227 Summary 227 §11.1. Share Prices as Random Variables 227 §11.2. Call Options 3 233 §11.3. Change of Measure 239 §11.4. Exercises 242 Chapter 12. Stochastic Integration 243 Summary 243 §12.1. Riemann Sums 243 §12.2. Convergence of Random Variables 245 §12.3. The Stochastic Riemann Integral 251 §12.4. The Itoˆ Integral 257 §12.5. Itô’s Lemma 265 §12.6. Call Options 4 274 §12.7. Epilogue 277 §12.8. Exercises 279 Solutions 281 Bibliography 299 Index 301 Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. Preface To doubt all or believe all are two equally convenient solutions, in that both dispense with thinking. Henri Poincaré, 1854-1912 Preface to the Second Edition. Comments from different sources, experience in using the first edition as a class text, and the opportunity to teach a preliminary course in analysis to studentswhowouldsubsequentlyusethisbookallcontributedtothechangesin thissecondedition. TheanalysisexperienceresultedinAnalysis: A Gateway to Understanding Mathematics published by World Scientific (Singapore) in 2012. While maintaining the original approach, format, and list of topics, I have revised to some extent most chapters. I found it convenient to rearrange some of the material and as a result to include an additional chapter (Chapter 9). This new chapter contains material from Chapters 6 and 7 in the first edition and, additionally, a construction of Lebesgue measure using dyadic rationals and a countable product of probability spaces. A brief paraphrasing of essentially one paragraph of the original preface is included here to help the reader navigate the second edition. Students of financial mathematics may wish to follow, as our students did, Chapters 1- 5; Sections 6.1, 6.2, 6.3 and 7.4; the statements of the main results in Sec- tions 6.3, 6.4, and 7.5; and Chapters 8 and 10-11. Students of mathematicsand ix Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community. x Preface statistics interested in probability theory could follow Chapters 3-7, Section 8.2 and Chapters 9 and 10. Students of mathematics could follow Chapters 3-6 and 9 as an introduction to measure theory. Chapter 12 is, modulo a modest backgroundinprobabilitytheory, aself-containedintroductiontostochasticin- tegrationandtheItoîntegral. Finallyanyonebeginningtheiruniversitystudies in mathematicsor merely interestedin modern mathematics, from a philosoph- icaloraestheticpointof view, willfind Chapters 1-5accessible, challengingand rewarding. It is a pleasure to thank once more Michael Mackey for all his help and patience and Sergei Gelfand for his constant encouragement. Preface to the First Edition. Mathematics occupies a unique place in modern society and education. It cannot be ignored and almost everyone has an opinion on its place and rel- evance. This has led to problems and questions that will never be solved or answered in a definitive fashion. At third level we have the perennial debate on the mathematics that is suitable for non-mathematics majors and the de- gree of abstraction with which it should be delivered. We mathematicians are still trusted with this task and our response has varied. Some institutions offer generic mathematics courses to all and sundry, and faculties, such as engi- neering and business, respond by directing their students to the courses they consider appropriate. In other institutions departments design specific courses for students who are not majoring in mathematics. The response of many departments lies somewhere in between. This can lead to tension between the professional mathematicians’ attitude to mathematics and the client faculties’ expectations. In the first case non-mathematics majors may find themselves obliged to accept without explanation an approach that is, in their experience, excessively abstract. In the second, a recipe-driven approach often produces students with skills they have difficulty using outside a limited number of well- defined settings. Some students, however, do arrive, by sheer endurance, at an intuitive feeling for mathematics. Clearly both extremes are unsatisfactoryand it is natural to ask if an alternative approach is possible. It is, and the difficulties to be overcome are not mathematical. The understanding of mathematics that we mathematicians have grown to appreci- ate and accept, often slowly and unconsciously, is not always shared by non- mathematicians, be they students or colleagues, and the benefits of abstract mathematics are not always obvious to academics from other disciplines. This is not their fault. They have, for the most part, been conditioned to think differently. They accept that mathematics is useful and for this reason are Copyright no copyright American Mathematical Society. Duplication prohibited. Please report unauthorized use to [email protected]. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.