Table Of ContentProbability Theory
in Finance
A Mathematical Guide to
the Black-Scholes Formula
Second ediTion
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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
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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,Sea´n,1944-
Probability theory in finance : a mathematical guide to the Black-Scholes formula / Sea´n
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
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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
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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
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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. Itˆo’s Lemma 265
§12.6. Call Options 4 274
§12.7. Epilogue 277
§12.8. Exercises 279
Solutions 281
Bibliography 299
Index 301
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Preface
To doubt all or believe all are two equally
convenient solutions, in that both dispense with
thinking.
Henri Poincar´e, 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
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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-
tegrationandtheItoˆintegral. 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 de-
partments 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 un-
derstanding 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
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