Probability and finance: it's only a game!

438 Pages·2001·21.752 MB·English

by  Shafer, Glenn;Vovk, Vladimir

Probability and
Finance
It’s Only a Game!
This Page Intentionally Left Blank
Probability and
Finance
WILEY SERIES IN PROBABILITY AND STATISTICS
FINANCIAL ENGINEERING SECTION
Established by WALTER A. SHEWHART and SAMUEL S. WILKS
Editors: Peter Bloorqfield, Noel A. C. Cressie, Nicholas 1. Fisher;
Iuin M. John.stone, J. B. Kudane, Louise M. Ryan, David W Scott,
Revnuid Silverman, Adrian E M. Smith, Jozef L. Teugels;
PY
Vic Burnett. Emeritus, Ralph A. Bradley, Emeritirs,
J. Stztul-t Hiinter; Emeritus, David G. Kenclall, Emel-itits
A complete list of the titles in this series appears at the end of this volume.
Probability and
Finance
It’s Only a Game!
GLENN SHAFER
Rzitgers University
Newark, New Jersey
VLADIMIR VOVK
Rqval Holloway, University of London
Egharn, Surrey, England
A Wiley-Interscience Publication
JOHN WILEY & SONS, INC.
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Library of Congress Catulojiing-iit-PNblicution Data:
Shafer, Glenn. I946
Probability and finance : it's only a game! /Glenn Shafer and Vladimir Vovk
p. cin. (Wiley series in probability and statistics. Financial engineering section)
~
Includes bibliographical references and index.
ISBN 0-471-40226-5 (acid-free paper)
1. Investments-Mathematics. 2. Statistical decision. 3. Financial engineering. 1. Vovk,
Vladimir, 1960-. 11. Title. 111. Series.
HG4S 15 3 .SS34 2001
332'.01'1 -dc21 2001024030
Printed in the United States of America
1 0 9 8 7 6 5 4 3 2 1
Contents
Preface ix
1 Probability and Finance as a Game 1
1.1 A Game with the World 4
1.2 The Protocol for a Probability Game 9
1.3 The Fundamental Interpretative Hypothesis 14
1.4 The Many Interpretations of Probability 19
1.5 Game-Theoretic Probability in Finance 22
Part I Probability without Measure 27
2 The Historical Context 29
2.1 Probability before Kolmogorov 30
2.2 Kolmogorov Measure-Theoretic Framework 39
's
2.3 Realized Randomness 46
2.4 What is a Martingale? 51
2.5 The Impossibility of a Gambling System 55
2.6 Neosubjectivism 59
2.7 Conclusion 60
V
Vi CONTENTS
3 The Bounded Strong Law of Large Numbers 61
3.1 The Fair-Coin Game 63
3.2 Forecasting a Bounded Variable 65
3.3 Who Sets the Prices? 70
3.4 Asymmetric Bounded Forecasting Games 72
3.5 Appendix: The Computation of Strategies 73
4 Kolmogorov’s Strong Law of Large Numbers 75
4.1 Two Statements of Kolmogorov Strong Law 77
’s
4.2 Skeptic’s Strategy 81
4.3 Reality’s Strategy 87
4.4 The Unbounded Upper Forecasting Protocol 89
4.5 A Martingale Strong Law 90
4.6 Appendix: Martin’s Theorem 94
5 The Law of the Iterated Logarithm 99
5.1 Unbounded Forecasting Protocols 101
5.2 The Validity of the Iterated-Logarithm Bound 104
5.3 The Sharpness of the Iterated-Logarithm Bound 108
5.4 A Martingale Law of the Iterated Logarithm 118
5.5 Appendix: Historical Comments 118
5.6 Appendix: Kolmogorov ’s Finitary Interpretation 120
6 The Weak Laws 121
6.1 Bernoulli’s Theorem 124
6.2 De Moivre’s Theorem 126
6.3 A One-sided Central Limit Theorem 133
6.4 Appendix: The Gaussian Distribution 143
6.5 Appendix: Stochastic Parabolic Potential Theory 144
7 Lindeberg ’s Theorem 147
7.1 Lindeberg Protocols 148
7.2 Statement and Proof of the Theorem 153
7.3 Examples of the Theorem 158
7.4 Appendix: The Classical Central Limit Theorem 164
CONTENTS vii
8 The Generality of Probability Games 167
8.1 Deriving the Measure-Theoretic Limit Theorems 168
8.2 Coin Tossing 177
8.3 Game-Theoretic Price and Probability 182
8.4 Open ScientiJc Protocols 189
8.5 Appendix: Ville’s Theorem 194
8.6 Appendix: A Brief Biography of Jean Ville 197
Part II Finance without Probability 199
9 Game-Theoretic Probability in Finance 201
9.1 The Behavior of Stock-Market Prices 203
9.2 The Stochastic Black-Scholes Formula 215
9.3 A Purely Game-Theoretic Black-Scholes Formula 221
9.4 Informational Eficiency 226
9.5 Appendix: Tweaking the Black-Scholes Model 229
9.6 Appendix: On the Stochastic Theory 231
10 Games for Pricing Options in Discrete Time 237
10.1 Bachelier’s Central Limit Theorem 239
10.2 Bachelier Pricing in Discrete Time 243
10.3 Black-Scholes Pricing in Discrete Time 249
10.4 Hedging Error in Discrete Time 252
10.5 Black-Scholes with Relative Variationsf or S 259
10.6 Hedging Error with Relative Variationsf or S 262
1 1 Gamesf or Pricing Options in Continuous Time 2 71
11. I The Variation Spectrum 2 73
11.2 Bachelier Pricing in Continuous Time 2 75
11.3 Black-Scholes Pricing in Continuous Time 2 79
11.4 The Game-Theoretic Source of the ddt Effect 281
11.5 Appendix: Elements of Nonstandard Analysis 283
11.6 Appendix: On the Diffusion Model 287