Doob Polya Kolmogorov Cramer Borel Levy Keynes Feller Contents PREFACE TO THE FOURTH EDITION xi PROLOGUE TO INTRODUCTION TO MATHEMATICAL FINANCE xiii 1 SET 1 1.1 Sample sets 1 1.2 Operations with sets 3 1.3 Various relations 7 1.4 Indicator 13 Exercises 17 2 PROBABILITY 20 2.1 Examples of probability 20 2.2 Definition and illustrations 24 2.3 Deductions from the axioms 31 2.4 Independent events 35 2.5 Arithmetical density 39 Exercises 42 3 COUNTING 46 3.1 Fundamental rule 46 3.2 Diverse ways of sampling 49 3.3 Allocation models; binomial coefficients 55 3.4 How to solve it 62 Exercises 70 vii viii Contents 4 RANDOM VARIABLES 74 4.1 What is a random variable? 74 4.2 How do random variables come about? 78 4.3 Distribution and expectation 84 4.4 Integer-valued random variables 90 4.5 Random variables with densities 95 4.6 General case 105 Exercises 109 APPENDIX 1: BOREL FIELDS AND GENERAL RANDOM VARIABLES 115 5 CONDITIONING AND INDEPENDENCE 117 5.1 Examples of conditioning 117 5.2 Basic formulas 122 5.3 Sequential sampling 131 5.4 Po´lya’s urn scheme 136 5.5 Independence and relevance 141 5.6 Genetical models 152 Exercises 157 6 MEAN, VARIANCE, AND TRANSFORMS 164 6.1 Basic properties of expectation 164 6.2 The density case 169 6.3 Multiplication theorem; variance and covariance 173 6.4 Multinomial distribution 180 6.5 Generating function and the like 187 Exercises 195 7 POISSON AND NORMAL DISTRIBUTIONS 203 7.1 Models for Poisson distribution 203 7.2 Poisson process 211 7.3 From binomial to normal 222 7.4 Normal distribution 229 7.5 Central limit theorem 233 7.6 Law of large numbers 239 Exercises 246 APPENDIX 2: STIRLING’S FORMULA AND DE MOIVRE–LAPLACE’S THEOREM 251 Contents ix 8 FROM RANDOM WALKS TO MARKOV CHAINS 254 8.1 Problems of the wanderer or gambler 254 8.2 Limiting schemes 261 8.3 Transition probabilities 266 8.4 Basic structure of Markov chains 275 8.5 Further developments 284 8.6 Steady state 291 8.7 Winding up (or down?) 303 Exercises 314 APPENDIX 3: MARTINGALE 325 9 MEAN-VARIANCE PRICING MODEL 329 9.1 An investments primer 329 9.2 Asset return and risk 331 9.3 Portfolio allocation 335 9.4 Diversification 336 9.5 Mean-variance optimization 337 9.6 Asset return distributions 346 9.7 Stable probability distributions 348 Exercises 351 APPENDIX 4: PARETO AND STABLE LAWS 355 10 OPTION PRICING THEORY 359 10.1 Options basics 359 10.2 Arbitrage-free pricing: 1-period model 366 10.3 Arbitrage-free pricing: N-period model 372 10.4 Fundamental asset pricing theorems 376 Exercises 377 GENERAL REFERENCES 379 ANSWERS TO PROBLEMS 381 VALUES OF THE STANDARD NORMAL DISTRIBUTION FUNCTION 393 INDEX 397 Preface to the Fourth Edition In this edition two new chapters, 9 and 10, on mathematical finance are added. They are written by Dr. Farid AitSahlia, ancien ´el`eve, who has taught such a course and worked on the research staff of several industrial and financial institutions. Thenewtextbeginswithameticulousaccountoftheuncommonvocab- ulary and syntax of the financial world; its manifold options and actions, withconsequentexpectationsandvariations,inthemarketplace.Theseare then expounded in clear, precise mathematical terms and treated by the methodsofprobabilitydevelopedintheearlierchapters.Numerousgraded and motivated examples and exercises are supplied to illustrate the appli- cability of the fundamental concepts and techniques to concrete financial problems. For the reader whose main interest is in finance, only a portion of the first eight chapters is a “prerequisite” for the study of the last two chapters. Further specific references may be scanned from the topics listed in the Index, then pursued in more detail. I have taken this opportunity to fill a gap in Section 8.1 and to expand Appendix 3 to include a useful proposition on martingale stopped at an optional time. The latter notion playsa basic role in more advanced finan- cial and other disciplines. However, the level of our compendium remains elementary,asbefittingthetitleandschemeofthistextbook.Wehavealso included some up-to-date financial episodes to enliven, for the beginners, the stratified atmosphere of “strictly business”. We are indebted to Ruth Williams, who read a draft of the new chapters with valuable suggestions for improvement; to Bernard Bru and Marc Barbut for information on the Pareto-L´evy laws originally designed for income distributions. It is hoped that a readable summary of this renowned work may be found in the new Appendix 4. Kai Lai Chung August 3, 2002 xi Prologue to Introduction to Mathematical Finance The two new chapters are self-contained introductions to the topics of mean-variance optimization and option pricing theory. The former covers a subject that is sometimes labeled “modern portfolio theory” and that is widely used by money managers employed by large financial institutions. To read this chapter, one only needs an elementary knowledge of prob- ability concepts and a modest familiarity with calculus. Also included is an introductory discussion on stable laws in an applied context, an of- ten neglected topic in elementary probability and finance texts. The latter chapter lays the foundations for option pricing theory, a subject that has fueledthedevelopmentoffinanceintoanadvancedmathematicaldiscipline as attested by the many recently published books on the subject. It is an initiation to martingale pricing theory, the mathematical expression of the so-called“arbitragepricingtheory”,inthecontextofthebinomialrandom walk. Despite its simplicity, this model captures the flavors of many ad- vanced theoretical issues. It is often used in practice as a benchmark for the approximate pricing of complex financial instruments. I would like to thank Professor Kai Lai Chung for inviting me to write the new material for the fourth edition. I would also like to thank my wife Unnur for her support during this rewarding experience. Farid AitSahlia November 1, 2002 xiii