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Doob
Polya
Kolmogorov
Cramer
Borel
Levy
Keynes
Feller
Contents
PREFACE TO THE FOURTH EDITION
PROLOGUE TO INTRODUCTION TO
MATHEMATICAL FINANCE
1 SET
1.1 Sample sets
1.2 Operations with sets
1.3 Various relations
1.4 Indicator
Exercises
2 PROBABILITY
2.1 Examples of probability
2.2 Definition and illustrations
2.3 Deductions from the axioms
2.4 Independent events
2.5 Arithmetical density
Exercises
3 COUNTING
3.1 Fundamental rule
3.2 Diverse ways of sampling
3.3 Allocation models; binomial coefficients
3.4 How to solve it
Exercises
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Contents
4 RANDOM VARIABLES
4.1 What is a random variable?
4.2 How do random variables come about?
4.3 Distribution and expectation
4.4 Integer-valued random variables
4.5 Random variables with densities
4.6 General case
Exercises
APPENDIX 1: BOREL FIELDS AND GENERAL
RANDOM VARIABLES
5 CONDITIONING AND INDEPENDENCE
5.1 Examples of conditioning
5.2 Basic formulas
5.3 Sequential sampling
5.4 P´olya’s urn scheme
5.5 Independence and relevance
5.6 Genetical models
Exercises
6 MEAN, VARIANCE, AND TRANSFORMS
6.1 Basic properties of expectation
6.2 The density case
6.3 Multiplication theorem; variance and covariance
6.4 Multinomial distribution
6.5 Generating function and the like
Exercises
7 POISSON AND NORMAL DISTRIBUTIONS
7.1 Models for Poisson distribution
7.2 Poisson process
7.3 From binomial to normal
7.4 Normal distribution
7.5 Central limit theorem
7.6 Law of large numbers
Exercises
APPENDIX 2: STIRLING’S FORMULA AND
DE MOIVRE–LAPLACE’S THEOREM
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Contents
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8 FROM RANDOM WALKS TO MARKOV CHAINS
8.1 Problems of the wanderer or gambler
8.2 Limiting schemes
8.3 Transition probabilities
8.4 Basic structure of Markov chains
8.5 Further developments
8.6 Steady state
8.7 Winding up (or down?)
Exercises
APPENDIX 3: MARTINGALE
9 MEAN-VARIANCE PRICING MODEL
9.1 An investments primer
9.2 Asset return and risk
9.3 Portfolio allocation
9.4 Diversification
9.5 Mean-variance optimization
9.6 Asset return distributions
9.7 Stable probability distributions
Exercises
APPENDIX 4: PARETO AND STABLE LAWS
10 OPTION PRICING THEORY
10.1 Options basics
10.2 Arbitrage-free pricing: 1-period model
10.3 Arbitrage-free pricing: N-period model
10.4 Fundamental asset pricing theorems
Exercises
GENERAL REFERENCES
ANSWERS TO PROBLEMS
VALUES OF THE STANDARD NORMAL
DISTRIBUTION FUNCTION
INDEX
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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.
The new text begins with a meticulous account of the uncommon vocab-
ulary and syntax of the financial world; its manifold options and actions,
with consequent expectations and variations, in the marketplace. These are
then expounded in clear, precise mathematical terms and treated by the
methods of probability developed in the earlier chapters. Numerous graded
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 plays a basic role in more advanced finan-
cial and other disciplines. However, the level of our compendium remains
elementary, as befitting the title and scheme of this textbook. We have also
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
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