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 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 P61ya'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 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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