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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
Contents4 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
5CONDITIONING AND INDEPENDENCE 117
5.1 Examples of conditioning 117
5.2 Basic formulas 122
5.3 Sequential sampling 131
5.4 P´olya’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 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
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
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