Preface xiii
1 A synthetic view 1
1.1 The world of derivatives 2
1.1.1 Different kinds of contracts 2
1.1.2 Vanilla options 3
1.1.3 Why options? 6
1.1.4 A variety of options 7
1.1.5 How to model asset prices 8
1.1.6 One step beyond 9
1.2 Bibliographical notes 10
References 10
2 Probability, random variables and statistics 13
2.1 Probability 13
2.1.1 Conditional probability 15
2.2 Bayes' rule 16
2.3 Random variables 18
2.3.1 Characteristic function 23
2.3.2 Moment generating function 24
2.3.3 Examples of random variables 24
2.3.4 Sum of random variables 35
2.3.5 Infinitely divisible distributions 37
2.3.6 Stable laws 38
2.3.7 Fast Fourier Transform 42
2.3.8 Inequalities 46
2.4 Asymptotics 48
2.4.1 Types of convergences 48
2.4.2 Law of large numbers 50
2.4.3 Central limit theorem 52
2.5 Conditional expectation 54
2.6 Statistics 57
2.6.1 Properties of estimators 57
2.6.2 The likelihood function 61
2.6.3 Efficiency of estimators 63
2.6.4 Maximum likelihood estimation 64
2.6.5 Moment type estimators 65
2.6.6 Least squares method 65
2.6.7 Estimating functions 66
2.6.8 Confidence intervals 66
2.6.9 Numerical maximization of the likelihood 68
2.6.10 The d-method 70
2.7 Solution to exercises 71
2.8 Bibliographical notes 77
References 77
3 Stochastic processes 79
3.1 Definition and first properties 79
3.1.1 Measurability and filtrations 81
3.1.2 Simple and quadratic variation of a process 83
3.1.3 Moments, covariance, and increments of stochastic processes 84
3.2 Martingales 84
3.2.1 Examples of martingales 85
3.2.2 Inequalities for martingales 88
3.3 Stopping times 89
3.4 Markov property 91
3.4.1 Discrete time Markov chains 91
3.4.2 Continuous time Markov processes 98
3.4.3 Continuous time Markov chains 99
3.5 Mixing property 101
3.6 Stable convergence 103
3.7 Brownian motion 104
3.7.1 Brownian motion and random walks 106
3.7.2 Brownian motion is a martingale 107
3.7.3 Brownian motion and partial differential equations 107
3.8 Counting and marked processes 108
3.9 Poisson process 109
3.10 Compound Poisson process 110
3.11 Compensated Poisson processes 113
3.12 Telegraph process 113
3.12.1 Telegraph process and partial differential equations 115
3.12.2 Moments of the telegraph process 117
3.12.3 Telegraph process and Brownian motion 118
3.13 Stochastic integrals 118
3.13.1 Properties of the stochastic integral 122
3.13.2 Itô formula 124
3.14 More properties and inequalities for the Itô integral 127
3.15 Stochastic differential equations 128
3.15.1 Existence and uniqueness of solutions 128
3.16 Girsanov's theorem for diffusion processes 130
3.17 Local martingales and semimartingales 131
3.18 Lévy processes 132
3.18.1 Lévy-Khintchine formula 134
3.18.2 Lévy jumps and random measures 135
3.18.3 Itô-Lévy decomposition of a Lévy process 137
3.18.4 More on the Lévy measure 138
3.18.5 The Itô formula for Lévy processes 139
3.18.6 Lévy processes and martingales 140
3.18.7 Stochastic differential equations with jumps 143
3.18.8 Itô formula for Lévy driven stochastic differential equations 144
3.19 Stochastic differential equations in R n 145
3.20 Markov switching diffusions 147
3.21 Solution to exercises 148
3.22 Bibliographical notes 155
References 155
4 Numerical methods 159
4.1 Monte Carlo method 159
4.1.1 An application 160
4.2 Numerical differentiati
