Martingales and Financial Mathematics in Discrete Time

Preface ix

Introduction xi

Chapter 1 Elementary Probabilities and an Introduction to Stochastic Processes 1

1.1 Measures and s-algebras 1

1.2 Probability elements 5

1.3. Stochastic processes 16

1.4. Exercises 19

Chapter 2 Conditional Expectation 21

2.1 Conditional probability with respect to an event 21

2.2. Conditional expectation 24

2.3. Geometric interpretation 37

2.4. Conditional expectation and independence 38

2.5. Exercises 41

Chapter 3 Random Walks 45

3.1 Trajectories of the random walk 45

3.2. Asymptotic behavior 52

3.3. The Gambler's ruin 58

3.4. Exercises 60

Chapter 4 Martingales 63

4.1 Definition 63

4.2. Martingale transform 66

4.3 The Doob decomposition 67

4.4 Stopping time 69

4.5 Stopped martingales 71

4.6. Exercises 75

Chapter 5 Financial Markets 81

5.1. Financial assets 82

5.2. Investment strategies 82

5.3. Arbitrage 84

5.4. The Cox, Ross and Rubinstein model 86

5.5. Exercises 88

5.6. Practical work 90

Chapter 6 European Options 95

6.1 Definition 95

6.2. Complete markets 96

6.3. Valuation and hedging 97

6.4. Cox, Ross and Rubinstein model 98

6.5. Exercises 104

6.6. Practical work: Simulating the value of a call option 106

Chapter 7 American Options 107

7.1 Definition 107

7.2 Optimal stopping 109

7.4. The Cox, Ross and Rubinstein model 115

7.5. Exercises 116

7.6. Practical work 117

Chapter 8 Solutions to Exercises and Practical Work 119

8.1. Solutions to exercises in Chapter1 119

8.2. Solutions to exercises in Chapter2 127

8.3 Solutions to exercises in Chapter3 143

8.4. Solutions to exercises in Chapter4 151

8.5. Solutions to exercises in Chapter5 170

8.6.Solutions to the practical exercises in Chapter5 175

8.7. Solutions to exercises in Chapter6 189

8.8. Solution to the practical exercise in Chapter6 (section6.6) 193

8.9. Solution to exercises in Chapter7 195

8.10. Solution to the practical exercise in Chapter7 (section7.6) 200

References 205

Index 207