This book is entirely devoted to discrete time and provides a detailed introduction to the construction of the rigorous mathematical tools required for the evaluation of options in financial markets. Both theoretical and practical aspects are explored through multiple examples and exercises, for which complete solutions are provided. Particular attention is paid to the Cox, Ross and Rubinstein model in discrete time.

The book offers a combination of mathematical teaching and numerous exercises for wide appeal. It is a useful reference for students at the master’s or doctoral level who are specializing in applied mathematics or finance as well as teachers, researchers in the field of economics or actuarial science, or professionals working in the various financial sectors.

Martingales and Financial Mathematics in Discrete Time is also for anyone who may be interested in a rigorous and accessible mathematical construction of the tools and concepts used in financial mathematics, or in the application of the martingale theory in finance

Benoite de Saporta is Professor of applied mathematics at the University of Montpellier, France.

Mounir Zili is Professor of mathematics and member of the scientific council within the Faculty of Sciences at the University of Monastir, Tunisia.

Preface ix

Introduction xi

Chapter 1. Elementary Probabilities and an Introduction to Stochastic Processes 1

1.1. Measures and σ-algebras 1

1.2. Probability elements 5

1.2.1. Probabilities 5

1.2.2. Random variables 8

1.2.3. σ-algebra generated by a random variable 12

1.2.4. Random vectors 13

1.2.5. Convergence of sequences of random variables 15

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

2.2.1. Definitions 24

2.2.2. Properties of conditional expectation 28

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

3.1.1. Definition 45

3.1.2. Graphical representation 47

3.1.3. Reflection principle 50

3.2. Asymptotic behavior 52

3.2.1. The Markov property and stationarity property 52

3.2.2. First return to0 55

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

5.6.1. Simulation of trajectories 90

5.6.2. Portfolio optimization 90

5.6.3. Portfolio optimization with withdrawal 91

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

6.4.1. Completeness 98

6.4.2. Value of European options 101

6.4.3. Hedging European options 103

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

7.2.1. Snell envelope and stopping time 109

7.2.2. Construction of optimal stopping times 111

7.3. Application to American options 114

7.4. The Cox, Ross and Rubinstein model 115

7.4.1. Value of American options 115

7.4.2. Hedging American options 116

7.5. Exercises 116

7.6. Practical work 117

Chapter 8. Solutions to Exercises and Practical Work 119

8.1. Solutions to exercises in Chapter 1 119

8.1.1. Exercise 1.1 119

8.1.2. Exercise 1.2 119

8.1.3. Exercise 1.3 120

8.1.4. Exercise 1.4 120

8.1.5. Exercise 1.5 121

8.1.6. Exercise 1.6 122

8.1.7. Exercise 1.7 123

8.1.8. Exercise 1.8 125

8.2. Solutions to exercises in Chapter 2 127

8.2.1. Exercise 2.1 127

8.2.2. Exercise 2.2 129

8.2.3. Exercise 2.3 131

8.2.4. Exercise 2.4 133

8.2.5. Exercise 2.5 133

8.2.6. Exercise 2.6 135

8.2.7. Exercise 2.7 135

8.2.8. Exercise 2.8 136

8.2.9. Exercise 2.9 138

8.2.10. Exercise 2.10 138

8.2.11. Exercise 2.11 141

8.2.12. Exercise 2.12 142

8.3. Solutions to exercises in Chapter 3 143

8.3.1. Exercise 3.1 143

8.3.2. Exercise 3.2 144

8.3.3. Exercise 3.3 144

8.3.4. Exercise 3.4 146

8.3.5. Exercise 3.5 147

8.3.6. Exercise 3.6 148

8.3.7. Exercise 3.7 148

8.3.8. Exercise 3.8 149

8.4. Solutions to exercises in Chapter 4 151

8.4.1. Exercise 4.1 151

8.4.2. Exercise 4.2 151

8.4.3. Exercise 4.3 152

8.4.4. Exercise 4.4 153

8.4.5. Exercise 4.5 154

8.4.6. Exercise 4.6 157

8.4.7. Exercise 4.7 160

8.4.8. Exercise 4.8 163

8.4.9. Exercise 4.9 164

8.4.10. Exercise 4.10 166

8.4.11. Exercise 4.11 167

8.5. Solutions to exercises in Chapter 5 170

8.5.1. Exercise 5.1 170

8.5.2. Exercise 5.2 171

8.5.3. Exercise 5.3 173

8.5.4. Exercise 5.4 174

8.5.5. Exercise 5.5 174

8.6. Solutions to the practical exercises in Chapter 5 175

8.6.1. Practical exercise 5.6.1 175

8.6.2. Practical exercise 5.6.2 176

8.6.3. Practical exercise 5.6.3 183

8.7. Solutions to exercises in Chapter 6 189

8.7.1. Exercise 6.1 189

8.7.2. Exercise 6.2 190

8.7.3. Exercise 6.3 191

8.7.4. Exercise 6.4 193

8.8. Solution to the practical exercise in Chapter 6 (section 6.6) 193

8.9. Solution to exercises in Chapter 7 195

8.9.1. Exercise 7.1 195

8.9.2. Exercise 7.2 196

8.9.3. Exercise 7.3 197

8.10. Solution to the practical exercise in Chapter 7 (section 7.6) 200

References 205

Index 207

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