Martingales and Financial Mathematics in Discrete Time

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 ixIntroduction xi Chapter 1 Elementary Probabilities and an Introduction to Stochastic Processes 11.1 Measures and σ-algebras 11.2 Probability elements 51.3. Stochastic processes 161.4. Exercises 19Chapter 2 Conditional Expectation 212.1 Conditional probability with respect to an event 212.2. Conditional expectation 242.3. Geometric interpretation 372.4. Conditional expectation and independence 382.5. Exercises 41Chapter 3 Random Walks 453.1 Trajectories of the random walk 453.2. Asymptotic behavior 523.3. The Gambler’s ruin 583.4. Exercises 60Chapter 4 Martingales 634.1 Definition 634.2. Martingale transform 664.3 The Doob decomposition 674.4 Stopping time 694.5 Stopped martingales 714.6. Exercises 75Chapter 5 Financial Markets 815.1. Financial assets 825.2. Investment strategies 825.3. Arbitrage 845.4. The Cox, Ross and Rubinstein model 865.5. Exercises 885.6. Practical work 90Chapter 6 European Options 956.1 Definition 956.2. Complete markets 966.3. Valuation and hedging 976.4. Cox, Ross and Rubinstein model 986.5. Exercises 1046.6. Practical work: Simulating the value of a call option 106Chapter 7 American Options 1077.1 Definition 1077.2 Optimal stopping 1097.4. The Cox, Ross and Rubinstein model 1157.5. Exercises 1167.6. Practical work 117Chapter 8 Solutions to Exercises and Practical Work 1198.1. Solutions to exercises in Chapter1 1198.2. Solutions to exercises in Chapter2 1278.3 Solutions to exercises in Chapter3 1438.4. Solutions to exercises in Chapter4 1518.5. Solutions to exercises in Chapter5 1708.6.Solutions to the practical exercises in Chapter5 1758.7. Solutions to exercises in Chapter6 1898.8. Solution to the practical exercise in Chapter6 (section6.6) 1938.9. Solution to exercises in Chapter7 1958.10. Solution to the practical exercise in Chapter7 (section7.6) 200References 205Index 207