Martingales and Financial Mathematics in Discrete Time. Benoîte de Saporta

      

Table of Contents

      1  Cover

      2  Title Page

      3  Copyright

      4  Preface

      5  Introduction

      6  1 Elementary Probabilities and an Introduction to Stochastic Processes 1.1. Measures and σ-algebras 1.2. Probability elements 1.3. Stochastic processes 1.4. Exercises

      7  2 Conditional Expectation 2.1. Conditional probability with respect to an event 2.2. Conditional expectation 2.3. Geometric interpretation 2.4. Conditional expectation and independence 2.5. Exercises

      8  3 Random Walks 3.1. Trajectories of the random walk 3.2. Asymptotic behavior 3.3. The Gambler’s ruin 3.4. Exercises

      9  4 Martingales 4.1. Definition 4.2. Martingale transform 4.3. The Doob decomposition 4.4. Stopping time 4.5. Stopped martingales 4.6. Exercises

      10  5 Financial Markets 5.1. Financial assets 5.2. Investment strategies 5.3. Arbitrage 5.4. The Cox, Ross and Rubinstein model 5.5. Exercises 5.6. Practical work

      11  6 European Options 6.1. Definition 6.2. Complete markets 6.3. Valuation and hedging 6.4. Cox, Ross and Rubinstein model 6.5. Exercises 6.6. Practical work: Simulating the value of a call option

      12  7 American Options 7.1. Definition 7.2. Optimal stopping 7.3. Application to American options 7.4. The Cox, Ross and Rubinstein model 7.5. Exercises 7.6. Practical work

      13  8 Solutions to Exercises and Practical Work 8.1. Solutions to exercises in Chapter 1 8.2. Solutions to exercises in Chapter 2 8.3. Solutions to exercises in Chapter 3 8.4. Solutions to exercises in Chapter 4 8.5. Solutions to exercises in Chapter 5 8.6. Solutions to the practical exercises in Chapter 5 8.7. Solutions to exercises in Chapter 6 8.8. Solution to the practical exercise in Chapter 6 (section 6.6) 8.9. Solution to exercises in Chapter 7 8.10. Solution to the practical exercise in Chapter 7 (section 7.6)

      14  References

      15  Index

      16  End User License Agreement

      List of Illustrations

      1 Chapter 3Figure 3.1. Graphical representation of a trajectory of a random walk between 0 ...Figure 3.2. Two paths from (1, 1) to (5, 3). For a color version of this figure,...Figure 3.3. A path from (0, 2) to (11, 1) passing through 0 (the unbroken blue l...

      2 Chapter 8Figure 8.1. Possible trajectories for the random walk of four steps starting fro...Figure 8.2. Possible paths from (0, 0) to (3, 1). For a color version of this fi...Figure 8.3. Event tree for the financial market in Exercise 5.1Figure 8.4. Event tree for the financial market in Exercise 5.3Figure 8.5. Trajectories of the risky asset for the Cox, Ross and Rubinstein mod...Figure 8.6. Trajectories of the risky