Martingales and Financial Mathematics in Discrete Time

Description

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 Cover Half-Title Page Title Page Copyright Page Contents Preface Introduction Chapter 1. Elementary Probabilities and an Introduction to Stochastic Processes 1.1. Measures and s-algebras 1.2. Probability elements 1.2.1. Probabilities 1.2.2. Random variables 1.2.3. σ-algebra generated by a random variable 1.2.4. Random vectors 1.2.5. Convergence of sequences of random variables 1.3. Stochastic processes 1.4. Exercises Chapter 2. Conditional Expectation 2.1. Conditional probability with respect to an event 2.2. Conditional expectation 2.2.1. Definitions 2.2.2. Properties of conditional expectation 2.3. Geometric interpretation 2.4. Conditional expectation and independence 2.5. Exercises Chapter 3. Random Walks 3.1. Trajectories of the random walk 3.1.1. Definition 3.1.2. Graphical representation 3.1.3. Reflection principle 3.2. Asymptotic behavior 3.2.1. The Markov property and stationarity property 3.2.2. First return to 0 3.3. The Gambler’s ruin 3.4. Exercises Chapter 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 Chapter 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 5.6.1. Simulation of trajectories 5.6.2. Portfolio optimization 5.6.3. Portfolio optimization with withdrawal Chapter 6. European Options 6.1. Definition 6.2. Complete markets 6.3. Valuation and hedging 6.4. Cox, Ross and Rubinstein model 6.4.1. Completeness 6.4.2. Value of European options 6.4.3. Hedging European options 6.5. Exercises 6.6. Practical work: Simulating the value of a call option Chapter 7. American Options 7.1. Definition 7.2. Optimal stopping 7.2.1. Snell envelope and stopping time 7.2.2. Construction of optimal stopping times 7.3. Application to American options 7.4. The Cox, Ross and Rubinstein model 7.4.1. Value of American options 7.4.2. Hedging American options 7.5. Exercises 7.6. Practical work Chapter 8. Solutions to Exercises and Practical Work 8.1. Solutions to exercises in Chapter 1 8.1.1. Exercise 1.1 8.1.2. Exercise 1.2 8.1.3. Exercise 1.3 8.1.4. Exercise 1.4 8.1.5. Exercise 1.5 8.1.6. Exercise 1.6 8.1.7. Exercise 1.7 8.1.8. Exercise 1.8 8.2. Solutions to exercises in Chapter 2 8.2.1. Exercise 2.1 8.2.2. Exercise 2.2 8.2.3. Exercise 2.3 8.2.4. Exercise 2.4 8.2.5. Exercise 2.5 8.2.6. Exercise 2.6 8.2.7. Exercise 2.7 8.2.8. Exercise 2.8 8.2.9. Exercise 2.9 8.2.10. Exercise 2.10 8.2.11. Exercise 2.11 8.2.12. Exercise 2.12 8.3. Solutions to exercises in Chapter 3 8.3.1. Exercise 3.1 8.3.2. Exercise 3.2 8.3.3. Exercise 3.3 8.3.4. Exercise 3.4 8.3.5. Exercise 3.5 8.3.6. Exercise 3.6 8.3.7. Exercise 3.7 8.3.8. Exercise 3.8 8.4. Solutions to exercises in Chapter 4 8.4.1. Exercise 4.1 8.4.2. Exercise 4.2 8.4.3. Exercise 4.3 8.4.4. Exercise 4.4 8.4.5. Exercise 4.5 8.4.6. Exercise 4.6 8.4.7. Exercise 4.7 8.4.8. Exercise 4.8 8.4.9. Exercise 4.9 8.4.10. Exercise 4.10 8.4.11. Exercise 4.11 8.5. Solutions to exercises in Chapter 5 8.5.1. Exercise 5.1 8.5.2. Exercise 5.2 8.5.3. Exercise 5.3 8.5.4. Exercise 5.4 8.5.5. Exercise 5.5 8.6. Solutions to the practical exercises in Chapter 5 8.6.1. Practical exercise 5.6.1 8.6.2. Practical exercise 5.6.2 8.6.3. Practical exercise 5.6.3 8.7. Solutions to exercises in Chapter 6 8.7.1. Exercise 6.1 8.7.2. Exercise 6.2 8.7.3. Exercise 6.3 8.7.4. Exercise 6.4 8.8. Solution to the practical exercise in Chapter 6 (section 6.6) 8.9. Solution to exercises in Chapter 7 8.9.1. Exercise 7.1 8.9.2. Exercise 7.2 8.9.3. Exercise 7.3 8.10. Solution to the practical exercise in Chapter 7 (section 7.6) References Index Other titles from iSTE in Mathematics and Statistics