Contents
Chapter 1. Introduction 1
1.1. Overview 1
1.2. Portfolio Theory 2
1.3. Fundamentals of Arbitrage Theory 3
1.4. The Black-Scholes Model 6
1.5. Summary 8
1.6. Problems 8
Chapter 2. Brownian Motion and Stochastic Integration 9
2.1. Definition of Brownian Motion 9
2.2. Construction of Brownian Motion 13
2.3. Problems 15
2.4. Definition of Stochastic Integration 16
2.5. Itˆo’s Formula 19
2.6. Itˆo Processes 22
2.7. Summary 25
2.8. Problems 25
Chapter 3. The Black-Scholes Analysis: Part I 27
3.1. The Black-Scholes PDE 27
3.2. The Black-Scholes Formula and Greeks 28
3.3. Summary 31
3.4. Problems 31
Chapter 4. Conditional Probability and Itˆo Processes 33
4.1. Conditional Probability 33
4.2. Conditioning with Itˆo Processes 35
4.3. Martingales 39
4.4. The Markov Property 41
4.5. Summary 42
4.6. Problems 43
Chapter 5. The Black-Scholes Analysis: Part II 44
5.1. The Feynman-Kaˇc Formula 44
5.2. Girsanov Transformation 48
ii
CONTENTS iii
5.3. Examples 53
5.4. Problems 55
Chapter 6. Complete Markets 57
6.1. Market Price of Risk 57
6.2. The Pricing Measure Q 59
6.3. Vector Itˆo Processes 61
6.4. Markets with Multiple Risky Securities 64
6.5. Martingale Representation 67
6.6. Decomposition 68
6.7. Problems 70
Chapter 7. Futures and Dividends 72
7.1. Futures 72
7.2. Dividends 75
7.3. Problems 78
Chapter 8. Computation 79
8.1. Simulation 79
8.2. Calibration 82
8.3. Problems 85
Chapter 9. Volatility Models 87
9.1. Local Volatility 87
9.2. Stochastic Volatility 89
Appendix A. Final Review 91
Appendix B. Solutions to Problems 92
Appendix. Bibliography 103
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