Preface vii

Chapter 1. Financial Markets and Derivatives 1

1.1. Financial Markets 1

1.2. Derivatives 2

1.3. Exercise 5

Chapter 2. Binomial Model 7

2.1. Binomial or CRR Model 7

2.2. Pricing a European Contingent Claim 10

2.3. Pricing an American Contingent Claim 19

2.4. Exercises 28

Chapter 3. Finite Market Model 31

3.1. Definition of the Finite Market Model 32

3.2. First Fundamental Theorem of Asset Pricing 34

3.3. Second Fundamental Theorem of Asset Pricing 39

3.4. Pricing European Contingent Claims 44

3.5. Incomplete Markets 47

3.6. Separating Hyperplane Theorem 51

3.7. Exercises 52

Chapter 4. Black-Scholes Model 55

4.1. Preliminaries 56

4.2. Black-Scholes Model 57

v

vi *Contents*

4.3. Equivalent Martingale Measure 61

4.4. European Contingent Claims 63

4.5. Pricing European Contingent Claims 65

4.6. European Call Option — Black-Scholes Formula 69

4.7. American Contingent Claims 74

4.8. American Call Option 80

4.9. American Put Option 83

4.10. Exercises 86

Chapter 5. Multi-dimensional Black-Scholes Model 89

5.1. Preliminaries 91

5.2. Multi-dimensional Black-Scholes Model 92

5.3. First Fundamental Theorem of Asset Pricing 99

5.4. Form of Equivalent Local Martingale Measures 101

5.5. Second Fundamental Theorem of Asset Pricing 110

5.6. Pricing European Contingent Claims 116

5.7. Incomplete Markets 120

5.8. Exercises 121

Appendix A. Conditional Expectation and *Lp*-Spaces 123

Appendix B. Discrete Time Stochastic Processes 127

Appendix C. Continuous Time Stochastic Processes 131

Appendix D. Brownian Motion and Stochastic Integration 135

D.1. Brownian Motion 135

D.2. Stochastic Integrals (with respect to Brownian motion) 136

D.3. Itˆo Process 139

D.4. Itˆo Formula 141

D.5. Girsanov Transformation 142

D.6. Martingale Representation Theorem 143

Bibliography 145

Index 149