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Preface ix
Frequently Used Notation xiii
Chapter 1. The Mean-Variance Approach in a One-Period Model 1
Chapter 2. The Continuous-Time Market Model 11
§2.1. Modeling the Security Prices 11
Excursion 1: Brownian Motion and Martingales 15
2.1. Continuation: Modeling the Security Prices 23
Excursion 2: The Ito Integral 26
Excursion 3: The Ito Formula 42
§2.2. Trading Strategy and Wealth Process 56
§2.3. Properties of the Continuous-Time Market Model 64
Excursion 4: The Martingale Representation Theorem 71
Exercises 76
Chapter 3. Option Pricing 79
§3.1. Introduction 79
§3.2. Option Pricing via the Replication Principle 83
Excursion 5: Girsanov's Theorem 93
3.2. Continuation: Option Pricing via the Replication Principle 99
§3.3. Option Pricing by the Partial Differential Approach 105
Excursion 6: The Feynman-Kac Representation 111
§3.4. Arbitrage Bounds for American and European Options 122
§3.5. Pricing of American Options 129
§3.6. Arbitrage, Equivalent Martingale Measures and Option
Pricing 134
§3.7. Market Numeraire and Numeraire Invariance 143
Exercises 148
Chapter 4. Pricing of Exotic Options and Numerical Algorithms 153
§4.1. Exotic Options with Explicit Pricing Formulae 155
a) Path independent options on one stock 155
b) Options on more than one underlying stock 162
c) Path dependent options 167
Excursion 7: Weak Convergence of Stochastic Processes 170
§4.2. Monte-Carlo Simulation 175
§4.3. Approximation via Binomial Trees 177
§4.4. Trinomial Trees and Explicit Finite-Difference Methods 187
§4.5. The Pathwise Binomial Approach of Rogers and Stapleton 191
Exercises 201
Chapter 5. Optimal Portfolios 203
§5.1. Introduction and Formulation of the Problem 203
§5.2. The Martingale Method 206
§5.3. Optimal Option Portfolios 215
Excursion 8: Stochastic Control 223
§5.4. Portfolio Optimization via the Stochastic Control Method 236
Exercises 244
Bibliography 247
Index 251
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