Preface v
List of Figures xi
1 Pricing by Arbitrage 1
1.1 Introduction: Pricing and Hedging . . . . . . . . . . . . . . 1
1.2 Single-Period Option Pricing Models . . . . . . . . . . . . . 9
1.3 A General Single-Period Model . . . . . . . . . . . . . . . . 12
1.4 A Single-Period Binomial Model . . . . . . . . . . . . . . . 13
1.5 Multi-Period Binomial Models . . . . . . . . . . . . . . . . 17
1.6 Bounds on Option Prices . . . . . . . . . . . . . . . . . . . 21
2 Martingale Measures 23
2.1 A General Discrete-Time Market Model . . . . . . . . . . . 23
2.2 Trading Strategies and Arbitrage Opportunities . . . . . . . 25
2.3 Martingales and Risk-Neutral Pricing . . . . . . . . . . . . 30
2.4 Arbitrage Pricing with Martingale Measures . . . . . . . . . 32
2.5 Example: Martingale Formulation of the Binomial Market
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6 From CRR to Black–Scholes . . . . . . . . . . . . . . . . . . 38
3 The Fundamental Theorem of Asset Pricing 45
3.1 The Separating Hyperplane Theorem in Rn . . . . . . . . . 45
3.2 Construction of Martingale Measures . . . . . . . . . . . . . 47
3.3 A Local Form of the ‘No Arbitrage’ Condition . . . . . . . . 49
3.4 Two Simple Examples . . . . . . . . . . . . . . . . . . . . . 56
3.5 Equivalent Martingale Measures
for Discrete Market Models . . . . . . . . . . . . . . . . . . 59
4 Complete Markets and Martingale Representation 63
4.1 Uniqueness of the EMM . . . . . . . . . . . . . . . . . . . . 63
4.2 Completeness and Martingale Representation . . . . . . . . 65
4.3 Martingale Representation in the CRR-Model . . . . . . . . 66
4.4 The Splitting Index and Completeness . . . . . . . . . . . . 70
4.5 Characterisation of Attainable Claims . . . . . . . . . . . . 73
5 Stopping Times and American Options 75
5.1 Hedging American Claims . . . . . . . . . . . . . . . . . . . 75
5.2 Stopping Times and Stopped Processes . . . . . . . . . . . 77
5.3 Uniformly Integrable Martingales . . . . . . . . . . . . . . . 80
5.4 Optimal Stopping: The Snell Envelope . . . . . . . . . . . . 86
5.5 Pricing and Hedging American Options . . . . . . . . . . . 93
5.6 Consumption–Investment Strategies . . . . . . . . . . . . . 96
6 A Review of Continuous-Time Stochastic Calculus 99
6.1 Continuous-Time Processes . . . . . . . . . . . . . . . . . . 99
6.2 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.3 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . 109
6.4 The Itˆo Calculus . . . . . . . . . . . . . . . . . . . . . . . . 118
6.5 Stochastic Differential Equations . . . . . . . . . . . . . . . 126
6.6 The Markov Property of Solutions of SDEs . . . . . . . . . 130
7 European Options in Continuous Time 135
7.1 Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
7.2 Girsanov’s Theorem . . . . . . . . . . . . . . . . . . . . . . 136
7.3 Martingale Representation . . . . . . . . . . . . . . . . . . . 142
7.4 Self-Financing Strategies . . . . . . . . . . . . . . . . . . . . 151
7.5 An Equivalent Martingale Measure . . . . . . . . . . . . . . 154
7.6 The Black–Scholes Formula . . . . . . . . . . . . . . . . . . 163
7.7 A Multi-Dimensional Situation . . . . . . . . . . . . . . . . 167
7.8 Barrier Options . . . . . . . . . . . . . . . . . . . . . . . . . 172
8 The American Option 187
8.1 Extended Trading Strategies . . . . . . . . . . . . . . . . . . 187
8.2 Analysis of American Put Options . . . . . . . . . . . . . . 190
8.3 The Perpetual Put Option . . . . . . . . . . . . . . . . . . . 196
8.4 Early Exercise Premium . . . . . . . . . . . . . . . . . . . . 199
8.5 Relation to Free Boundary Problems . . . . . . . . . . . . . 202
8.6 An Approximate Solution . . . . . . . . . . . . . . . . . . . 208
9 Bonds and Term Structure 211
9.1 Market Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 211
9.2 Future Price and Futures Contracts . . . . . . . . . . . . . 215
9.3 Changing Num´eraire . . . . . . . . . . . . . . . . . . . . . . 219
9.4 A General Option Pricing Formula . . . . . . . . . . . . . . 222
9.5 Term Structure Models . . . . . . . . . . . . . . . . . . . . 227
9.6 Diffusion Models for the Short-Term Rate Process . . . . . 229
9.7 The Heath–Jarrow–Morton Model . . . . . . . . . . . . . . 242
9.8 A Markov Chain Model . . . . . . . . . . . . . . . . . . . . 247
10 Consumption-Investment Strategies 251
10.1 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . 251
10.2 Admissible Strategies . . . . . . . . . . . . . . . . . . . . . . 253
10.3 Utility Maximization from Consumption . . . . . . . . . . . 258
10.4 Maximization of Terminal Utility . . . . . . . . . . . . . . . 263
10.5 Utility Maximization for Both Consumption
and Terminal Wealth . . . . . . . . . . . . . . . . . . . . . . 266
References 271
Index 289 | 