1 Pricing by Arbitrage 1
1.1 Introduction:Pricing and Hedging 1
1.2 Single-Period Option Pricing Models 10
1.3 A General Single-Period Model 12
1.4 A Single-Period Binomial Model 14
1.5 Multi-period Binomial Models 20
1.6 Bounds on Option Prices 24
2 Martingale Measures 27
2.1 A General Discrete-Time Market Model 27
2.2 Trading Strategies 29
2.3 Martingales and Risk-Neutral Pricing 35
2.4 Arbitrage Pricing:Martingale Measures 38
2.5 Strategies Using Contingent Claims 43
2.6 Example:The Binomial Model 48
2.7 From CRR to Black-Scholes 50
3 The First Fundamental Theorem 57
3.1 The Separating Hyperplane Theorem in Rn 57
3.2 Construction of Martingale Measures 59
3.3 Pathwise Description 61
3.4 Examples 69
3.5 General Discrete Models 71
4 Complete Markets 87
4.1 Completeness and Martingale Representation 88
4.2 Completeness for Finite Market Models 89
4.3 The CRR Model 91
4.4 The Splitting Index and Completeness 94
4.5 Incomplete Models:The Arbitrage Interval 97
4.6 Characterisation of Complete Models 101
5 Discrete-time American Options 105
5.1 Hedging American Claims 105
5.2 Stopping Times and Stopped Processes 107
5.3 Uniformly Integrable Martingales 110
5.4 Optimal Stopping:The Snell Envelope 116
5.5 Pricing and Hedging American Options 124
5.6 Consumption-Investment Strategies 126
6 Continuous-Time Stochastic Calculus 131
6.1 Continuous-Time Processes 131
6.2 Martingales 135
6.3 Stochastic Integrals 141
6.4 The It？ Calculus 149
6.5 Stochastic Differential Equations 158
6.6 Markov Property of Solutions of SDEs 162
7 Continuous-Time European Options 167
7.1 Dynamics 167
7.2 Girsanov's Theorem 168
7.3 Martingale Representation 174
7.4 Self-Financing Strategies 183
7.5 An Equivalent Martingale Measure 185
7.6 Black-Scholes Prices 193
7.7 Pricing in a Multifactor Model 198
7.8 Barrier Options 204
7.9 The Black-Scholes Equation 214
7.10 The Greeks 217
8 The American Put Option 223
8.1 Extended Trading Strategies 223
8.2 Analysis of American Put Options 226
8.3 The Perpetual Put Option 231
8.4 Early Exercise Premium 234
8.5 Relation to Free Boundary Problems 238
8.6 An Approximate Solution 243
9 Bonds and Term Structure 247
9.1 Market Dynamics 247
9.2 Future Price and Futures Contracts 252
9.3 Changing Numéraire 255
9.4 A General Option Pricing Formula 258
9.5 Term Structure Models 262
9.6 Short-rate Diffusion Models 264
9.7 The Heath-Jarrow-Morton Model 277
9.8 A Markov Chain Model 282
10 Consumption-Investment Strategies 285
10.1 Utility Functions 285
10.2 Admissible Strategies 287
10.3 Maximising Utility of Consumption 291
10.4 Maximisation of Terminal Utility 296
10.5 Consumption and Terminal Wealth 299
11 Measures of Risk 303
11.1 Value at Risk 304
11.2 Coherent Risk Measures 308
11.3 Deviation Measures 316
11.4 Hedging Strategies with Shortfall Risk 320
Bibliography 329
Index 349