Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . vii
1 Pricing by Arbitrage 1
1.1 Introduction: Pricing and Hedging . . . . . . . . . . . . . . 1
1.2 Single-Period Option PricingModels . . . . . . . . . . . . . 10
1.3 A General Single-PeriodModel . . . . . . . . . . . . . . . . 12
1.4 A Single-Period BinomialModel . . . . . . . . . . . . . . . 14
1.5 Multi-period BinomialModels . . . . . . . . . . . . . . . . . 20
1.6 Bounds on Option Prices . . . . . . . . . . . . . . . . . . . 24
2 Martingale Measures 27
2.1 A General Discrete-TimeMarketModel . . . . . . . . . . . 27
2.2 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 Martingales and Risk-Neutral Pricing . . . . . . . . . . . . 35
2.4 Arbitrage Pricing: MartingaleMeasures . . . . . . . . . . . 38
2.5 Strategies Using Contingent Claims . . . . . . . . . . . . . . 43
2.6 Example: The BinomialModel . . . . . . . . . . . . . . . . 48
2.7 FromCRR to Black-Scholes . . . . . . . . . . . . . . . . . . 50
3 The First Fundamental Theorem 57
3.1 The Separating Hyperplane Theorem in Rn . . . . . . . . . 57
3.2 Construction ofMartingaleMeasures . . . . . . . . . . . . . 59
3.3 Pathwise Description . . . . . . . . . . . . . . . . . . . . . . 61
3.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.5 General DiscreteModels . . . . . . . . . . . . . . . . . . . . 71
4 Complete Markets 87
4.1 Completeness and Martingale Representation . . . . . . . . 88
4.2 Completeness for FiniteMarketModels . . . . . . . . . . . 89
4.3 The CRRModel . . . . . . . . . . . . . . . . . . . . . . . . 91
4.4 The Splitting Index and Completeness . . . . . . . . . . . . 94
4.5 IncompleteModels: The Arbitrage Interval . . . . . . . . . 97
4.6 Characterisation of CompleteModels . . . . . . . . . . . . . 101
ix
x CONTENTS
5 Discrete-time American Options 105
5.1 Hedging American Claims . . . . . . . . . . . . . . . . . . . 105
5.2 Stopping Times and Stopped Processes . . . . . . . . . . . 107
5.3 Uniformly IntegrableMartingales . . . . . . . . . . . . . . . 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 Itˆo 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 EquivalentMartingaleMeasure . . . . . . . . . . . . . . 185
7.6 Black-Scholes Prices . . . . . . . . . . . . . . . . . . . . . . 193
7.7 Pricing in aMultifactorModel . . . . . . . . . . . . . . . . 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 Num´eraire . . . . . . . . . . . . . . . . . . . . . . 255
9.4 A General Option Pricing Formula . . . . . . . . . . . . . . 258
9.5 TermStructureModels . . . . . . . . . . . . . . . . . . . . 262
9.6 Short-rate DiffusionModels . . . . . . . . . . . . . . . . . . 264
9.7 The Heath-Jarrow-MortonModel . . . . . . . . . . . . . . . 277
9.8 AMarkov ChainModel . . . . . . . . . . . . . . . . . . . . 282
CONTENTS xi
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 TerminalWealth . . . . . . . . . . . . . . 299
11 Measures of Risk 303
11.1 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . 304
11.2 Coherent RiskMeasures . . . . . . . . . . . . . . . . . . . . 308
11.3 DeviationMeasures . . . . . . . . . . . . . . . . . . . . . . . 316
11.4 Hedging Strategies with Shortfall Risk . . . . . . . . . . . . 320
Bibliography 329
Index 349