Contents
Preface to the second edition v
Preface to the first edition vii
I Mathematical finance in one period 1
1 Arbitrage theory 3
1.1 Assets, portfolios, and arbitrage opportunities . . . . . . . . . . . . . 3
1.2 Absence of arbitrage and martingale measures . . . . . . . . . . . . . 6
1.3 Derivative securities . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Complete market models . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Geometric characterization of arbitrage-free models . . . . . . . . . . 27
1.6 Contingent initial data . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Preferences 44
2.1 Preference relations and their numerical representation . . . . . . . . 45
2.2 Von Neumann–Morgenstern representation . . . . . . . . . . . . . . . 51
2.3 Expected utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.4 Uniform preferences . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.5 Robust preferences on asset profiles . . . . . . . . . . . . . . . . . . 86
2.6 Probability measures with given marginals . . . . . . . . . . . . . . . 99
3 Optimality and equilibrium 108
3.1 Portfolio optimization and the absence of arbitrage . . . . . . . . . . 108
3.2 Exponential utility and relative entropy . . . . . . . . . . . . . . . . . 116
3.3 Optimal contingent claims . . . . . . . . . . . . . . . . . . . . . . . 125
3.4 Microeconomic equilibrium . . . . . . . . . . . . . . . . . . . . . . 137
4 Monetary measures of risk 152
4.1 Risk measures and their acceptance sets . . . . . . . . . . . . . . . . 153
4.2 Robust representation of convex risk measures . . . . . . . . . . . . . 161
4.3 Convex risk measures on L∞ . . . . . . . . . . . . . . . . . . . . . . 171
4.4 Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
4.5 Law-invariant risk measures . . . . . . . . . . . . . . . . . . . . . . 183
4.6 Concave distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.7 Comonotonic risk measures . . . . . . . . . . . . . . . . . . . . . . . 195
4.8 Measures of risk in a financial market . . . . . . . . . . . . . . . . . 203
4.9 Shortfall risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
x Contents
II Dynamic hedging 221
5 Dynamic arbitrage theory 223
5.1 The multi-period market model . . . . . . . . . . . . . . . . . . . . . 223
5.2 Arbitrage opportunities and martingale measures . . . . . . . . . . . 227
5.3 European contingent claims . . . . . . . . . . . . . . . . . . . . . . . 234
5.4 Complete markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
5.5 The binomial model . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.6 Exotic derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.7 Convergence to the Black–Scholes price . . . . . . . . . . . . . . . . 259
6 American contingent claims 277
6.1 Hedging strategies for the seller . . . . . . . . . . . . . . . . . . . . 277
6.2 Stopping strategies for the buyer . . . . . . . . . . . . . . . . . . . . 282
6.3 Arbitrage-free prices . . . . . . . . . . . . . . . . . . . . . . . . . . 292
6.4 Stability under pasting . . . . . . . . . . . . . . . . . . . . . . . . . 297
6.5 Lower and upper Snell envelopes . . . . . . . . . . . . . . . . . . . . 300
7 Superhedging 308
7.1 P-supermartingales . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
7.2 Uniform Doob decomposition . . . . . . . . . . . . . . . . . . . . . 310
7.3 Superhedging of American and European claims . . . . . . . . . . . . 313
7.4 Superhedging with liquid options . . . . . . . . . . . . . . . . . . . . 322
8 Efficient hedging 333
8.1 Quantile hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
8.2 Hedging with minimal shortfall risk . . . . . . . . . . . . . . . . . . 339
9 Hedging under constraints 350
9.1 Absence of arbitrage opportunities . . . . . . . . . . . . . . . . . . . 350
9.2 Uniform Doob decomposition . . . . . . . . . . . . . . . . . . . . . 357
9.3 Upper Snell envelopes . . . . . . . . . . . . . . . . . . . . . . . . . 362
9.4 Superhedging and risk measures . . . . . . . . . . . . . . . . . . . . 369
10 Minimizing the hedging error 372
10.1 Local quadratic risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
10.2 Minimal martingale measures . . . . . . . . . . . . . . . . . . . . . . 382
10.3 Variance-optimal hedging . . . . . . . . . . . . . . . . . . . . . . . . 392
Appendix 399
A.1 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
A.2 Absolutely continuous probability measures . . . . . . . . . . . . . . 403
A.3 Quantile functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
A.4 The Neyman–Pearson lemma . . . . . . . . . . . . . . . . . . . . . . 414
Contents xi
A.5 The essential supremum of a family of random variables . . . . . . . 417
A.6 Spaces of measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
A.7 Some functional analysis . . . . . . . . . . . . . . . . . . . . . . . . 428
Notes 433
Bibliography 439
List of symbols 449
Index 451


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