Stochastic Finance (eBook)

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Hans Föllmer, Humboldt-Universität zu Berlin, Germany; Alexander Schied, University of Mannheim, Germany.

Hans Föllmer, Humboldt-Universität zu Berlin; Alexander Schied, Universität Mannheim.

Preface to the third edition 6
Preface to the second edition 7
Preface to the first edition 8
Contents 10
I Mathematical finance in one period 14
1 Arbitrage theory 16
1.1 Assets, portfolios, and arbitrage opportunities 16
1.2 Absence of arbitrage and martingale measures 20
1.3 Derivative securities 29
1.4 Complete market models 40
1.5 Geometric characterization of arbitrage-free models 46
1.6 Contingent initial data 50
2 Preferences 63
2.1 Preference relations and their numerical representation 64
2.2 Von Neumann–Morgenstern representation 70
2.3 Expected utility 80
2.4 Stochastic dominance 96
2.5 Robust preferences on asset profiles 107
2.6 Probability measures with given marginals 126
3 Optimality and equilibrium 134
3.1 Portfolio optimization and the absence of arbitrage 134
3.2 Exponential utility and relative entropy 143
3.3 Optimal contingent claims 152
3.4 Optimal payoff profiles for uniform preferences 161
3.5 Robust utility maximization 164
3.6 Microeconomic equilibrium 172
4 Monetary measures of risk 188
4.1 Risk measures and their acceptance sets 189
4.2 Robust representation of convex risk measures 199
4.3 Convex risk measures on L1 212
4.4 Value at Risk 220
4.5 Law-invariant risk measures 226
4.6 Concave distortions 232
4.7 Comonotonic risk measures 241
4.8 Measures of risk in a financial market 249
4.9 Utility-based shortfall risk and divergence risk measures 259
II Dynamic hedging 272
5 Dynamic arbitrage theory 274
5.1 The multi-period market model 274
5.2 Arbitrage opportunities and martingale measures 279
5.3 European contingent claims 287
5.4 Complete markets 300
5.5 The binomial model 303
5.6 Exotic derivatives 309
5.7 Convergence to the Black–Scholes price 315
6 American contingent claims 334
6.1 Hedging strategies for the seller 334
6.2 Stopping strategies for the buyer 340
6.3 Arbitrage-free prices 350
6.4 Stability under pasting 355
6.5 Lower and upper Snell envelopes 360
7 Superhedging 367
7.1 P -supermartingales 367
7.2 Uniform Doob decomposition 369
7.3 Superhedging of American and European claims 372
7.4 Superhedging with liquid options 381
8 Efficient hedging 393
8.1 Quantile hedging 393
8.2 Hedging with minimal shortfall risk 400
8.3 Efficient hedging with convex risk measures 409
9 Hedging under constraints 417
9.1 Absence of arbitrage opportunities 417
9.2 Uniform Doob decomposition 425
9.3 Upper Snell envelopes 430
9.4 Superhedging and risk measures 437
10 Minimizing the hedging error 441
10.1 Local quadratic risk 441
10.2 Minimal martingale measures 451
10.3 Variance-optimal hedging 462
11 Dynamic risk measures 469
11.1 Conditional risk measures and their robust representation 469
11.2 Time consistency 478
Appendix 489
A.1 Convexity 489
A.2 Absolutely continuous probability measures 493
A.3 Quantile functions 497
A.4 The Neyman–Pearson lemma 506
A.5 The essential supremum of a family of random variables 509
A.6 Spaces of measures 510
A.7 Some functional analysis 520
Notes 525
Bibliography 530
List of symbols 546
Index 548