Föllmer, Hans / Schied, AlexanderStochastic Finance:An Introduction in Discrete TimeThird revised and extended editionPublished in January 2011
by Walter de Gruyter,
Berlin. xii + 544 pages. Paperback 

This third revised and extended edition now contains more than one hundred exercises. It also includes new material on risk measures and the related issue of model uncertainty, in particular a new chapter on dynamic risk measures and new sections on robust utility maximization and on efficient hedging with convex risk measures.
Contents:
Part I: Mathematical finance in one period
1. Arbitrage theory
1.1 Assets, portfolios, and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures
1.3 Derivative securities
1.4 Complete market models
1.5 Geometric characterization of arbitragefree
models
1.6 Contingent initial data
2. Preferences
2.1 Preference relations and their numerical
representation
2.2 Von NeumannMorgenstern representation
2.3 Expected utility
2.4 Uniform preference
2.5 Robust preferences on asset profiles
2.6 Probability measures with given marginals
3. Optimality and equilibrium
3.1 Portfolio optimization and the absence
of arbitrage
3.2 Exponential utility and relative entropy
3.3 Optimal contingent claims
3.4 Optimal payoff profiles for uniform preferences
3.5 Robust utility maximization
3.6 Microeconomic equilibrium
4. Monetary measures of risk
4.1 Risk measures and their acceptance sets
4.2 Robust representation of convex risk measures
4.3 Convex risk measures on L^{∞}
4.4 Value at Risk
4.5 Lawinvariant risk measures
4.6 Concave distortions
4.7 Comonotonic risk measures
4.8 Measures of risk in a financial market
4.9 Utilitybased shortfall risk and divergence risk
measures
Part II: Dynamic hedging
5. Dynamic arbitrage theory
5.1 The multiperiod market model
5.2 Arbitrage opportunities and martingale measures
5.3 European contingent claims
5.4 Complete markets
5.5 The binomial model
5.6 Exotiv derivatives
5.7 Convergence to the BlackScholes price
6. American contingent claims
6.1 Hedging strategies for the seller
6.2 Stopping strategies for the buyer
6.3 Arbitragefree prices
6.4 Stability under pasting
6.5 Lower Snell envelopes
7. Superhedging
7.1 supermartingales and
upper
Snell envelopes
7.2 Uniform Doob decomposition
7.3 Superhedging of American and European claims
7.4 Superhedging with liquid options
8. Efficient hedging
8.1 Quantile hedging
8.2 Hedging with minimal shortfall risk
8.3 Efficient hedging with convex risk measures
9. Hedging under constraints
9.1 Absence of arbitrage opportunities
9.2 Uniform Doob decomposition
9.3 Upper Snell envelopes
9.4 Superhedging and risk measures
10. Minimizing the hedging error
10.1 Local quadratic risk
10.2 Minimal martingale measures
10.3 Varianceoptimal hedging
11. Dynamic risk measures
11.1 Conditional risk measures and their robust
representation
11.2 Time consistency
Appendix
A.1 Convexity
A.2 Absolutely continuous probability measures
A.3 Quantile functions
A.4 The NeymanPearson lemma
A.5 The essential supremum of a family of random
variables
A.6 Spaces of measures
A.7 Some functional analysis
Notes
References
List of symbols
Index