ABOUT THIS BOOK
This is the third, revised and extended edition of the classical introduction to the mathematics of finance, based on stochastic models in discrete time. In the first part of the book simple one-period models are studied, in the second part the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Due to the strong appeal and wide use of this book, it is now available as a textbook with exercises. It will be of value for a broad community of students and researchers. It may serve as basis for graduate courses and be also interesting for those who work in the financial industry and want to get an idea about the mathematical methods of risk assessment.


TABLE OF 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 arbitrage-free models
    1.6 Contingent initial data
2. Preferences
    2.1 Preference relations and their numerical representation
    2.2 Von Neumann-Morgenstern 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 Law-invariant risk measures
    4.6 Concave distortions
    4.7 Comonotonic risk measures
    4.8 Measures of risk in a financial market
    4.9 Utility-based shortfall risk and divergence risk measures
Part II: Dynamic hedging
5. Dynamic arbitrage theory
    5.1 The multi-period 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 Black-Scholes price
6. American contingent claims
    6.1 Hedging strategies for the seller
    6.2 Stopping strategies for the buyer
    6.3 Arbitrage-free prices
    6.4 Stability under pasting
    6.5 Lower Snell envelopes
7. Superhedging
    7.1 {mathcal P}-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 Variance-optimal 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 Neyman-Pearson 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