Föllmer, Hans / Schied, AlexanderStochastic Finance:An Introduction in Discrete TimePublished in July 2002 by Walter
de Gruyter
, Berlin. ix + 422 pages. Hardcover 24 x 17 cm. ISBN 3110171198 

This book is an introduction to financial mathematics. It is intended
for graduate students in mathematics and for researchers working in academia
and industry.
The focus on stochastic models in discrete time has two immediate
benefits. First, the probabilistic machinery is simpler, and one can
discuss right away some of the key problems in the theory of pricing
and hedging of financial derivatives. Second, the paradigm of a complete
financial market, where all derivatives admit a perfect hedge, becomes
the exception rather than the rule. Thus, the need to confront the problems
arising in incomplete financial market models appears at a very
early stage.
The first part of the book studies a simple oneperiod model which
serves as a building stone for later developments. Topics include
the characterization of arbitragefree markets, the representation
of preferences on asset profiles by expected utility theory and its
robust extensions, monetary measures of risk, and an introduction to
equilibrium analysis.
In the second part, the idea of dynamic hedging of contingent claims
is developed in a multiperiod framework. Such models are typically
incomplete: They involve intrinsic risks which cannot be hedged away
completely. Topics include martingale measures, pricing formulas for
derivatives, American options, superhedging, and hedging strategies with
minimal shortfall risk. Markets are modeled on general probability spaces.
Thus, the text captures the interplay between probability theory
and functional
analysis which has been crucial for recent advances in
mathematical finance.
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 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 Measures of risk in a financial market
4.6 Shortfall risk
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 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 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 derivatives
8. Efficient hedging
8.1 Quantile hedging
8.2 Hedging with minimal shortfall risk
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
Appendix
A.1 Convexity
A.2 Absolutely continuous probability measures
A.3 The NeymanPearson lemma
A.4 The essential supremum of a family of random
variables
A.5 Spaces of measures
A.6 Some functional analysis
Notes
References
List of symbols
Index