Föllmer/Schied, Stochastic Finance

Föllmer, Hans / Schied, Alexander

Stochastic Finance:

An Introduction in Discrete Time

Second revised and extended edition

Published in November 2004 by Walter de Gruyter, Berlin.
First edition July 2002.

xi + 459 pages. Hardcover 24 x 17 cm.
€ 58,- / sFr 93,- / for USA, Canada, Mexico US$ 59,95

ISBN 3-11-018346-3

de Gruyter Studies in Mathematics 27

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For download:: Preface to the second edition Preface to the first edition Table of contents Errata sheet

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 intrinsic risks arising from market incompleteness appears at a very early stage.

The first part of the book contains a study of a simple one-period model, which also serves as a building block for later developments. Topics include the characterization of arbitrage-free markets, preferences on asset profiles, an introduction to equilibrium analysis, and
monetary measures of financial risk.

In the second part, the idea of dynamic hedging of contingent claims is developed in a multiperiod framework. Topics include martingale measures, pricing formulas for derivatives, American options, superhedging, and hedging strategies with minimal shortfall risk.

In this second edition major parts have been improved or entirely rewritten. Among them are those on robust representations of risk measures, arbitrage-free pricing of contingent claims, exotic derivatives in the CRR model, convergence to Black-Scholes prices, and stability under pasting with its connections to dynamically consistent coherent risk measures. Moreover, new sections have been added, including a systematic discussion of law-invariant risk measures, of concave distortions, and of the relations between risk measures and Choquet integration.

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 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 Shortfall risk

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

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

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