Stochastic Finance: An Introduction in Discrete Time

by Hans Föllmer, Alexander Schied

Overview

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.

Product Details

ISBN-13:	9783110218046
Publisher:	De Gruyter
Publication date:	01/28/2011
Series:	De Gruyter Textbook
Edition description:	3rd rev. and extend. ed.
Pages:	555
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

About the Author

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

Table of Contents

Preface to the third edition v

Preface to the second edition vi

Preface to the first edition vii

I Mathematical finance in one period 1

1 Arbitrage theory 3

1.1 Assets, portfolios, and arbitrage opportunities 3

1.2 Absence of arbitrage and martingale measures 7

1.3 Derivative securities 16

1.4 Complete market models 27

1.5 Geometric characterization of arbitrage-free models 33

1.6 Contingent initial data 37

2 Preferences 50

2.1 Preference relations and their numerical representation 51

2.2 Von Neumann-Morgenstern representation 57

2.3 Expected utility 67

2.4 Uniform preferences 83

2.5 Robust preferences on asset profiles 94

2.6 Probability measures with given marginals 113

3 Optimality and equilibrium 121

3.1 Portfolio optimization and the absence of arbitrage 121

3.2 Exponential utility and relative entropy 130

3.3 Optimal contingent claims 139

3.4 Optimal payoff profiles for uniform preferences 148

3.5 Robust utility maximization 151

3.6 Microeconomic equihbrium 159

4 Monetary measures of risk 175

4.1 Risk measures and their acceptance sets 176

4.2 Robust representation of convex risk measures 186

4.3 Convex risk measures on L∞ 199

4.4 Value at Risk 207

4.5 Law-invariant risk measures 213

4.6 Concave distortions 219

4.7 Comonotonic risk measures 228

4.8 Measures of risk in a financial market 236

4.9 Utility-based shortfall risk and divergence risk measures 246

II Dynamic hedging 259

5 Dynamic arbitrage theory 261

5.1 The multi-period market model 261

5.2 Arbitrage opportunities and martingale measures 266

5.3 European contingent claims 274

5.4 Complete markets 287

5.5 The binomial model 290

5.6 Exotic derivatives 296

5.7 Convergence to the Black-Scholes price 302

6 American contingent claims 321

6.1 Hedging strategies for the seller 321

6.2 Stopping strategies for the buyer 327

6.3 Arbitrage-free prices 337

6.4 Stability under pasting 342

6.5 Lower and upper Snell envelopes 347

7 Superhedging 354

7.1 P-supermartingales 354

7.2 Uniform Doob decomposition 356

7.3 Superhedging of American and European claims 359

7.4 Superhedging with liquid options 368

8 Efficient hedging 380

8.1 Quantile hedging 380

8.2 Hedging with minimal shortfall risk 387

8.3 Efficient hedging with convex risk measures 396

9 Hedging under constraints 404

9.1 Absence of arbitrage opportunities 404

9.2 Uniform Doob decomposition 412

9.3 Upper Snell envelopes 417

9.4 Superhedging and risk measures 424

10 Minimizing the hedging error 428

10.1 Local quadratic risk 428

10.2 Minimal martingale measures 438

10.3 Variance-optimal hedging 449

11 Dynamic risk measures 456

11.1 Conditional risk measures and their robust representation 456

11.2 Time consistency 465

Appendix 476

A.1 Convexity 476

A.2 Absolutely continuous probability measures 480

A.3 Quantile functions 484

A.4 The Neyman-Pearson lemma 493

A.5 The essential supremum of a family of random variables 496

A.6 Spaces of measures 497

A.7 Some functional analysis 507

Notes 512

Bibliography 517

List of symbols 533

Index 535

Editorial Reviews

"This book provides a fairly complete treatment of the most important probabilistic aspects of financial mathematics (or stochastic finance). [...] It is a worthwhile addition to the literature and will serve as highly recommended reading for students in the subject area for some years to come." Mathematical Reviews (review of the first edition)

"Since the appearance of the first edition in 2002, this book has become a classic in mathematical finance and has served in numerous courses as a basic textbook and as a basic source and introduction to the field for graduate students and researchers. [...] Altogether this is an extraordinarily well-done book concerning the subjects chosen, with a clear and readable description of its aims and material and a precise and explicit mathematical presentation. [...] This third edition has benefited from some new focus and material and is an enjoyable read and a most relevant textbook." Mathematical Reviews (review of the third edition)

From the Publisher