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Stochastic Finance, 4th Edition

Book Description

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 incomleteness 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.
This fourth, newly revised edition contains more than one hundred exercises. It also includes material on risk measures and the related issue of model uncertainty, in particular a chapter on dynamic risk measures and sections on robust utility maximization and on efficient hedging with convex risk measures.

Contents:
Part I: Mathematical finance in one period

Arbitrage theory
Preferences
Optimality and equilibrium
Monetary measures of risk
Part II: Dynamic hedging
Dynamic arbitrage theory
American contingent claims
Superhedging
Efficient hedging
Hedging under constraints
Minimizing the hedging error
Dynamic risk measures

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Preface to the fourth edition
  5. Preface to the third edition
  6. Preface to the second edition
  7. Preface to the first edition
  8. Contents
  9. Part I: Mathematical finance in one period
    1. 1 Arbitrage theory
      1. 1.1 Assets, portfolios, and arbitrage opportunities
      2. 1.2 Absence of arbitrage and martingale measures
      3. 1.3 Derivative securities
      4. 1.4 Complete market models
      5. 1.5 Geometric characterization of arbitrage-free models
      6. 1.6 Contingent initial data
    2. 2 Preferences
      1. 2.1 Preference relations and their numerical representation
      2. 2.2 Von Neumann–Morgenstern representation
      3. 2.3 Expected utility
      4. 2.4 Stochastic dominance
      5. 2.5 Robust preferences on asset profiles
      6. 2.6 Probability measures with given marginals
    3. 3 Optimality and equilibrium
      1. 3.1 Portfolio optimization and the absence of arbitrage
      2. 3.2 Exponential utility and relative entropy
      3. 3.3 Optimal contingent claims
      4. 3.4 Optimal payoff profiles for uniform preferences
      5. 3.5 Robust utility maximization
      6. 3.6 Microeconomic equilibrium
    4. 4 Monetary measures of risk
      1. 4.1 Risk measures and their acceptance sets
      2. 4.2 Robust representation of convex risk measures
      3. 4.3 Convex risk measures on L∞
      4. 4.4 Value at Risk
      5. 4.5 Law-invariant risk measures
      6. 4.6 Concave distortions
      7. 4.7 Comonotonic risk measures
      8. 4.8 Measures of risk in a financial market
      9. 4.9 Utility-based shortfall risk and divergence risk measures
  10. Part II: Dynamic hedging
    1. 5 Dynamic arbitrage theory
      1. 5.1 The multi-period market model
      2. 5.2 Arbitrage opportunities and martingale measures
      3. 5.3 European contingent claims
      4. 5.4 Complete markets
      5. 5.5 The binomial model
      6. 5.6 Exotic derivatives
      7. 5.7 Convergence to the Black–Scholes price
    2. 6 American contingent claims
      1. 6.1 Hedging strategies for the seller
      2. 6.2 Stopping strategies for the buyer
      3. 6.3 Arbitrage-free prices
      4. 6.4 Stability under pasting
      5. 6.5 Lower and upper Snell envelopes
    3. 7 Superhedging
      1. 7.1 P-supermartingales
      2. 7.2 Uniform Doob decomposition
      3. 7.3 Superhedging of American and European claims
      4. 7.4 Superhedging with liquid options
    4. 8 Efficient hedging
      1. 8.1 Quantile hedging
      2. 8.2 Hedging with minimal shortfall risk
      3. 8.3 Efficient hedging with convex risk measures
    5. 9 Hedging under constraints
      1. 9.1 Absence of arbitrage opportunities
      2. 9.2 Uniform Doob decomposition
      3. 9.3 Upper Snell envelopes
      4. 9.4 Superhedging and risk measures
    6. 10 Minimizing the hedging error
      1. 10.1 Local quadratic risk
      2. 10.2 Minimal martingale measures
      3. 10.3 Variance-optimal hedging
    7. 11 Dynamic risk measures
      1. 11.1 Conditional risk measures and their robust representation
      2. 11.2 Time consistency
  11. Appendix
    1. A.1 Convexity
    2. A.2 Absolutely continuous probability measures
    3. A.3 Quantile functions
    4. A.4 The Neyman–Pearson lemma
    5. A.5 The essential supremum of a family of random variables
    6. A.6 Spaces of measures
    7. A.7 Some functional analysis
  12. Bibliographical notes
  13. References
  14. List of symbols
  15. Index