The Mathematics of Arbitrage by Freddy Delbaen and Walter Schachermayer

The Mathematics of Arbitrage

The Mathematics of Arbitrage by Freddy Delbaen and Walter Schachermayer

Contents of The Mathematics of Arbitrage

  • Part I A Guided Tour to Arbitrage Theory
  • The Story in a Nutshell
  • Arbitrage
  • An Easy Model of a Financial Market
  • Pricing by No-Arbitrage
  • Variations of the Example
  • Martingale Measures
  • The Fundamental Theorem of Asset Pricing
  • Models of Financial Markets on Finite Probability Spaces
  • Description of the Model
  • No-Arbitrage and the Fundamental Theorem of Asset Pricing
  • Equivalence of Single-period with Multiperiod Arbitrage
  • Pricing by No-Arbitrage
  • Change of Num´eraire
  • Kramkov’s Optional Decomposition Theorem
  • Utility Maximisation on Finite Probability Spaces
  • The Complete Case
  • The Incomplete Case
  • The Binomial and the Trinomial Model
  • Bachelier and Black-Scholes
  • Introduction to Continuous Time Models
  • Models in Continuous Time
  • Bachelier’s Model
  • The Black-Scholes Model
  • The Kreps-Yan Theorem
  • A General Framework
  • No Free Lunch
  • The Dalang-Morton-Willinger Theorem
  • Statement of the Theorem
  • The Predictable Range
  • The Selection Principle
  • The Closedness of the Cone C
  • Proof of the Dalang-Morton-Willinger Theorem for T =
  • A Utility-based Proof of the DMW Theorem for T =
  • Proof of the Dalang-Morton-Willinger Theorem for T ≥ by Induction on T
  • Proof of the Closedness of K in the Case T ≥
  • Proof of the Closedness of C in the Case T ≥ under the NA Condition
  • Proof of the Dalang-Morton-Willinger Theorem for T ≥ using the Closedness of C
  • Interpretation of the L∞-Bound in the DMW Theorem
  • A Primer in Stochastic Integration
  • The Set-up
  • Introductory on Stochastic Processes
  • Strategies, Semi-martingales and Stochastic Integration
  • Arbitrage Theory in Continuous Time: an Overview
  • Notation and Preliminaries
  • The Crucial Lemma
  • Sigma-martingales and the Non-locally Bounded Case
  • Part II The Original Papers
  • A General Version of the Fundamental Theorem of Asset Pricing
  • Introduction
  • Definitions and Preliminary Results
  • No Free Lunch with Vanishing Risk
  • Proof of the Main Theorem
  • The Set of Representing Measures
  • No Free Lunch with Bounded Risk
  • Simple Integrands
  • Appendix: Some Measure Theoretical Lemmas
  • A Simple Counter-Example to Several Problems in the Theory of Asset Pricing
  • Introduction and Known Results
  • Construction of the Example
  • Incomplete Markets
  • The No-Arbitrage Property under a Change of Num´eraire
  • Introduction
  • Basic Theorems
  • Duality Relation
  • Hedging and Change of Num´eraire
  • The Existence of Absolutely Continuous
  • Local Martingale Measures
  • Introduction
  • The Predictable Radon-Nikod´ym Derivative
  • The No-Arbitrage Property and Immediate Arbitrage
  • The Existence of an Absolutely Continuous
  • Local Martingale Measure
  • The Banach Space of Workable Contingent Claims in Arbitrage Theory
  • Introduction
  • Maximal Admissible Contingent Claims
  • The Banach Space Generated by Maximal Contingent Claims
  • Some Results on the Topology of G
  • The Value of Maximal Admissible Contingent Claims on the Set Me
  • The Space G under a Num´eraire Change
  • The Closure of G∞ and Related Problems
  • The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes
  • Introduction
  • Sigma-martingales
  • One-period Processes
  • The General Rd-valued Case
  • Duality Results and Maximal Elements
  • A Compactness Principle for Bounded Sequences of Martingales with Applications
  • Introduction
  • Notations and Preliminaries
  • An Example
  • A Substitute of Compactness for Bounded Subsets of H
  • Proof of Theorem A
  • Proof of Theorem C
  • Proof of Theorem B
  • A proof of M Yor’s Theorem
  • Proof of Theorem D
  • Application
  • Part III Bibliography
  • References

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