**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