# 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 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
• 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