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Part Ⅰ A Guided Tour to Arbitrage Theory3

1 The Story in a Nutshell3

1.1 Arbitrage3

1.2 An Easy Model of a Financial Market4

1.3 Pricing by No-Arbitrage5

1.4 Variations of the Example7

1.5 Martingale Measures7

1.6 The Fundamental Theorem of Asset Pricing8

2 Models of Financial Markets on Finite Probability Spaces11

2.1 Description of the Model11

2.2 No-Arbitrage and the Fundamental Theorem of Asset Pricing16

2.3 Equivalence of Single-period with Multiperiod Arbitrage22

2.4 Pricing by No-Arbitrage23

2.5 Change of Numéraire27

2.6 Kramkov's Optional Decomposition Theorem31

3 Utility Maximisation on Finite Probability Spaces33

3.1 The Complete Case34

3.2 The Incomplete Case41

3.3 The Binomial and the Trinomial Model45

4 Bachelier and Black-Scholes57

4.1 Introduction to Continuous Time Models57

4.2 Models in Continuous Time57

4.3 Bachelier's Model58

4.4 The Black-Scholes Model60

5 The Kreps-Yan Theorem71

5.1 A General Framework71

5.2 No Free Lunch76

6 The Dalang-Morton-Willinger Theorem85

6.1 Statement of the Theorem85

6.2 The Predictable Range86

6.3 The Selection Principle89

6.4 The Closedness of the Cone C92

6.5 Proof of the Dalang-Morton-Willinger Theorem for T=194

6.6 A Utility-based Proof of the DMW Theorem for T=196

6.7 Proof of the Dalang-Morton-Willinger Theorem for T≥1 by Induction on T102

6.8 Proof of the Closedness of K in the Case T≥1103

6.9 Proof of the Closedness of C in the Case T≥1 under the(NA)Condition105

6.10 Proof of the Dalang-Morton-Willinger Theorem for T≥1 using the Closedness of C107

6.11 Interpretation of the L∞-Bound in the DMW Theorem108

7 A Primer in Stochastic Integration111

7.1 The Set-up111

7.2 Introductory on Stochastic Processes112

7.3 Strategies,Semi-martingales and Stochastic Integration117

8 Arbitrage Theory in Continuous Time:an Overview129

8.1 Notation and Preliminaries129

8.2 The Crucial Lemma131

8.3 Sigma-martingales and the Non-locally Bounded Case140

Part Ⅱ The Original Papers149

9 A General Version of the Fundamental Theorem of Asset Pricing(1994)149

9.1 Introduction149

9.2 Definitions and Preliminary Results155

9.3 No Free Lunch with Vanishing Risk160

9.4 Proof of the Main Theorem164

9.5 The Set of Representing Measures181

9.6 No Free Lunch with Bounded Risk186

9.7 Simple Integrands190

9.8 Appendix:Some Measure Theoretical Lemmas202

10 A Simple Counter-Example to Several Problems in the Theory of Asset Pricing(1998)207

10.1 Introduction and Known Results207

10.2 Construction of the Example210

10.3 Incomplete Markets212

11 The No-Arbitrage Property under a Change of Numéraire(1995)217

11.1 Introduction217

11.2 Basic Theorems219

11.3 Duality Relation222

11.4 Hedging and Change of Numéraire225

12 The Existence of Absolutely Continuous Local Martingale Measures(1995)231

12.1 Introduction231

12.2 The Predictable Radon-Nikod？m Derivative235

12.3 The No-Arbitrage Property and Immediate Arbitrage239

12.4 The Existence of an Absolutely Continuous Local Martingale Measure244

13 The Banach Space of Workable Contingent Claims in Arbitrage Theory(1997)251

13.1 Introduction251

13.2 Maximal Admissible Contingent Claims255

13.3 The Banach Space Generated by Maximal Contingent Claims261

13.4 Some Results on the Topology of G266

13.5 The Value of Maximal Admissible Contingent Claims on the Set Me272

13.6 The Space G under a Numéraire Change274

13.7 The Closure of G∞ and Related Problems276

14 The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes(1998)279

14.1 Introduction279

14.2 Sigma-martingales280

14.3 One-period Processes284

14.4 The General Rd-valued Case294

14.5 Duality Results and Maximal Elements305

15 A Compactness Principle for Bounded Sequences of Martingales with Applications(1999)319

15.1 Introduction319

15.2 Notations and Preliminaries326

15.3 An Example332

15.4 A Substitute of Compactness for Bounded Subsets of H1334

15.4.1 Proof of Theorem 15.A335

15.4.2 Proof of Theorem 15.C337

15.4.3 Proof of Theorem 15.B339

15.4.4 A proof of M.Yor's Theorem345

15.4.5 Proof of Theorem 15.D346

15.5 Application352

Part Ⅲ Bibliography359

References359