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Part Ⅰ A Guided Tour to Arbitrage Theory 3

1 The Story in a Nutshell 3

1.1 Arbitrage 3

1.2 An Easy Model of a Financial Market 4

1.3 Pricing by No-Arbitrage 5

1.4 Variations of the Example 7

1.5 Martingale Measures 7

1.6 The Fundamental Theorem of Asset Pricing 8

2 Models of Financial Markets on Finite Probability Spaces 11

2.1 Description of the Model 11

2.2 No-Arbitrage and the Fundamental Theorem of Asset Pricing 16

2.3 Equivalence of Single-period with Multiperiod Arbitrage 22

2.4 Pricing by No-Arbitrage 23

2.5 Change of Numéraire 27

2.6 Kramkov's Optional Decomposition Theorem 31

3 Utility Maximisation on Finite Probability Spaces 33

3.1 The Complete Case 34

3.2 The Incomplete Case 41

3.3 The Binomial and the Trinomial Model 45

4 Bachelier and Black-Scholes 57

4.1 Introduction to Continuous Time Models 57

4.2 Models in Continuous Time 57

4.3 Bachelier's Model 58

4.4 The Black-Scholes Model 60

5 The Kreps-Yan Theorem 71

5.1 A General Framework 71

5.2 No Free Lunch 76

6 The Dalang-Morton-Willinger Theorem 85

6.1 Statement of the Theorem 85

6.2 The Predictable Range 86

6.3 The Selection Principle 89

6.4 The Closedness of the Cone C 92

6.5 Proof of the Dalang-Morton-Willinger Theorem for T=1 94

6.6 A Utility-based Proof of the DMW Theorem for T=1 96

6.7 Proof of the Dalang-Morton-Willinger Theorem for T≥1 by Induction on T 102

6.8 Proof of the Closedness of K in the Case T≥1 103

6.9 Proof of the Closedness of C in the Case T≥1 under the(NA)Condition 105

6.10 Proof of the Dalang-Morton-Willinger Theorem for T≥1 using the Closedness of C 107

6.11 Interpretation of the L∞-Bound in the DMW Theorem 108

7 A Primer in Stochastic Integration 111

7.1 The Set-up 111

7.2 Introductory on Stochastic Processes 112

7.3 Strategies,Semi-martingales and Stochastic Integration 117

8 Arbitrage Theory in Continuous Time:an Overview 129

8.1 Notation and Preliminaries 129

8.2 The Crucial Lemma 131

8.3 Sigma-martingales and the Non-locally Bounded Case 140

Part Ⅱ The Original Papers 149

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

9.1 Introduction 149

9.2 Definitions and Preliminary Results 155

9.3 No Free Lunch with Vanishing Risk 160

9.4 Proof of the Main Theorem 164

9.5 The Set of Representing Measures 181

9.6 No Free Lunch with Bounded Risk 186

9.7 Simple Integrands 190

9.8 Appendix:Some Measure Theoretical Lemmas 202

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

10.1 Introduction and Known Results 207

10.2 Construction of the Example 210

10.3 Incomplete Markets 212

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

11.1 Introduction 217

11.2 Basic Theorems 219

11.3 Duality Relation 222

11.4 Hedging and Change of Numéraire 225

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

12.1 Introduction 231

12.2 The Predictable Radon-Nikod？m Derivative 235

12.3 The No-Arbitrage Property and Immediate Arbitrage 239

12.4 The Existence of an Absolutely Continuous Local Martingale Measure 244

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

13.1 Introduction 251

13.2 Maximal Admissible Contingent Claims 255

13.3 The Banach Space Generated by Maximal Contingent Claims 261

13.4 Some Results on the Topology of G 266

13.5 The Value of Maximal Admissible Contingent Claims on the Set Me 272

13.6 The Space G under a Numéraire Change 274

13.7 The Closure of G∞ and Related Problems 276

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

14.1 Introduction 279

14.2 Sigma-martingales 280

14.3 One-period Processes 284

14.4 The General Rd-valued Case 294

14.5 Duality Results and Maximal Elements 305

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

15.1 Introduction 319

15.2 Notations and Preliminaries 326

15.3 An Example 332

15.4 A Substitute of Compactness for Bounded Subsets of H1 334

15.4.1 Proof of Theorem 15.A 335

15.4.2 Proof of Theorem 15.C 337

15.4.3 Proof of Theorem 15.B 339

15.4.4 A proof of M.Yor's Theorem 345

15.4.5 Proof of Theorem 15.D 346

15.5 Application 352

Part Ⅲ Bibliography 359

References 359