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