The Mathematics of Arbitrage (eBook)

Walter Schachermeyer, born in 1950 in Linz, Austria, has received--as the first mathematician--the 1998 Wittgenstein Award, Austria's highest honor for scienctific achievement. Since 1998 he holds the Chair for Actuarial and Financial Mathematics at the Vienna University of Technolgoy. Among his achievements is the proof of the "Fundamental Theorem of Asset Pricing" in its general form, which was done in joint work with Freddy Delbaen. Freddy Delbaen, born in 1946 in Duffel/Antwerpen, Belgium, is Professor for Financial Mathematics at the ETH in Zurich since 1995.

Preface 7
Contents 13
Part I A Guided Tour to Arbitrage Theory 17
1 The Story in a Nutshell 19
1.1 Arbitrage 19
1.2 An Easy Model of a Financial Market 20
1.3 Pricing by No-Arbitrage 21
1.4 Variations of the Example 23
1.5 Martingale Measures 23
1.6 The Fundamental Theorem of Asset Pricing 24
2 Models of Financial Markets on Finite Probability Spaces 27
2.1 Description of the Model 27
2.2 No-Arbitrage and the Fundamental Theorem of Asset Pricing 32
2.3 Equivalence of Single-period with Multiperiod Arbitrage 38
2.4 Pricing by No-Arbitrage 39
2.5 Change of Numéraire 43
2.6 Kramkov’s Optional Decomposition Theorem 47
3 Utility Maximisation on Finite Probability Spaces 49
3.1 The Complete Case 50
3.2 The Incomplete Case 57
3.3 The Binomial and the Trinomial Model 61
4 Bachelier and Black-Scholes 73
4.1 Introduction to Continuous Time Models 73
4.2 Models in Continuous Time 73
4.3 Bachelier's Model 74
4.4 The Black-Scholes Model 76
5 The Kreps-Yan Theorem 87
5.1 A General Framework 87
5.2 No Free Lunch 92
6 The Dalang-Morton-Willinger Theorem 101
6.1 Statement of the Theorem 101
6.2 The Predictable Range 102
6.3 The Selection Principle 105
6.4 The Closedness of the Cone C 108
6.5 Proof of the Dalang-Morton-Willinger Theorem for T = 1 110
6.6 A Utility-based Proof of the DMW Theorem for T = 1 112
6.7 Proof of the Dalang-Morton-Willinger Theorem for T = 1 by Induction on T 118
6.8 Proof of the Closedness of K in the Case T = 1 119
6.9 Proof of the Closedness of C in the Case T = 1 under the (NA) Condition 121
6.10 Proof of the Dalang-Morton-Willinger Theorem for T = 1 using the Closedness of C 123
6.11 Interpretation of the L8-Bound in the DMW Theorem 124
7 A Primer in Stochastic Integration 127
7.1 The Set-up 127
7.2 Introductory on Stochastic Processes 128
7.3 Strategies, Semi-martingales and Stochastic Integration 133
8 Arbitrage Theory in Continuous Time: an Overview 145
8.1 Notation and Preliminaries 145
8.2 The Crucial Lemma 147
8.3 Sigma-martingales and the Non-locally Bounded Case 156
Part II The Original Papers 163
9 A General Version of the Fundamental Theorem of Asset Pricing (1994) 165
9.1 Introduction 165
9.2 De.nitions and Preliminary Results 171
9.3 No Free Lunch with Vanishing Risk 176
9.4 Proof of the Main Theorem 180
9.5 The Set of Representing Measures 197
9.6 No Free Lunch with Bounded Risk 202
9.7 Simple Integrands 206
9.8 Appendix: Some Measure Theoretical Lemmas 218
10 A Simple Counter-Example to Several Problems in the Theory of Asset Pricing (1998) 223
10.1 Introduction and Known Results 223
10.2 Construction of the Example 226
10.3 Incomplete Markets 228
11 The No-Arbitrage Property under a Change of Numéraire (1995) 233
11.1 Introduction 233
11.2 Basic Theorems 235
11.3 Duality Relation 238
11.4 Hedging and Change of Numéraire 241
12 The Existence of Absolutely Continuous Local Martingale Measures (1995) 247
12.1 Introduction 247
12.2 The Predictable Radon-Nikod´ym Derivative 251
12.3 The No-Arbitrage Property and Immediate Arbitrage 255
12.4 The Existence of an Absolutely Continuous Local Martingale Measure 260
13 The Banach Space of Workable Contingent Claims in Arbitrage Theory (1997) 267
13.1 Introduction 267
13.2 Maximal Admissible Contingent Claims 271
13.3 The Banach Space Generated by Maximal Contingent Claims 277
13.4 Some Results on the Topology of G 282
13.5 The Value of Maximal Admissible Contingent Claims on the Set Me 288
13.6 The Space G under a Num´eraire Change 290
13.7 The Closure of G8 and Related Problems 292
14 The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes (1998) 295
14.1 Introduction 295
14.2 Sigma-martingales 296
14.3 One-period Processes 300
14.4 The General Rd-valued Case 310
14.5 Duality Results and Maximal Elements 321
15 A Compactness Principle for Bounded Sequences of Martingales with Applications (1999) 335
15.1 Introduction 335
15.2 Notations and Preliminaries 342
15.3 An Example 348
15.4 A Substitute of Compactness for Bounded Subsets of H1 350
15.4.1 Proof of Theorem 15.A. 351
15.4.2 Proof of Theorem 15.C. 353
15.4.3 Proof of Theorem 15.B. 355
15.4.4 A proof of M. Yor’s Theorem 361
15.4.5 Proof of Theorem 15.D 362
15.5 Application 368
Part III Bibliography 373
References 375

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

9.1 Introduction

A basic result in mathematical .nance, sometimes called the fundamental theorem of asset pricing (see [DR 87]), is that for a stochastic process (St)t ¸R +, the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. In .nance the process (St)t ¸R + describes the random evolution of the discounted price of one or several .nancial assets. The equivalence of no-arbitrage with the existence of an equivalent probability martingale measure is at the basis of the entire theory of "pricing by arbitrage". Starting from the economically meaningful assumption that S does not allow arbitrage pro.ts (di.erent variants of this concept will be de.ned below), the theorem allows the probability P on the underlying probability space (.,F,P) to be replaced by an equivalent measure Q such that the process S becomes a martingale under the new measure. This makes it possible to use the rich machinery of martingale theory. In particular the problem of fair pricing of contingent claims is reduced to taking expected values with respect to the measure Q. This method of pricing contingent claims is known to actuaries since the introduction of actuarial skills, centuries ago and known by the name of "equivalence principle".

The theory of martingale representation allows to characterise those assets that can be reproduced by buying and selling the basic assets. One might get the impression that martingale theory and the general theory of stochastic processes were tailor-made for .nance (see [HP 81]). The change of measure from P to Q can also be seen as a result of risk aversion. By changing the physical probability measure from P to Q, one can attribute more weight to unfavourable events and less weight to more favourable ones.

As an example that this technique has in fact a long history, we quote the use of mortality tables in insurance. The actual mortality table is replaced by a table re.ecting more mortality if a life insurance premium is calculated but is replaced by a table re.ecting a lower mortality rate if e.g. a lump sum buying a pension is calculated. Changing probabilities is common practice in actuarial sciences. It is therefore amazing to notice that today’s actuaries are introducing these modern .nancial methods at such a slow pace.

The present paper focuses on the question: "What is the precise meaning of the word essentially in the .rst paragraph of the paper?" The question has a twofold interest. From an economic point of view one wants to understand the precise relation between concepts of no-arbitrage type and the existence of an equivalent martingale measure in order to understand the exact limitations up to which the above sketched approach may be extended. From a purely mathematical point of view it is also of natural interest to get a better understanding of the question which stochastic processes are martingales after an appropriate change to an equivalent probability measure. We refer to the well-known fact that a semi-martingale becomes a quasi-martingale under a well-chosen equivalent law (see [P 90]); from here to the question whether we can obtain a martingale, or more generally a local martingale, is natural.