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The Mathematics of Arbitrage

Publisher: 
Springer Verlag
Number of Pages: 
373
Price: 
79.95
ISBN: 
3540219927

What is arbitrage? Following the authors' definition, arbitrage is “...a possibility to make a profit in a financial market without risk and without net investment of capital. ” We can relate to no arbitrage principle by the old saying “there ain't no such thing as a free lunch.” The notion of arbitrage is a key principle in financial theory today. Mathematics of Arbitrage deals with the theories of pricing and hedging financial derivatives given a “no arbitrage” principle.

The book is divided into two parts. Part I, A Guided Tour to Arbitrage Theory , is an introduction to the mathematics of arbitrage (the second part of the book). It covers the basic principles of the Fundamental Theorem of Asset Pricing, general models of financial markets on finite probability spaces, Bachelier's and the “Black-Scholes” models in continuous time, and a short account on stochastic integration. Part II consists of reprinted and updated research papers by the authors. The papers included are:

  • A General Version of the Fundamental Theorem of Asset Pricing
  • A Simple Counter-Example to Several Problems in the Theory of Asset Pricing
  • The No-Arbitrage Property under a Change of Numéraire
  • The Existence of Absolutely Continuous Local Martingale Measures
  • The Banach Space of Workable Contingent Claims in Arbitrage Theory
  • The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes
  • A Compactness Principle for Bounded Sequences of Martingales with Applications

This is by all means a research book and as such it is written. Previous in depth knowledge of the concept of arbitrage and the basic models of financial market is not necessary, since Part I of the book covers these concepts from the basics level. One would need to be versatile in graduate level probability and stochastic processes to follow the book efficiently. I believe this book to be quite useful to those involved in research on the Fundamental Theorem of Asset Pricing and the concept of arbitrage in general.


Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.

Date Received: 
Friday, February 17, 2006
Reviewable: 
Yes
Include In BLL Rating: 
No
Freddy Delbaen and Walter Schachermayer
Series: 
Springer Finance
Publication Date: 
2006
Format: 
Hardcover
Category: 
Monograph
Tags: 
Ita Cirovic Donev
06/13/2006

Part I A Guided Tour to Arbitrage Theory

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 Num´eraire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

XIV Contents

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

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

Contents XV

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 Num´eraire (1995) . . . . . . . . . . . . . . . . . . . . . . 217

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

11.2 Basic Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

11.3 Duality Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

11.4 Hedging and Change of Num´eraire. . . . . . . . . . . . . . . . . . . . . . . . . 225

12 The Existence of Absolutely Continuous

Local Martingale Measures (1995) . . . . . . . . . . . . . . . . . . . . . . . . . 231

12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

12.2 The Predictable Radon-Nikod´ym 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 Num´eraire 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

XVI Contents

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

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

Publish Book: 
Modify Date: 
Tuesday, June 13, 2006

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