The mathematics of arbitrage

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Springer Finance Editorial Board M Avellaneda G Barone-Adesi M Broadie M.H.A Davis E Derman C Klüppelberg E Kopp W Schachermayer Springer Finance Springer Finance is a programme of books aimed at students, academics and practitioners working on increasingly technical approaches to the analysis of financial markets It aims to cover a variety of topics, not only mathematical finance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and financial economics Ammann M., Credit Risk Valuation: Methods, Models, and Application (2001) Back K., A Course in Derivative Securities: Introduction to Theory and Computation (2005) Barucci E., Financial Markets Theory Equilibrium, Efficiency and Information (2003) Bielecki T.R and Rutkowski M., Credit Risk: Modeling, Valuation and Hedging (2002) Bingham N.H and Kiesel R., Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives (1998, 2nd ed 2004) Brigo D and Mercurio F., Interest Rate Models: Theory and Practice (2001) Buff R., Uncertain Volatility Models-Theory and Application (2002) Dana R.A and Jeanblanc M., Financial Markets in Continuous Time (2002) Deboeck G and Kohonen T (Editors), Visual Explorations in Finance with Self-Organizing Maps (1998) Delbaen F and Schachermayer W., The Mathematics of Arbitrage (2005) Elliott R.J and Kopp P.E., Mathematics of Financial Markets (1999, 2nd ed 2005) Fengler M.R., Semiparametric Modeling of Implied Volatility (200) Geman H., Madan D., Pliska S.R and Vorst T (Editors), Mathematical Finance–Bachelier Congress 2000 (2001) Gundlach M., Lehrbass F (Editors), CreditRisk+ in the Banking Industry (2004) Kellerhals B.P., Asset Pricing (2004) Külpmann M., Irrational Exuberance Reconsidered (2004) Kwok Y.-K., Mathematical Models of Financial Derivatives (1998) Malliavin P and Thalmaier A., Stochastic Calculus of Variations in Mathematical Finance (2005) Meucci A., Risk and Asset Allocation (2005) Pelsser A., Efficient Methods for Valuing Interest Rate Derivatives (2000) Prigent J.-L., Weak Convergence of Financial Markets (2003) Schmid B., Credit Risk Pricing Models (2004) Shreve S.E., Stochastic Calculus for Finance I (2004) Shreve S.E., Stochastic Calculus for Finance II (2004) Yor M., Exponential Functionals of Brownian Motion and Related Processes (2001) Zagst R., Interest-Rate Management (2002) Zhu Y.-L., Wu X., Chern I.-L., Derivative Securities and Difference Methods (2004) Ziegler A., Incomplete Information and Heterogeneous Beliefs in Continuous-time Finance (2003) Ziegler A., A Game Theory Analysis of Options (2004) Freddy Delbaen · Walter Schachermayer The Mathematics of Arbitrage 123 Freddy Delbaen ETH Zürich Departement Mathematik, Lehrstuhl für Finanzmathematik Rämistr 101 8092 Zürich Switzerland E-mail: delbaen@math.ethz.ch Walter Schachermayer Technische Universität Wien Institut für Finanz- und Versicherungsmathematik Wiedner Hauptstr 8-10 1040 Wien Austria E-mail: wschach@fam.tuwien.ac.at Mathematics Subject Classification (2000): M13062, M27004, M12066 Library of Congress Control Number: 2005937005 ISBN-10 3-540-21992-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-21992-7 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Cover design: design & production, Heidelberg Typesetting by the authors using a Springer LATEX macro package Produktion: LE-TEX Jelonek, Schmidt & Vöckler GbR Printed on acid-free paper 41/3142YL - To Rita and Christine with love Preface In 1973 F Black and M Scholes published their pathbreaking paper [BS 73] on option pricing The key idea — attributed to R Merton in a footnote of the Black-Scholes paper — is the use of trading in continuous time and the notion of arbitrage The simple and economically very convincing “principle of noarbitrage” allows one to derive, in certain mathematical models of financial markets (such as the Samuelson model, [S 65], nowadays also referred to as the “Black-Scholes” model, based on geometric Brownian motion), unique prices for options and other contingent claims This remarkable achievement by F Black, M Scholes and R Merton had a profound effect on financial markets and it shifted the paradigm of dealing with financial risks towards the use of quite sophisticated mathematical models It was in the late seventies that the central role of no-arbitrage arguments was crystallised in three seminal papers by M Harrison, D Kreps and S Pliska ([HK 79], [HP 81], [K 81]) They considered a general framework, which allows a systematic study of different models of financial markets The Black-Scholes model is just one, obviously very important, example embedded into the framework of a general theory A basic insight of these papers was the intimate relation between no-arbitrage arguments on one hand, and martingale theory on the other hand This relation is the theme of the “Fundamental Theorem of Asset Pricing” (this name was given by Ph Dybvig and S Ross [DR 87]), which is not just a single theorem but rather a general principle to relate no-arbitrage with martingale theory Loosely speaking, it states that a mathematical model of a financial market is free of arbitrage if and only if it is a martingale under an equivalent probability measure; once this basic relation is established, one can quickly deduce precise information on the pricing and hedging of contingent claims such as options In fact, the relation to martingale theory and stochastic integration opens the gates to the application of a powerful mathematical theory VIII Preface The mathematical challenge is to turn this general principle into precise theorems This was first established by M Harrison and S Pliska in [HP 81] for the case of finite probability spaces The typical example of a model based on a finite probability space is the “binomial” model, also known as the “CoxRoss-Rubinstein” model in finance Clearly, the assumption of finite Ω is very restrictive and does not even apply to the very first examples of the theory, such as the Black-Scholes model or the much older model considered by L Bachelier [B 00] in 1900, namely just Brownian motion Hence the question of establishing theorems applying to more general situations than just finite probability spaces Ω remained open Starting with the work of D Kreps [K 81], a long line of research of increasingly general — and mathematically rigorous — versions of the “Fundamental Theorem of Asset Pricing” was achieved in the past two decades It turned out that this task was mathematically quite challenging and to the benefit of both theories which it links As far as the financial aspect is concerned, it helped to develop a deeper understanding of the notions of arbitrage, trading strategies, etc., which turned out to be crucial for several applications, such as for the development of a dynamic duality theory of portfolio optimisation (compare, e.g., the survey paper [S 01a]) Furthermore, it also was fruitful for the purely mathematical aspects of stochastic integration theory, leading in the nineties to a renaissance of this theory, which had originally flourished in the sixties and seventies It would go beyond the framework of this preface to give an account of the many contributors to this development We refer, e.g., to the papers [DS 94] and [DS 98], which are reprinted in Chapters and 14 In these two papers the present authors obtained a version of the “Fundamental Theorem of Asset Pricing”, pertaining to general Rd -valued semimartingales The arguments are quite technical Many colleagues have asked us to provide a more accessible approach to these results as well as to several other of our related papers on Mathematical Finance, which are scattered through various journals The idea for such a book already started in 1993 and 1994 when we visited the Department of Mathematics of Tokyo University and gave a series of lectures there Following the example of M Yor [Y 01] and the advice of C Byrne of Springer-Verlag, we finally decided to reprint updated versions of seven of our papers on Mathematical Finance, accompanied by a guided tour through the theory This guided tour provides the background and the motivation for these research papers, hopefully making them more accessible to a broader audience The present book therefore is organised as follows Part I contains the “guided tour” which is divided into eight chapters In the introductory chapter we present, as we did before in a note in the Notices of the American Mathematical Society [DS 04], the theme of the Fundamental Theorem of As- Preface IX set Pricing in a nutshell This chapter is very informal and should serve mainly to build up some economic intuition In Chapter we then start to present things in a mathematically rigourous way In order to keep the technicalities as simple as possible we first restrict ourselves to the case of finite probability spaces Ω This implies that all the function spaces Lp (Ω, F , P) are finite-dimensional, thus reducing the functional analytic delicacies to simple linear algebra In this chapter, which presents the theory of pricing and hedging of contingent claims in the framework of finite probability spaces, we follow closely the Saint Flour lectures given by the second author [S 03] In Chapter we still consider only finite probability spaces and develop the basic duality theory for the optimisation of dynamic portfolios We deal with the cases of complete as well as incomplete markets and illustrate these results by applying them to the cases of the binomial as well as the trinomial model In Chapter we give an overview of the two basic continuous-time models, the “Bachelier” and the “Black-Scholes” models These topics are of course standard and may be found in many textbooks on Mathematical Finance Nevertheless we hope that some of the material, e.g., the comparison of Bachelier versus Black-Scholes, based on the data used by L Bachelier in 1900, will be of interest to the initiated reader as well Thus Chapters 1–4 give expositions of basic topics of Mathematical Finance and are kept at an elementary technical level From Chapter on, the level of technical sophistication has to increase rather steeply in order to build a bridge to the original research papers We systematically study the setting of general probability spaces (Ω, F , P) We start by presenting, in Chapter 5, D Kreps’ version of the Fundamental Theorem of Asset Pricing involving the notion of “No Free Lunch” In Chapter we apply this theory to prove the Fundamental Theorem of Asset Pricing for the case of finite, discrete time (but using a probability space that is not necessarily finite) This is the theme of the Dalang-Morton-Willinger theorem [DMW 90] For dimension d ≥ 2, its proof is surprisingly tricky and is sometimes called the “100 meter sprint” of Mathematical Finance, as many authors have elaborated on different proofs of this result We deal with this topic quite extensively, considering several different proofs of this theorem In particular, we present a proof based on the notion of “measurably parameterised subsequences” of a sequence (fn )∞ n=1 of functions This technique, due to Y Kabanov and C Stricker [KS 01], seems at present to provide the easiest approach to a proof of the Dalang-MortonWillinger theorem In Chapter we give a quick overview of stochastic integration Because of the general nature of the models we draw attention to general stochastic integration theory and therefore include processes with jumps However, a systematic development of stochastic integration theory is beyond the scope of the present “guided tour” We suppose (at least from Chapter onwards) that the reader is sufficiently familiar with this theory as presented in sev- X Preface eral beautiful textbooks (e.g., [P 90], [RY 91], [RW 00]) Nevertheless, we highlight those aspects that are particularly important for the applications to Finance Finally, in Chapter 8, we discuss the proof of the Fundamental Theorem of Asset Pricing in its version obtained in [DS 94] and [DS 98] These papers are reprinted in Chapters and 14 The main goal of our “guided tour” is to build up some intuitive insight into the Mathematics of Arbitrage We have refrained from a logically well-ordered deductive approach; rather we have tried to pass from examples and special situations to the general theory We did so at the cost of occasionally being somewhat incoherent, for instance when applying the theory with a degree of generality that has not yet been formally developed A typical example is the discussion of the Bachelier and Black-Scholes models in Chapter 4, which is introduced before the formal development of the continuous time theory This approach corresponds to our experience that the human mind works inductively rather than by logical deduction We decided therefore on several occasions, e.g., in the introductory chapter, to jump right into the subject in order to build up the motivation for the subsequent theory, which will be formally developed only in later chapters In Part II we reproduce updated versions of the following papers We have corrected a number of typographical errors and two mathematical inaccuracies (indicated by footnotes) pointed out to us over the past years by several colleagues Here is the list of the papers Chapter 9: [DS 94] A General Version of the Fundamental Theorem of Asset Pricing Chapter 10: [DS 98a] A Simple Counter-Example to Several Problems in the Theory of Asset Pricing Chapter 11: [DS 95b] The No-Arbitrage Property under a Change of Num´eraire Chapter 12: [DS 95a] The Existence of Absolutely Continuous Local Martingale Measures Chapter 13: [DS 97] The Banach Space of Workable Contingent Claims in Arbitrage Theory Chapter 14: [DS 98] The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes Chapter 15: [DS 99] A Compactness Principle for Bounded Sequences of Martingales with Applications Our sincere thanks go to Catriona Byrne from Springer-Verlag, who encouraged us to undertake the venture of this book and provided the logistic background We also thank Sandra Trenovatz from TU Vienna for her infinite patience in typing and organising the text References [AS 93] [AS 94] [AH 95] [A 97] [A 65] [BP 91] [B 00] [B 12] [B 14] [Be 01] [B 32] [B 72] [BF 02] [B 81] J.P Ansel, C Stricker, (1993), Lois de martingale, densit´ es et decomposition de Fă ollmer-Schweizer Annales de lInstitut Henri Poincare – Probabilit´es et Statistiques, vol 28, no 3, pp 375–392 J.P Ansel, C Stricker, (1994), Couverture des actifs contingents et prix maximum Annales de l’Institut Henri Poincar´e – Probabilit´es et Statistiques, vol 30, pp 303–315 Ph Artzner, D Heath, (1995), Approximate Completeness with Multiple Martingale Measures Mathematical Finance, vol 5, pp 1–11 Ph Artzner, (1997), On the numeraire portfolio Mathematics of Derivative Securities (M Dempster, S Pliska, editors), Cambridge University Press, pp 53–60 R Aumann, (1965), Integrals of Set-Valued Functions Journal of Mathematical 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Guided Tour to Arbitrage Theory The Story in a Nutshell 1.1 Arbitrage The notion of arbitrage is crucial to the modern theory of Finance It is the corner-stone of the option pricing theory due to... integral The mathematical challenge of the above story consists of getting rid of the word “essentially” and to turn this program into precise theorems The central piece of the theory relating the. .. Proof of the Dalang-Morton-Willinger Theorem for T ≥ by Induction on T 102 6.8 Proof of the Closedness of K in the Case T ≥ 103 6.9 Proof of the
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