Ngày đăng: 23/03/2018, 09:01
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 Analysis and Applications, vol 12, pp 1–12 K Back, S Pliska, (1991), On the fundamental theorem of asset pricing with an inﬁnite state space Journal of Mathematical Economics, vol 20, pp 1–18 L Bachelier, (1900), Th´eorie de la Sp´eculation Ann Sci Ecole Norm Sup., vol 17, pp 21–86 English translation in: The Random Character of stock market prices (P Cootner, editor), MIT Press, 1964 L Bachelier, (1912), Calcul des Probabilit´es Gauthier-Villars, Paris L Bachelier, (1914), Le Jeu, la Chance et le Hasard Ernest Flammarion, Paris D Becherer, (2001), The numeraire portfolio for unbounded semimartingales Finance and Stochastics, vol 5, no 3, pp 327–341 S Banach, (1932), Th´eorie des op´ erations lin´eaires Monogr Mat., Warszawa Reprint by: Chelsea Scientiﬁc Books (1963) T Bewley, D Heath, (1972), Existence of equilibria in economics with inﬁnitely many commodities Journal of Economic Theory, vol 4, pp 514– 540 F Bellini, M Frittelli, (2002), On the existence of minimax martingale measures Mathematical Finance, vol 12, no 1, pp 1–21 K Bichteler, (1981), Stochastic integration and Lp -theory of semi martingales Annals of Probability, vol 9, pp 49–89 360 References [BKT 98] N.H Bingham, R Kiesel, (1998), Risk-Neutral Valuation Springer-Verlag, London [BF 04] S Biagini, M Frittelli, (2004), On the super replication price of unbounded claims Annals of Applied Probability, vol 14, no 4, pp 19701991 [BJ 00] T Bjă ork (2000), Arbitrage Theory in Continuous Time Oxford University Press [BS 73] F Black, M Scholes, (1973), The pricing of options and corporate liabilities Journal of Political Economy, vol 81, pp 637–659 [BKT 01] B Bouchard, Y.M Kabanov, N Touzi, (2001), Option pricing by large risk aversion utility under transaction costs Decisions in Economics and Finance, vol 24, no 1, pp 127–136 [BT 00] B Bouchard, N Touzi, (2000), Explicit solution of the multivariate superreplication problem under transaction costs Annals of Applied Probability, vol 10, pp 685–708 [B 79] J Bourgain, (1979), The Komlos Theorem for Vector Valued Functions Manuscript, Vrije Universiteit Brussel, pp 1–12 [BR 97] W Brannath, (1997), On fundamental theorems in mathematical ﬁnance Doctoral Thesis, University of Vienna [BS 99] W Brannath, W Schachermayer, (1999), A Bipolar Theorem for Subsets of L0+ (Ω, F, P ) S´eminaire de Probabilit´es XXXIII, Springer Lecture Notes in Mathematics 1709, pp 349–354 [B 73] D Burkholder, (1973), Distribution Function Inequalities for Martingales Annals of Probability, vol 1, pp 19–42 [BG 70] D Burkholder, R.F Gundy, (1970), Extrapolation and Interpolation and Quasi-Linear Operators on Martingales Acta Mathematica, vol 124, pp 249–304 [CS 05] L Campi, W Schachermayer, (2005), A Super-Replication Theorem in Kabanov’s Model of Transaction Costs Preprint [C 77] C.S Chou, (1977/78), Caract´erisation d’une classe de semimartingales S´eminaire de Probabilit´es XIII, Springer Lecture Notes in Mathematics 721, pp 250–252 [CMS 80] C.S Chou, P.A Meyer, S Stricker, (1980), Sur les int´egrales stochastiques de processus pr´ evisibles non born´es In: J Az´ema, M Yor (eds.), S´eminaire de Probabilit´es XIV, Springer Lecture Notes in Mathematics 784, pp 128– 139 [Cl 93] S.A Clark, (1993), The valuation problem in arbitrage price theory Journal of Mathematical Economics, vol 22, pp 463–478 [Cl 00] S.A Clark, (2000), Arbitrage approximation theory Journal of Mathematical Economics, vol 33, pp 167–181 [CH 89] J.C Cox, C.F Huang, (1989), Optimal consumption and portfolio policies when asset prices follow a diﬀusion process Journal of Economic Theory, vol 49, pp 33–83 [CH 91] J.C Cox, C.F Huang, (1991), A variational problem arising in ﬁnancial economics Journal of Mathematical Economics, vol 20, no 5, pp 465– 487 [CRR 79] J Cox, S Ross, M Rubinstein, (1979), Option pricing: a simpliﬁed approch Journal of Financial Economics, vol 7, pp 229–263 [CK 00] J.-M Courtault, Y Kabanov, B Bru, P Cr´epel, I Lebon, A Le Marchand, (2000), Louis Bachelier: On the Centenary of “Th´ eorie de la Sp´eculation” Mathematical Finance, vol 10, no 3, pp 341–353 References 361 I Csiszar, (1975), I-Divergence Geometry of Probability Distributions and Minimization Problems Annals of Probability, vol 3, no 1, pp 146–158 [C 99] J Cvitanic, (1999), On minimizing expected loss of hedging in incomplete and constrained market SIAM Journal on Control and Optimization [C 00] J Cvitanic, (2000), Minimizing expected loss of hedging in incomplete and constrained markets SIAM Journal on Control and Optimization, vol 38, no 4, pp 1050–1066 [CK 96] J Cvitanic, I Karatzas, (1996), Hedging and portfolio optimization under transaction costs: A martingale approach Mathematical Finance, vol 6, no 2, pp 133–165 [CPT 99] J Cvitanic, H Pham, N Touzi, (1999), A closed-form solution to the problem of super-replication under transaction costs Finance and Stochastics, vol 3, pp 35–54 [CSW 01] J Cvitanic, W Schachermayer, H Wang, (2001), Utility Maximization in Incomplete Markets with Random Endowment Finance and Stochastics, vol 5, no 2, pp 259–272 [CW 01] J Cvitanic, H Wang, (2001), On optimal terminal wealth under transaction costs Journal of Mathematical Economics, vol 35, no 2, pp 223–231 [DMW 90] R.C Dalang, A Morton, W Willinger, (1990), Equivalent Martingale measures and no-arbitrage in stochastic securities market model Stochastics and Stochastic Reports, vol 29, pp 185–201 [D 97] M Davis, (1997), Option pricing in incomplete markets Mathematics of Derivative Securities (M.A.H Dempster, S.R Pliska, editors), Cambridge University Press, pp 216–226 [D 00] M Davis, (2000), Optimal valuation and hedging with basis risk System theory: modeling analysis and control (Cambridge, MA, 1999), Kluwer International Series in Engineering and Computer Science, vol 518, pp 245–254 [DN 90] M.H.A Davis, A Norman, (1990), Portfolio selection with transaction costs Math Operation Research, vol 15, pp 676–713 [DST 01a] M Davis, W Schachermayer, R Tompkins, (2001), Installment Options and Static Hedging Mathematical Finance: Trends in Mathematics (M Kohlmann, S Tang, editors), pp 130–139 Reprint: Risk Finance, vol 3, no 2, pp 46–52 (2002) [DST 01] M Davis, W Schachermayer, R Tompkins, (2001), Pricing, No-arbitrage Bounds and Robust Hedging of Installment Options Quantitative Finance, vol 1, pp 597–610 [DPT 01] G Deelstra, H Pham, N Touzi, (2001), Dual formulation of the utility maximisation problem under transaction costs Annals of Applied Probability, vol 11, no 4, pp 1353–1383 [D 92] F Delbaen, (1992), Representing Martingale Measures when Asset Prices are Continuous and Bounded Mathematical Finance, vol 2, pp 107–130 [De 00] F Delbaen, (2000), Coherent Risk Measures Notes of the Scuola Normale Superiore Cattedra Galileiana, Pisa [DGRSSS 02] F Delbaen, P Grandits, T Rheinlăander, D Samperi, M Schweizer, C Stricker, (2002), Exponential hedging and entropic penalties Mathematical Finance, vol 12, no 2, pp 99–123 [DKV 02] F Delbaen, Y.M Kabanov, E Valkeila, (2002), Hedging under transaction costs in currency markets: a discrete-time model Mathematical Finance, vol 12, no 1, pp 45–61 [C 75] 362 References [DMSSS 94] F Delbaen, P Monat, W Schachermayer, M Schweizer, C Stricker, (1994), In´egalit´e de normes avec poids et fermeture d’un espace d’int´egrales stochastiques C.R Acad Sci Paris, vol 319, no 1, pp 1079– 1081 [DMSSS 97] F Delbaen, P Monat, W Schachermayer, M Schweizer, C Stricker, (1997), Weighted Norm Inequalities and Closedness of a Space of Stochastic Integrals Finance and Stochastics, vol 1, no 3, pp 181–227 [DS 93] F Delbaen, W Schachermayer, (1993), Non-arbitrage and the fundamental theorem of asset pricing In: Abstracts of the Meeting on Stochastic Processes and Their Applications, Amsterdam, June 21-25, 1993, pp 37– 38 [DS 94] F Delbaen, W Schachermayer, (1994), A General Version of the Fundamental Theorem of Asset Pricing Mathematische Annalen, vol 300, pp 463–520 First reprint: The International Library of Critical Writings in Financial Economics — Option Markets (G.M Constantinides, A.G Malliaris, editors) Second reprint: Chap of this book [DS 94a] F Delbaen, W Schachermayer, (1994), Arbitrage and free lunch with bounded Risk for unbounded continuous Processes Mathematical Finance, vol 4, pp 343–348 [DS 95a] F Delbaen, W Schachermayer, (1995), The Existence of Absolutely Continuous Local Martingale Measures Annals of Applied Probability, vol 5, no 4, pp 926–945 Reprint: Chap 12 of this book [DS 95b] F Delbaen, W Schachermayer, (1995), The No-Arbitrage Property under a Change of Num´ eraire Stochastics and Stochastic Reports, vol 53, pp 213–226 Reprint: Chap 11 of this book [DS 95c] F Delbaen, W Schachermayer, (1995), Arbitrage Possibilities in Bessel processes and their relations to local martingales Probability Theory and Related Fields, vol 102, pp 357–366 [DS 95d] F Delbaen, W Schachermayer, (1995), An Inequality for the Predictable Projection of an Adapted Process S´eminaire de Probabilit´es XXIX, ´ Springer Lecture Notes in Mathematics 1613, (J Az´ema, M Emery, P.A Meyer, M Yor, editors), pp 17–24 [DS 96] F Delbaen, W Schachermayer, (1996), Attainable Claims with p’th Moments Annales de l’Institut Henri Poincar´e – Probabilit´es et Statistiques, vol 32, no 6, pp 743–763 [DS 96a] F Delbaen, W Schachermayer, (1996), The Variance-Optimal Martingale Measure for Continuous Processes Bernoulli, vol 2, no 1, pp 81–105 [DS 97] F Delbaen, W Schachermayer, (1997), The Banach Space of Workable Contingent Claims in Arbitrage Theory Annales de l’Institut Henri Poincar´e – Probabilit´es et Statistiques, vol 33, no 1, pp 113–144 Reprint: Chap 13 of this book [DS 98] F Delbaen, W Schachermayer, (1998), The Fundamental Theorem of Asset Pricing for Unbounded Stochastic Processes Mathematische Annalen, vol 312, pp 215–250 Reprint: Chap 14 of this book [DS 98a] F Delbaen, W Schachermayer, (1998), A Simple Counter-Example to Several Problems in the Theory of Asset Pricing, which arises in many incomplete markets Mathematical Finance, vol 8, pp 1–12 Reprint: Chap 10 of this book [DS 99] F Delbaen, W Schachermayer, (1999), A Compactness Principle for Bounded Sequences of Martingales with Applications Proceedings of References [DS 99a] [DS 00] [DS 04] [DSh 96] [D 72] [DM 80] [DMY 78] [D 75] [DRS 93] [DU 77] [D 53] [Du 92] [DFS 03] [DH 86] [DS 58] [DR 87] [ET 76] [E 05] 363 the Seminar of Stochastic Analysis, Random Fields and Applications, Progress in Probability, vol 45, pp 137–173 Reprint: Chap 15 of this book F Delbaen, W Schachermayer, (1999), Non-Arbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Introduction to Mathematical Finance (D.C Heath, G Swindle, editors), “Proceedings of Symposia in Applied Mathematics” of the AMS, vol 57, pp 49–58 F Delbaen, W Schachermayer, (2000), Applications to Mathematical Finance Handbook of the Geometry of Banach Spaces (W Johnson, J Lindenstrauss, editors), vol 1, pp 367–391 F Delbaen, W Schachermayer, (2004), What is a Free Lunch? Notices of the AMS, vol 51, no 5, pp 526–528 F Delbaen, H Shirakawa, (1996), A Note on the No-Arbitrage Condition for International Financial Markets Financial Engineering and the Japanese Markets, vol 3, pp 239–251 C Dellacherie, (1972), Capacit´es et Processus Stochastiques Ergebnisse der Mathematik und ihrer Grenzgebiete, vol 67, Springer, Berlin C Dellacherie, P.A Meyer, (1980), Probabilit´es et Potentiel, Chapitres V a VIII Th´eorie des martingales Hermann, Paris ` C Dellacherie, P.A Meyer, M Yor, (1978), Sur certaines propri´ et´es des espaces de Banach H1 et BM O S´eminaire de Probabilit´es XII, Springer Lecture Notes in Mathematics 649, pp 98–113 J Diestel, (1975), Geometry of Banach spaces — selected topics Springer Lecture Notes in Mathematics 485, Springer, Berlin, Heidelberg, New York J Diestel, W Ruess, W Schachermayer, (1993), On weak compactness in L1 (µ, X) Proc Am Math Soc., vol 118, pp 447–453 J Diestel, J.J Uhl, (1977), Vector Measures Mathematical Surveys, vol 15 Providence, R.I.: American Mathematical Society (AMS) J.L Doob, (1953), Stochastic Processes Wiley, New York D Duﬃe, (1992), Dynamic asset pricing theory Princeton University Press D Duﬃe, D Filipovic, W Schachermayer, (2003), Aﬃne Processes and Applications in Finance Annals of Applied Probability, vol 13, no 3, pp 984–1053 D Duﬃe, C.F Huang, (1986), Multiperiod security markets with diﬀerential information; martingales and resolution times Journal of Mathematical Economics, vol 15, pp 283–303 N Dunford, J Schwartz, (1958), Linear Operators I General theory Pure and Applied Mathematics, vol 6, New York and London: Interscience Publishers Ph Dybvig, S Ross, (1987), Arbitrage In: J Eatwell, M Milgate, P Newman (eds.), The new Palgrave dictionary of economics, vol l, pp 100–106, Macmillan, London I Ekeland, R Temam, (1976), Convex Analysis and Variational Problems North Holland, Amsterdam Reprint: 1999, SIAM Classics in Applied Mathematics 38 ă A Einstein, (1905), Uber die von der molekularkinetischen Theorie der Wă arme geforderte Bewegung von in ruhenden Flă ussigkeiten suspendierten Teilchen Annalen der Physik, vol IV, no 17, pp 549–560 364 References [E 81] [EGR 95] [EQ 95] [EJ 98] [ER 00] [EK 99] [ELY 99] [E 79] [E 80] [FH 97] [FK 98] [FK 97] [FS 91] [FL 00] [FS 86] [FWY 99] [FWY 00] N El Karoui, (1981), Les aspects probabilistes du contrˆ ole stochastique Ecole d’Et´e de Probabilit´es de Saint-Flour IX-1979 (P.L Hennequin, editor), Springer Lecture Notes in Mathematics 876, pp 74–238 N El Karoui, H Geman, J.-C Rochet, (1995), Changes of Num´eraire, Changes of Probability Measure and Option Pricing Journal of Applied Probability, vol 32, no 2, pp 443–458 N El Karoui, M.-C Quenez, (1995), Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market SIAM Journal on Control and Optimization, vol 33, no 1, pp 29–66 N El Karoui, M Jeanblanc, (1998), Optimization of consumptions with labor income Finance and Stochastics, vol 4, pp 409–440 N El Karoui, R Rouge, (2000), Pricing via utility maximization and entropy Mathematical Finance, vol 10, no 2, pp 259–276 R Elliott, P.E Kopp, (1999), Mathematics of ﬁnancial markets Springer Finance, Springer New York D Elworthy, X.-M Li, M Yor, (1999), The Importance of Strictly Local Martingales; applications to radial Ornstein-Uhlenbeck processes Probab Theory Relat Fields, vol 115, no 3, pp 325–355 ´ M Emery, (1979), Une topologie sur l’espace des semi-martingales In: C Dellacherie et al (eds.), S´eminaire de Probabilit´es XIII, Springer Lecture Notes in Mathematics 721, pp 260–280 ´ M Emery, (1980), Compensation de processus a ` variation ﬁnie non localement int´egrables In: J Az´ema, M Yor (eds.), S´eminaire de Probabilit´es XIV, Springer Lecture Notes in Mathematics 784, pp 152–160 B Flesaker, L.P Hughston, (1997), International Models for Interest Rates and Foreign Exchange Net Exposure, vol 3, pp 55–79 Reprinted in: The New Interest Rate Models, L.P Hughston (ed.), Risk Publications (2000) H Fă ollmer, Y.M Kabanov, (1998), Optional decomposition and Lagrange multipliers Finance and Stochastics, vol 2, no 1, pp 69–81 H Fă ollmer, D Kramkov, (1997), Optional Decompositions under Constraints Probability Theory and Related Fields, vol 109, pp 125 H Fă ollmer, M Schweizer, (1991), Hedging of Contingent Claims Under Incomplete Information In: M.H.A Davis, R.J Elliott (eds.), Applied Stochastic Analysis, Stochastic Monogr., vol 5, pp 389–414, Gordon and Breach, London, New York H Fă ollmer, P Leukert, (2000), Ecient Hedging: Cost versus Shortfall Risk Finance and Stochastics, vol 4, no 2, pp 117146 H Fă ollmer, D Sondermann, (1986), Hedging of Non-redundant Contingent Claims Contributions to Mathematical Economics in honor of G Debreu (Eds W Hildenbrand and A Mas-Colell), Elsevier Science Publ., North-Holland, pp 205223 H Fă ollmer, C.-T Wu, M Yor, (1999), Canonical decomposition of linear transformations of two independent Brownian motions motivated by models of insider trading Stochastic Processes and Their Applications, vol 84, no 1, pp 137–164 H Fă ollmer, C.-T Wu, M Yor, (2000), On weak Brownian motions of arbitrary order Annales de l’Institut Henri Poincar´e – Probabilit´es et Statistiques, vol 36, no 4, pp 447–487 References [F 90] 365 L.P Foldes, (1990), Conditions for optimality in the inﬁnite-horizon portfolio-cum-savings problem with semimartingale investments Stochastics and Stochastics Reports, vol 29, pp 133–171 [F 00] M Frittelli, (2000), The minimal entropy martingale measure and the valuation problem in incomplete markets Mathematical Finance, vol 10, no 1, pp 39–52 [F 00a] M Frittelli, (2000), Introduction to a Theory of Value Coherent with the No-Arbitrage Principle Finance and Stochastics, vol 4, no 3, pp 275– 297 [G 66] F.R Gantmacher, (1966), Matrizentheorie Springer, Berlin, Heidelberg, New York [G 73] A.M Garsia, (1973), Martingale Inequalities, Seminar Notes on Recent Progress Mathematics Lecture Notes Series, W.A Benjamin, Inc, Reading, Massachusetts [G 77] R Geske, (1977), The valuation of corporate liabilities as compound options Journal of Financial and Quantitative Analysis, vol 12, pp 541– 562 [G 79] R Geske, (1979), The valuation of compound options Journal of Financial Economics, vol 7, pp 63–81 [GHR 96] E Ghysets, A.C Harvey and E Renault, (1996), Stochastic volatility G.S Maddala and C.R Rao (eds.), Handbook of Statistics, vol 14, Elsevier, Amsterdam [G 60] I.V Girsanov, (1960), On transforming a certain class of stochastic processes by absolutely continuous substitution of measures Theory Prob and Appl., vol 5, pp 285–301 [GK 00] T Goll, J Kallsen, (2000), Optimal portfolios for logarithmic utility Stochastic Processes and Their Applications, vol 89, pp 31–48 [GR 01] T Goll, L Ră uschendorf, (2001), Minimax and minimal distance martingale measures and their relationship to portfolio optimization Finance and Stochastics, vol 5, no 4, pp 557581 [GR 02] P Grandits, T Rheinlă ander, (2002), On the minimal Entropy Martingale Measure Annals of Probability, vol 30, no 3, pp 1003–1038 [G 54] A Grothendieck, (1954), Espaces vectoriels topologiques Sociedade de Matematica de S˜ ao Paulo, S˜ ao Paulo [G 79] S Guerre, (1979/80), La propri´ et´e de Banach-Saks ne passe pas de E es J Bourgain S´eminaire d’Analyse Fonctionnelle, Ecole a L2 (E) d’apr` ` Polytechnique, Paris [G 05] P Guasoni, (2005), No-Arbitrage with Transaction Costs, with Fractional Brownian Motion and Beyond Mathematical Finance, forthcoming 2006 [HS 49] P.R Halmos, L.J Savage, (1949), Application of the Radon-Nikod´ ym Theorem to the theory of suﬃcient statistics Ann Math Statist., vol 20, pp 225–241 [HK 79] J.M Harrison, D.M Kreps, (1979), Martingales and Arbitrage in Multiperiod Securities Markets Journal of Economic Theory, vol 20, pp 381– 408 [HP 81] J.M Harrison, S.R Pliska, (1981), Martingales and Stochastic Integrals in the Theory of Continous Trading Stochastic Processes and their Applications, vol 11, pp 215–260 [H 79] O Hart, (1979), Monopolistic competition in a large economy with diﬀerentiated commodities Review of Economic Studies, vol 46, pp 1–30 366 [HP 91] References H He, N.D Pearson, (1991), Consumption and Portfolio Policies with Incomplete Markets and Short-Sale Constraints: The Finite-Dimensional Case Mathematical Finance, vol 1, pp 1–10 [HP 91a] H He, N.D Pearson, (1991), Consumption and Portfolio Policies with Incomplete Markets and Short- Sale Constraints: The Inﬁnite-Dimensional Case Journal of Economic Theory, vol 54, pp 239–250 [HJM 92] D Heath, R Jarrow, A Morton, (1992), Bond pricing and the term structure of interest rates: a new methodology for contingent claim valuation Econometrica, vol 60, pp 77–105 [H 86] J.R Hicks, (1986, First Edition 1956), A Revision of Demand Theory Oxford University Press, Oxford [HN 89] S.D Hodges, A Neuberger, (1989), Optimal replication of contingent claims under transaction costs Review of Futures Markets, vol 8, pp 222–239 [HL 88] C.-F Huang, R.H Litzenberger, (1988), Foundations for Financial Economics North-Holland Publishing Co New York [HS 98] F Hubalek, W Schachermayer, (1998), When does Convergence of Asset Price Processes Imply Convergence of Option Prices? Mathematical Finance, vol 8, no 4, pp 215–233 [HS 01] F Hubalek, W Schachermayer, (2001), The Limitations of No-Arbitrage Arguments for Real Options International Journal of Theoretical and Applied Finance, vol 4, no 2, pp 361–373 [HK 04] J Hugonnier, D Kramkov, (2004), Optimal investment with random endowments in incomplete markets Annals of Applied Probability, vol 14, no 2, pp 845–864 [HKS 05] J Hugonnier, D Kramkov, W Schachermayer, (2005), On Utility Based Pricing of Contingent Claims in Incomplete Markets Mathematical Finance, vol 15, no 2, pp 203–212 [HW 87] J Hull, A White, (1987), The Pricing of Options on Assets with Stochastic Volatilities Journal of Finance, vol 42, pp 281–300 [HW 88] J Hull, A White, (1988), The Use of the Control Variate Technique in Option Pricing Journal of Financial and Quantitative Analysis, vol 23, pp 237–252 [H 99] J Hull, (1999), Options, Futures, and Other Derivatives 4th Edition, Prentice-Hall, Englewood Cliﬀs, New Jersey [I 44] K Itˆ o, (1944), Stochastic integral Proc Imperial Acad Tokyo, vol 20, pp 519–524 [J 91] S.D Jacka, (1991), Optimal stopping and the American put Mathematical Finance, vol 1, pp 1–14 [J 92] S.D Jacka, (1992), A Martingale Representation Result and an Application to Incomplete Financial Markets Mathematical Finance, vol 2, pp 239–250 [J 93] S.D Jacka, (1993), Local times, optimal stopping and semimartingales Annals of Probability, vol 21, pp 329–339 [J 79] J Jacod, (1979), Calcul Stochastique et Probl`emes de Martingales Springer Lecture Notes in Mathematics 714, Springer, Berlin, Heidelberg, New York [JS 87] J Jacod, A Shiryaev, (1987), Limit Theorems for Stochastic Processes Springer, Berlin, Heidelberg, New York References [J 87] 367 F Jamshidian, (1987), Pricing of Contingent Claims in the One Factor Term Structure Model Merrill Lynch Capital Markets, Research Paper [JS 00] M Jonsson, K.R Sircar, (2000), Partial hedging in a stochastic volatility environment Mathematical Finance, vol 12, pp 375–409 [JK 95] E Jouini, H Kallal, (1995), Martingales and arbitrage in securities markets with transaction costs Journal of Economic Theory, vol 66, pp 178– 197 [JK 95a] E Jouini, H Kallal, (1995), Arbitrage in securities markets with shortsales constraints Mathematical Finance, vol 3, pp 237–248 [JK 99] E Jouini, H Kallal, (1999), Viability and Equilibrium in Securities Markets with Frictions Mathematical Finance, vol 9, no 3, pp 275–292 [JNS 05] E Jouini, C Napp, W Schachermayer, (2005), Arbitrage and state price deﬂators in a general intertemporal framework Journal of Mathematical Economics, vol 41, pp 722–734 [K 97] Y.M Kabanov, (1997), On the FTAP of Kreps-Delbaen-Schachermayer (English) Y.M Kabanov (ed.) et al., Statistics and control of stochastic processes The Liptser Festschrift Papers from the Steklov seminar held in Moscow, Russia, 1995-1996 Singapore: World Scientiﬁc pp 191–203 [K 99] Y.M Kabanov, (1999), Hedging and liquidation under transaction costs in currency markets Finance and Stochastics, vol 3, no 2, pp 237–248 [K 01a] Y.M Kabanov, (2001), Arbitrage Theory Handbooks in Mathematical Finance Option Pricing: Theory and Practice, pp 3–42 [KK 94] Y.M Kabanov, D Kramkov, (1994), No-arbitrage and equivalent martingale measures: An elementary proof of the Harrison–Pliska theorem Theory Prob Appl., vol 39, no 3, pp 523–527 [KL 02] Y.M Kabanov, G Last, (2002), Hedging under transaction costs in currency markets: a continuous-time model Mathematical Finance, vol 12, no 1, pp 63–70 [KRS 02] Y.M Kabanov, M R´ asonyi, Ch Stricker, (2002), No-arbitrage criteria for ﬁnancial markets with eﬃcient friction Finance and Stochastics, vol 6, no 3, pp 371–382 [KRS 02a] Y.M Kabanov, M R´ asonyi, Ch Stricker, (2003), On the closedness of sums of convex cones in L0 and the robust no-arbitrage property Finance and Stochastics, vol 7, no 3, pp 403–411 [KS 01] Y.M Kabanov, Ch Stricker, (2001), A teachers’ note on no-arbitrage criteria S´eminaire de Probabilit´es XXXV, Springer Lecture Notes in Mathematics 1755, pp 149–152 [KS 01a] Y.M Kabanov, Ch Stricker, (2001), The Harrison-Pliska arbitrage pricing theorem under transaction costs Journal of Mathematical Economics, vol 35, no 2, pp 185–196 [KS 02] Y.M Kabanov, Ch Stricker, (2002), On the optimal portfolio for the exponential utility maximization: Remarks to the six-author paper Mathematical Finance, vol 12, no 2, pp 125–134 [KS 02a] Y.M Kabanov, Ch Stricker, (2002), Hedging of contingent claims under transaction costs Sandmann, Klaus (ed.) et al., Advances in ﬁnance and stochastics Essays in honour of Dieter Sondermann Berlin: Springer, pp 125–136 [KS 03] Y.M Kabanov, Ch Stricker, (2003), On the true submartingale property d’apr` es Schachermayer S´eminaire de Probabilit´es XXXVI, Springer Lecture Notes in Mathematics 1801, pp 413–414 368 [KP 65] References M Kadeˇc, A Pelczy´ nski, (1965), Basic sequences, biorthogonal systems and norming sets in Banach and Fr´echet spaces Studia Mathematica, vol 25, pp 297–323 [K 00] J Kallsen, (2000), Optimal portfolios for exponential L´evy processes Mathematical Methods of Operation Research, vol 51, no 3, pp 357– 374 [K 01] J Kallsen, (2001), Utility-Based Derivative Pricing in Incomplete Markets Mathematical Finance: Bachelier Congress 2000 (H Geman, D Madan, S.R Pliska, T Vorst, editors), Springer, pp 313–338 [KLS 87] I Karatzas, J.P Lehoczky, S.E Shreve, (1987), Optimal portfolio and consumption decisions for a “small investor” on a ﬁnite horizon SIAM Journal of Control and Optimisation, vol 25, pp 1557–1586 [KLS 90] I Karatzas, J.P Lehoczky, S.E Shreve, (1990), Existence and uniqueness of multi-agent equilibrium in a stochastic, dynamic consumption/investment model Mathematics of Operations Research, vol 15, pp 80–128 [KLS 91] I Karatzas, J.P Lehoczky, S.E Shreve, (1991), Equilibrium models with singular asset prices Mathematical Finance, vol 1, pp 11–29 [KLSX 91] I Karatzas, J.P Lehoczky, S.E Shreve, G.L Xu, (1991), Martingale and duality methods for utility maximisation in an incomplete Market SIAM Journal of Control and Optimisation, vol 29, pp 702–730 [KS 88] I Karatzas, S.E Shreve, (1988), Brownian motion and stochastic calculus Springer, Berlin, Heidelberg, New York [KSh 98] I Karatzas, S.E Shreve, (1998), Methods of Mathematical Finance Springer-Verlag, New York [KS 96a] I Klein, W Schachermayer, (1996), Asymptotic Arbitrage in NonComplete Large Financial Markets Theory of Probability and its Applications, vol 41, no 4, pp 927–934 [KS 96b] I Klein, W Schachermayer, (1996), A Quantitative and a Dual Version of the Halmos-Savage Theorem with Applications to Mathematical Finance Annals of Probability, vol 24, no 2, pp 867–881 [KPT 99] P.-F Koehl, H Pham, N Touzi, (1999), On super-replication under Transaction costs in general discrete-time models Theory of Probability and its Applications, vol 45, pp 783–788 [K 33] A.N Kolmogorov, (1933), Grundbegriﬀe der Wahrscheinlichkeitsrechnung Ergebnisse der Mathematik und ihrer Grenzgebiete 3, Springer, Berlin [K 67] J Komlos, (1967), A generalisation of a theorem of Steinhaus Acta Math Acad Sci Hungar., vol 18, pp 217–229 [K 96a] D Kramkov, (1996), Optional decomposition of supermartingales and hedging contingent claims in incomplete security markets Probability Theory and Related Fields, vol 105, pp 459–479 [K 96b] D Kramkov, (1996), On the Closure of the Family of Martingale Measures and an Optional Decomposition of Supermartingales Theory Probab Appl., vol 41, no 4, pp 788–791 [KS 99] D Kramkov, W Schachermayer, (1999), The Asymptotic Elasticity of Utility Functions and Optimal Investment in Incomplete Markets Annals of Applied Probability, vol 9, no 3, pp 904–950 References 369 [KS 03a] D Kramkov, W Schachermayer, (2003), Necessary and suﬃcient conditions in the problem of optimal investment in incomplete markets Annals of Applied Probability, vol 13, no 4, pp 1504–1516 [K 81] D.M Kreps, (1981), Arbitrage and Equilibrium in Economics with inﬁnitely many Commodities Journal of Mathematical Economics, vol 8, pp 15–35 [KW 67] H Kunita, S Watanabe, (1967), On square integrable martingales Nagoya Mathematical Journal, vol 30, pp 209–245 [K 93] S Kusuoka, (1993), A remark on arbitrage and martingale measures Publ Res Inst Math Sci., vol 19, pp 833–840 [L 92] P Lakner, (1992), Martingale measures for a class of right-continuous processes Mathematical Finance, vol 3, no 1, pp 43–53 [LL 96] D Lamberton, B Lapeyre, (1996), Introduction to Stochastic Calculus Applied to Finance Chapman & Hall, London [L 78] D L´epingle, (1978), Une inegalit´e de martingales In: C Dellacherie et al (eds.), S´emin de Probab XII (Lect Notes Math., vol 649, pp 134–137), Springer, Berlin, Heidelberg, New York [L 91] D L´epingle, (1991), Orthogonalit´ e et ´equi-int´egrabilit´e de martingales discr`etes S´eminaire de Probabilit´es XXVI, Springer Lecture Notes in Mathematics 1526, pp 167–170 [L 77] E Lenglart, (1977), Transformation des martingales locales par changement absolument continu de probabilites Zeitschrift fă ur Wahrscheinlichkeitstheorie und verwandte Gebiete vol 39, pp 65–70 [LW 01] S.F LeRoy, J Werner, (2001), Principles of Financial Economics Cambridge University Press [LS 94] S Levental, A.V Skorohod, (1994), A necessary and suﬃcient condition for absence of arbitrage with tame portfolios Annals of Applied Probability, vol 5, no 4, pp 906–925 [LS 97] S Levental, A.V Skorohod, (1997), On the possibility of hedging options in the presence of transaction costs Annals of Applied Probability, vol 7, pp 410–443 [Lo 78] M Lo`eve, (1978), Probability theory 4th edn., Springer, Berlin, Heidelberg, New York [L 90] J.B Long, (1990), The numeraire portfolio Journal of Financial Economics, vol 26, pp 29–69 [MZ 38] J Marcinkiewicz, A Zygmund, (1938), Quelques th´eor`emes sur les fonctions ind´ependantes Studia Mathematica, vol 7, pp 104–120 [M 78a] W Margrabe, (1978), The value of an option to exchange one asset for another Journal of Finance, vol 33, no 1, pp 177–186 [M 78b] W Margrabe, (1978), A theory of forward and future prices Preprint Wharton School, University of Pennsylvania [M 75] A Mas-Colell, (1975), A model of equilibrium with diﬀerentiated commodities Journal of Mathematical Economics, vol 2, pp 263–296 [MB 91] D.W McBeth, (1991), On the existence of equivalent martingale measures Thesis Cornell University [MK 69] H.P McKean, (1969), Stochastic Integrals Wiley, New York [M 80] J M´emin, (1980), Espaces de semi Martingales et changement de probabilite Zeitschrift fă ur Wahrscheinlichkeitstheorie und verwandte Gebiete, vol 52, pp 9–39 370 [M 69] [M 71] [M 73] [M 73b] [M 76a] [M 80a] [M 90] [M 76] [M 62] [M 63] [M 94] [MR 97] [N 75] [PT 99] [PY 82] [P 86] [P 97] [P 90] [R 98] [RY 91] [R 70] [R 93] References R.C Merton, (1969), Lifetime portfolio selection under uncertainty: the continuous-time model Rev Econom Statist., vol 51, pp 247–257 R.C Merton, (1971), Optimum consumption and portfolio rules in a continuous-time model Journal of Economic Theory, vol 3, pp 373–413 R.C Merton, (1973), The theory of rational option pricing Bell J Econ Manag Sci., vol 4, pp 141–183 R.C Merton, (1973), An intertemporal capital asset pricing model Econometrica, vol 41, pp 867–888 R.C Merton, (1976), Option pricing when underlying stock returns are discontinuous Journal of Financial Economics, vol 3, pp 125–144 R.C Merton, (1980), On estimating the expected return on the market: an exploratory investigation Journal of Financial Economics, vol 8, pp 323–361 R.C Merton, (1990), Continuous-Time Finance Basil Blackwell, Oxford P.A Meyer, (1976), Un cours sur les int´ egrales stochastiques In: P.A Meyer (ed.), S´eminaire de Probabilit´es X, Springer Lecture Notes in Mathematics 511, pp 245–400 P.A Meyer, (1962), A decomposition theorem for supermartingales Illinois J Math., vol 6, pp 193–205 P.A Meyer, (1963), Decomposition of supermartingales: the uniqueness theorem Illinois J Math., vol 7, pp 1–17 P Monat, (1994), Remarques sur les in´ egalit´es de Burkholder-DavisGundy S´eminaire de Probabilit´es XXVIII, Springer Lecture Notes in Mathematics 1583, pp 92–97 M Musiela, M Rutkowski, (1997), Martingale Methods in Financial Modelling Springer-Verlag, Berlin J Neveu, (1975), Discrete Parameter Martingales North-Holland, Amsterdam H Pham, N Touzi, (1999), The fundamental theorem of asset pricing with cone constraints Journal of Mathematical Economics, vol 31, pp 265–279 J Pitman, M Yor, (1982), A decomposition of Bessel Bridges Zeitschrift f Wahrscheinlichkeit u Verw Gebiete, vol 59, no 4, pp 425–457 S.R Pliska, (1986), A stochastic calculus model of continuous trading: optimal portfolios Math Oper Res., vol 11, pp 371–382 S.R Pliska, (1997), Introduction to Mathematical Finance Blackwell Publishers P Protter, (1990), Stochastic Integration and Diﬀerential Equations A new approach Applications of Mathematics, vol 21, Springer-Verlag, Berlin, Heidelberg, New York (second edition: 2003, corrected third printing: 2005) R Rebonato, (1998), Interest-rate Option models 2nd ed., Wiley, Chichester D Revuz, M Yor, (1991), Continuous Martingales and Brownian Motion Grundlehren der Mathematischen Wissenschaften, vol 293, Springer (third edition: 1999, corrected third printing: 2005) R.T Rockafellar, (1970), Convex Analysis Princeton University Press, Princeton, New Jersey L.C.G Rogers, (1993), Notebook, private communication Dec 20, 1993 References [R 94] [RW 00] [R 04] [RS 05] [R 84] [Sa 00] [S 65] [S 69] [S 70] [S 73] [SM 69] [S 81] [S 92] [S 93] [S 94] [S 00] [S 01] [S 01a] [S 02] [S 03] 371 L.C.G Rogers, (1994), Equivalent martingale measures and no-arbitrage Stochastics and Stochastic Reports, vol 51, no 1–2, pp 41–49 L.C.G Rogers, D Williams, (2000), Diﬀusions, Markov Processes and Martingales Volume and 2, Cambridge University Press D Rokhlin, (2004) The Kreps-Yan Theorem for L∞ Preprint D Rokhlin, W Schachermayer, (2005), A note on lower bounds of martingale measure densities Preprint L Ră uschendorf, (1984), On the minimum discrimination information theorem Statistics and Decisions Supplement Issue, vol 1, pp 263–283 D Samperi, (2000), Entropy and Model Calibration for Asset Pricing and Risk Management Preprint P.A Samuelson, (1965), Proof that properly anticipated prices ﬂuctuate randomly Industrial Management Review, vol 6, pp 41–50 P.A Samuelson, (1969), Lifetime portfolio selection by dynamic stochastic programming Rev Econom Statist., vol 51, pp 239–246 P.A Samuelson, (1970), The fundamental approximation theorem of portfolio analysis in terms of means, variances, and higher moments Rev Econom Stud., vol 37, pp 537–542 P.A Samuelson, (1973), Mathematics of speculative prices SIAM Review, vol 15, pp 1–42 P.A Samuelson, R.C Merton, (1969), A complete model of warrant pricing that maximizes utility Industrial Management Review, vol 10, pp 17–46 W Schachermayer, (1981), The Banach-Saks property is not L2 hereditary Israel Journal of Mathematics, vol 40, pp 340–344 W Schachermayer, (1992), A Hilbert space proof of the fundamental theorem of asset pricing in ﬁnite discrete time Insurance: Mathematics and Economics, vol 11, no 4, pp 249–257 W Schachermayer, (1993), A Counter-Example to several Problems in the Theory of Asset Pricing Mathematical Finance, vol 3, pp 217–229 W Schachermayer, (1994), Martingale Measures for Discrete time Processes with Inﬁnite Horizon Mathematical Finance, vol 4, no 1, pp 25–56 W Schachermayer, (2000), Die Rolle der Mathematik auf den Finanzmă arkten In: Alles Mathematik, Die Urania Vortrăage auf dem Weltkongress fă ur Mathematik, Berlin 1998 (M Aigner, E Behrends, editors), pp 99–111 W Schachermayer, (2001), Optimal Investment in Incomplete Markets when Wealth may Become Negative Annals of Applied Probability, vol 11, no 3, pp 694–734 W Schachermayer, (2001), Optimal Investment in Incomplete Financial Markets Mathematical Finance: Bachelier Congress 2000 (H Geman, D Madan, S.R Pliska, T Vorst, editors), Springer, pp 427–462 W Schachermayer, (2002), No-Arbitrage: On the Work of David Kreps Positivity, vol 6, pp 359–368 W Schachermayer, (2003), Introduction to the Mathematics of Financial Markets In: S Albeverio, W Schachermayer, M Talagrand: Springer Lecture Notes in Mathematics 1816 — Lectures on Probability Theory and Statistics, Saint-Flour summer school 2000 (Pierre Bernard, editor), Springer Verlag, Heidelberg, pp 111–177 372 [S 03a] References W Schachermayer, (2003), A Super-Martingale Property of the Optimal Portfolio Process Finance and Stochastics, vol 7, no 4, pp 433–456 [S 04] W Schachermayer, (2004), The Fundamental Theorem of Asset Pricing under Proportional Transaction Costs in Finite Discrete Time Mathematical Finance, vol 14, no 1, pp 19–48 [S 04a] W Schachermayer, (2004), Utility Maximisation in Incomplete Markets In: Stochastic Methods in Finance, Lectures given at the CIME-EMS Summer School in Bressanone/Brixen, Italy, July 6-12, 2003 (M Frittelli, W Runggaldier, eds.), Springer Lecture Notes in Mathematics 1856, pp 225–288 [S 04b] W Schachermayer, (2004), Portfolio Optimization in Incomplete Financial Markets Notes of the Scuola Normale Superiore Cattedra Galileiana, Pisa [S 05] W Schachermayer, (2005), A Note on Arbitrage and Closed Convex Cones Mathematical Finance, vol 15, no 1, pp 183–189 [ST 05] W Schachermayer, J Teichmann, (2005), How close are the Option Pricing Formulas of Bachelier and Black-Merton-Scholes? Preprint [Sch 99] H.H Schaefer, (1999), Topological Vector Spaces Graduate Texts in Mathematics, Springer New York [SH 87] M.J.P Selby, S.D Hodges, (1987), On the evaluation of compound options Management Science, vol 33, pp 347–355 [Sh 99] A.N Shiryaev, (1999), Essentials of Stochastic Finance Facts, Models, Theory World Scientiﬁc [Sh 04a] S.E Shreve, (2004), Stochastic Calculus for Finance I: The Binomial Asset Pricing Model Springer Finance, Springer-Verlag, New York [Sh 04b] S.E Shreve, (2004), Stochastic Calculus for Finance II: Continuous-Time Models Springer Finance, Springer-Verlag, New York [SSC 95] H.M Soner, S.E Shreve, J Cvitanic, (1995), There is no nontrivial hedging portfolio for option pricing with transaction costs Annals of Applied Probability, vol 5, pp 327–355 [St 70] E Stein, (1970), Topics in harmonic analysis Ann Math., Princeton University Press [St 03] E Strasser, (2003), Necessary and suﬃcient conditions for the supermartingale property of a stochastic integral with respect to a local martingale S´eminaire de Probabilit´es XXXVII, Springer Lecture Notes in Mathematics 1832, pp 385–393 [St 85] H Strasser, (1985), Mathematical theory of statistics: statistical experiments and asymptotic decision theory De Gruyter studies in mathematics, vol [St 97] H Strasser, (1997), On a Lemma of Schachermayer Preprint, Vienna University of Economics and Business Administration [Str 90] C Stricker, (1990), Arbitrage et Lois de Martingale Annales de l’Institut Henri Poincar´e – Probabilit´es et Statistiques, vol 26, pp 451–460 [Str 02] Ch Stricker, (2002), Simple strategies in exponential utility maximization Preprint of the Universit´e de Franche-Comt´e [SV 69a] D.W Stroock, S.R.S Varadhan, (1969), Diﬀusion processes with continuous coeﬃcients I Comm Pure & Appl Math., vol 22, pp 345–400 [SV 69b] D.W Stroock, S.R.S Varadhan, (1969), Diﬀusion processes with continuous coeﬃcients II Comm Pure & Appl Math., vol 22, pp 479–530 References [T 00] [T 99] [W 23] [W 91] [W 99a] [W 99b] [WY 02] [XY 00] [Y 80] [Y 05] [Y 78a] [Y 78b] [Y 01] [Z 05] 373 M.S Taqqu, (2000), Bachelier and his Times: A Conversation with Bernard Bru Finance and Stochastics, vol 5, no 1, pp 2–32 N Touzi, (1999), Super-replication under proportional transaction costs: from discrete to continuous-time models Mathematical Methods of Operation Research, vol 50, pp 297–320 N Wiener, (1923), Diﬀerential space J Math Phys., vol 2, pp 131–174 D Williams, (1991), Probability with Martingales Cambridge University Press C.-T Wu, (1999), Construction of Brownian motions in enlarged ﬁltrations and their role in mathematical models of insider trading Dissertation Humboldt-Universită at zu Berlin C.-T Wu, (1999), Brownian motions in enlarged ﬁltrations and a case study in insider trading Preprint C.-T Wu, M Yor, (2002), Linear transformations of two independent Brownian motions and orthogonal decompositions of Brownian ﬁltrations Publications Matem` atiques, vol 46, pp 237–256 J Xia, J.A Yan, (2000), Martingale measure method for expected utility maximisation and valuation in incomplete markets Preprint J.A Yan, (1980), Caract´erisation d’ une classe d’ensembles convexes de L1 ou H In: J Az´ema, M Yor (eds.), S´eminaire de Probabilit´es XIV, Springer Lecture Notes in Mathematics 784, pp 220–222 J.A Yan, (2005), A Num´eraire-free and Original Probability Based Framework for Financial Markets In: Proceedings of the ICM 2002, vol III, Beijing, pp 861–874, World Scientiﬁc Publishers M Yor, (1978), Sous-espaces denses dans L1 ou H et repr´esentation des martingales In: C Dellacherie et al (eds.), S´eminaire de Probabilit´es XII, Springer Lecture Notes in Mathematics 649, pp 265–309 M Yor, (1978), In´egalit´es entre processus minces et applications C.R Acad Sci., Paris, Ser A 286, pp 799–801 M Yor, (2001), Exponential Functionals of Brownian Motion and Related Processes Springer Finance G Zitkovic, (2005), A ﬁltered version of the Bipolar Theorem of Brannath and Schachermayer Journal of Theoretical Probability, vol 15, no 1, pp 41–61 ... 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
- Xem thêm -
Xem thêm: The mathematics of arbitrage , The mathematics of arbitrage