214 GRADUATE STUDIES IN MATHEMATICS Portfolio Theory and Arbitrage A Course in Mathematical Finance Ioannis Karatzas Constantinos Kardaras Portfolio Theory and Arbitrage 214 GRADUATE STUDIES IN MATHEMATICS Portfolio Theory and Arbitrage A Course in Mathematical Finance Ioannis Karatzas Constantinos Kardaras EDITORIAL COMMITTEE Marco Gualtieri Bjorn Poonen Gigliola Staffilani (Chair) Jeff A. Viaclovsky Rachel Ward 2020 Mathematics Subject Classification. Primary 60H05, 91G10,91G80; Secondary 60H30, 60G44, 46N10. For additional informationand updates on this book, visit www.ams.org/bookpages/gsm-214 Library of Congress Cataloging-in-Publication Data Names: Karatzas,Ioannis,author. |Kardaras,Constantinos,1977-author. Title: Portfolio theory and arbitrage : a course in mathematical finance / Ioannis Karatzas, ConstantinosKardaras. Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2021]|Series: Gradu- atestudiesinmathematics,1065-7339;214|Includesbibliographicalreferencesandindex. Identifiers: LCCN2021008084|ISBN9781470460143(hardcover)|ISBN9781470465988(paper- back)|ISBN9781470465971(ebook) Subjects: LCSH: Portfolio management. | Arbitrage. | AMS: Probability theory and stochastic processes{Foradditionalapplications,see11Kxx,62-XX,90-XX,91-XX,92-XX,93-XX,94- XX} – Stochastic analysis [See also 58J65] – Stochastic integrals. | Game theory, economics, social and behavioral sciences – Mathematical finance – Portfolio theory. | Game theory, economics, social and behavioral sciences – Mathematical finance – Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems). | Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX,91-XX,92-XX,93-XX,94-XX}–Stochasticanalysis[Seealso58J65]–Applicationsof stochastic ana | Probability theory and stochastic processes {For additional applications, see 11Kxx, 62-XX, 90-XX, 91-XX, 92-XX, 93-XX, 94-XX} – Stochastic processes – Martingales withcontinuousparameter. Classification: LCCHG4529.5.K3652021|DDC332.64/5–dc23 LCrecordavailableathttps://lccn.loc.gov/2021008084 Copying and reprinting. Individual readersofthispublication,andnonprofit librariesacting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews,providedthecustomaryacknowledgmentofthesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthispublication ispermittedonlyunderlicensefromtheAmericanMathematicalSociety. Requestsforpermission toreuseportionsofAMSpublicationcontentarehandledbytheCopyrightClearanceCenter. For moreinformation,pleasevisitwww.ams.org/publications/pubpermissions. Sendrequestsfortranslationrightsandlicensedreprintstoreprint-permission@ams.org. (cid:2)c 2021bytheAmericanMathematicalSociety. Allrightsreserved. TheAmericanMathematicalSocietyretainsallrights exceptthosegrantedtotheUnitedStatesGovernment. PrintedintheUnitedStatesofAmerica. (cid:2)∞ Thepaperusedinthisbookisacid-freeandfallswithintheguidelines establishedtoensurepermanenceanddurability. VisittheAMShomepageathttps://www.ams.org/ 10987654321 262524232221 To Eleni, for all her love and her patience. —I.K. To my parents, for their unconditional love and support; and to Ellie, for bearing with me throughout. —C.K. Contents Preface ix Preview x Prerequisites xiii Further topics xiii Suggested reading pathways xiv Acknowledgements xv Chapter 1. The Market 1 §1.1. Probabilistic setup 1 §1.2. Assets and investment 9 §1.3. Proportional investment 16 §1.4. Relative performance 22 §1.5. Functional generation of stock portfolios 30 Notes and Complements 38 Chapter 2. Num´eraires and Market Viability 41 §2.1. Supermartingale num´eraires 42 §2.2. Market viability 51 §2.3. Optimality properties of supermartingale num´eraires 66 §2.4. The local martingale num´eraire for stock portfolios 81 §2.5. Capital Asset Pricing Model 87 Notes and Complements 93 vii viii Contents Chapter 3. Financing, Optimization, Maximality 99 §3.1. Optional Decomposition 101 §3.2. Financing 108 §3.3. Contingent claims; Completeness 127 §3.4. Utility Maximization 148 §3.5. Maximality 168 Notes and Complements 184 Chapter 4. Ramifications and Extensions 189 §4.1. Drawdown-constrained investment 190 §4.2. Simple trading and semimartingales 205 §4.3. Models with infinitely many assets 217 Notes and Complements 245 Appendix A. Elements of Functional and Convex Analysis 249 §A.1. A minimax theorem 250 §A.2. The space L0 253 §A.3. Concave optimization and duality on L0 264 + §A.4. The space L∞ 275 §A.5. Reproducing kernel Hilbert space 284 Notes and Complements 291 Bibliography 295 Index 307 Preface Thisbookdevelopsamathematicaltheoryforfinancebasedonthefollowing “viability” principle: that it should not be possible to fund cumulative cap- ital withdrawal streams which are nontrivial, starting with arbitrarily small amounts of initial wealth. The underlying framework consists of a given, finite number of risky assets, already discounted by a money-market or “cash”. The asset prices are modelled via continuous semimartingales with respect to an arbitrary right-continuous filtration, or flow of information. In this context, proscrib- ing such egregious forms of what is commonly called arbitrage, as the one described above, turns out to be equivalent to the existence of a portfolio with wealth process possessing the local martingale num´eraire and growth optimality properties; whereas, the reciprocal of this wealth process is a particular case of a so-called local martingale deflator. A precise meaning is assigned to these terms, and it is shown that the aboveequivalentconditionscanbeformulatedentirely,actuallyverysimply, intermsofthelocal characteristics oftheunderlyingassetprices: theirdrift and covariation components. Inthisframework,full-fledgedtheoriesaredevelopedforthemostcentral problems of the field: the hedging of liabilities, and portfolio/consumption optimization. The important notion of market completeness is also intro- duced, and is characterized equivalently as the martingale representation property for the underlying flow of information, in terms of the fundamen- tal local martingales that represent the assets’ noise components. As it turns out, the semimartingale property of these processes is nec- essary and sufficient for viability, when investment occurs only along a ix