Optimal Trading Strategies Under Arbitrage Johannes Karl Dominik Ruf Submitted in partial fulfillment of the Requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2011 (cid:13)c 2011 Johannes Karl Dominik Ruf All Rights Reserved ABSTRACT Optimal Trading Strategies Under Arbitrage Johannes Karl Dominik Ruf This thesis analyzes models of financial markets that incorporate the possibility of arbitrage opportunities. The first part demonstrates how explicit formulas for opti- mal trading strategies in terms of minimal required initial capital can be derived in order to replicate a given terminal wealth in a continuous-time Markovian context. Towards this end, only the existence of a square-integrable market price of risk (rather than the existence of an equivalent local martingale measure) is assumed. A new measure under which the dynamics of the stock price processes simplify is constructed. It is shown that delta hedging does not depend on the “no free lunch with vanishing risk” assumption. However, in the presence of arbitrage opportu- nities, finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. In order to apply these analytic tools, sufficient conditions are derived for the necessary differentiability of expectations indexed over the initial market configuration. The phenomenon of “bubbles,” which has been a popular topic in the recent academic literature, appears as a special case of the setting in the first part of this thesis. Several examples at the end of the first part illustrate the techniques contained therein. In the second part, a more general point of view is taken. The stock price processes, which again allow for the possibility of arbitrage, are no longer assumed to be Markovian, but rather only Itˆo processes. We then prove the Second Funda- mental Theorem of Asset Pricing for these markets: A market is complete, meaning that any bounded contingent claim is replicable, if and only if the stochastic dis- count factor is unique. Conditions under which a contingent claim can be perfectly replicated in an incomplete market are established. Then, precise conditions un- der which relative arbitrage and strong relative arbitrage with respect to a given trading strategy exist are explicated. In addition, it is shown that if the market is quasi-complete, meaning that any bounded contingent claim measurable with respect to the stock price filtration is replicable, relative arbitrage implies strong relative arbitrage. It is further demonstrated that markets are quasi-complete, sub- ject to the condition that the drift and diffusion coefficients are measurable with respect to the stock price filtration. Contents Contents Acknowledgments iii Chapter 1: Outline of Thesis 1 Chapter 2: The Markovian Case 4 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Market model and market price of risk . . . . . . . . . . . . . . . . 9 2.3 Strategies, wealth processes and arbitrage opportunities . . . . . . . 17 2.4 Optimal strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Optimizing a given strategy . . . . . . . . . . . . . . . . . . 23 2.4.2 Hedging of contingent claims . . . . . . . . . . . . . . . . . . 31 2.4.3 Smoothness of hedging price . . . . . . . . . . . . . . . . . . 36 2.5 Change of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.8 Condition that hedging price solves a PDE . . . . . . . . . . . . . . 66 Chapter 3: Completeness and Relative Arbitrage 70 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.2.1 Market model . . . . . . . . . . . . . . . . . . . . . . . . . . 75 i Contents 3.2.2 Market prices of risk and stochastic discount factors . . . . . 76 3.2.3 Trading strategies and claims . . . . . . . . . . . . . . . . . 77 3.3 Existence of (super-)replicating trading strategies . . . . . . . . . . 78 3.4 Completeness and Second Fundamental Theorem of Asset Pricing . 91 3.5 Relative arbitrage and strong relative arbitrage . . . . . . . . . . . 94 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Bibliography 100 ii Acknowledgments Acknowledgments I am very grateful to Professor Ioannis Karatzas for serving as my advisor in my doctoral studies. His scholarship and integrity have set an ideal to which I inspire. Professor Karatzas is not only an excellent researcher with deep insights, but also a great instructor. I thank him for always sharing his ideas, for his generosity in offering fast, helpful, and constructive feedback, for introducing me to other junior and senior researchers, for supporting me in all my academic projects, and, last but not least, for his friendliness in our academic and non-academic interactions. I am also deeply indebted to Professors Rama Cont, Dilip Madan, Philip Protter, and Josef Teichmann for agreeing to serve on my dissertation committee. I have had many enlightening discussions with each of them over the course of my graduate studies, which sharpened my understanding of the subject matter of this thesis. Without Dr. Bob Fernholz at Intech, there would be no Stochastic Portfolio Theory, which is the fundamental building block of this thesis. I therefore thank Bob Fernholz for sharing his insights and for further creating the space and atmo- sphere for our many meetings at Intech, where research is done at its finest: a free flow and exchange of ideas. Dr. Adrian Banner, Dr. Daniel Fernholz, Dr. Tomoyuki Ichiba, Dr. Phi Long iii Acknowledgments Nguen-Tranh, Dr. Vassilios Papathanakos, Radka Pickova, Subhankar Sadhukhan, and Dr. Phillip Whitman have regularly attended these meetings. Their feedback on my presentations and the exchange of ideas with them have greatly improved the results presented here. Besides the previously mentioned colleagues, I have been discussing this the- sis with friends and colleagues within the Department of Statistics at Columbia University, mainly with Emilio Seijo and Dr. Li Song, and outside of Columbia University, primarily with Professor Erhan Bayraktar, Professor Kostas Kardaras, Dr. Sergio Pulido, Winslow Strong, Professor Johan Tysk, and Dr. Hao Xing. Dur- ing a visit for the Minerva lectures at Columbia University, Professor Hans F¨ollmer spent valuable time reviewing an early draft of the first part of this thesis. Two referees and an associate editor of the Journal of Mathematical Finance also con- tributed very helpful comments on a paper that serves as the foundation for the first part of this thesis. I am deeply indebted to all of them and to the audiences in various conferences and seminars for giving me the opportunity to discuss this research. It would be harder to read this thesis if its English had not been proofread by five very patient friends. I thank Michael Agne, Rachel Schutt, Zach Shahn, Ekaterina Vinkovskaya, and, in particular, Ashley Griffith. This thesis is not directly, but indirectly, influenced by joint work with amaz- ing co-authors. In particular, I thank Professor Peter Carr, Dr. Travis Fisher, Professor Carlos Oyarzun, Professor Matthias Scherer, Amal Moussa, Tyler Mc- Cormick, Professor Andrew Gelman, Professor Tian Zheng, and my other co- authors for sharing their knowledge with me and for many good collaborations. iv Acknowledgments I am grateful to my colleagues at Commerzbank, D-Fine, J.P. Morgan, and Morgan Stanley for making my internships such a valuable experience. In partic- ular, I express my gratitude to Dr. Ju¨rgen Hakala, Andreas Weber, Dr. Jochen Alberty, Dr. J¨orn Rank, Dr. Lei Fang, Dr. Xiaolan Zhang, Daniel Leiter, Michael Jansen, Michael Sternberg, Dr. Len Umantz, and Tyler Ward. Without seeing the passion of my mathematics and finance teachers, I would never have started my doctoral studies. I am indebted to Inge Schneider, Siegfried Ranz, Dr. Josef Krutel, Helmut Freischmidt, Herbert Sittenberger, Ottmar Fis- cher, Elvira Rendle, and Professors Werner Balser, Werner Lu¨tkebohmert, Volker Schmidt, Franz Schweiggert, Frank Stehling, Karsten Urban, Ru¨diger Kiesel, Ul- rich Stadtmu¨ller, Amanda Adkisson, Kerry Back, Dante DeBlassie, and Jean-Luc Guermond. I am grateful to the Cusanuswerk, the Fulbright Commission, the National Science Foundation (DMS Grant 09-05754), and ColumbiaUniversity for both their financial and intellectual support. This thesis ends my doctoral studies at the Department of Statistics at Columbia University. I am grateful to Professors David Madigan and Richard Davis for creating such a friendly and creative environment in the department. I thank Dood Kalicharan, Faiza Bellounis, and Anthony Cruz for their always helpful support in all personal and administrative matters. All faculty members at Columbia University have always been very supportive; in particular, I would like to thank Professors Jan Vecer and Zhiliang Ying, both of whom have served on my oral committee, and Professors Jose Blanchet, Mark Brown, Rama Cont, Michael Hogan, Ioannis Karatzas, Steve Kou, Martin Lindquist, Shaw-Hwa Lo, v Acknowledgments Panagiota Daskalopoulos, Michael Sobel, Victor De la Pen˜a, Philip Protter, Daniel Rabinowitz, Bodhisattva Sen, and Tian Zheng. I also would like to thank all of the intelligent and passionate students whom I had the opportunity to teach or to instruct in the capacity of their teaching assistant. My friends at Columbia University have challenged me with interesting ques- tions and with their achievements, and most importantly, have been excellent friends. Each encounter has been very important to me. I thank all of them and mention a necessarily very incomplete, but hopefully representative, list: Xiaodong Li, Amal Moussa, Shawn Simpson, Li Song, Xiaoru Wu, Priya Dev, Stacey Han- cock, SamanthaSartori, MaggieWang, TomoyukiIchiba, AlexandraChronopoulou, Georgios Fellouris, Ragnheidur Haraldsdottir, Petr Novotny, Libor Pospisil, Rachel Schutt, Tyler McCormick, Ivy Ng, Subhankar Sadhukhan, Emilio Seijo, Heng Liu, Radka Pickova, Tony Sit, Ekaterina Vinkovskaya, Junyi Zhang, Michael Agne, Stephanie Zhang, Zach Shahn, Gerado Hernandez, Aleks Jakulin, Siliva Ardila, Juan Campos, Rodrigo Carrasco, Romain Deguest, Rouba Ibrahim, Xuedong He, Tulia Herrera, Yu Hang Kan, Olga Kolesnikova, Arseniy Kukanov, Xianhua Peng, Matthieu Plumettaz, Daniela Wachholz, Cecilia Zenteno, Yori Zwols, Avishek Ad- hikari, Danielle Cohen, Amit Duvshani, Hugo Escobar, Mei-Ho Lee, Adam Shavit, Joan Shavit, Casandra Silva Sibil´ın, Esther Quintero, Fabio Dominguez, Ashley Griffith, Olympia Hadjiliadis, Eva Jozsef, Gabor Jozsef, Helena Kauppila, Robert Neel, Oya Okman, Aly Sanoh, Bjarni Torfason, Olger Twyner, Jin Wang, and Yun Zhou. Over the last several years, I have also been supported by many friends out- side of Columbia University. I am especially grateful to Erhan Bayraktar, Matthias vi