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Some Optimal Control Problems in Mat Hematical Finance

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Some Optimal Control Problems in Mat Hematical Finance Some Optimal Control Problems in Mat hematical Finance Chaoyang Guo A thesis submitted to the Faculty of Graduate Studies in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Programme in Mathematics and Statistics York University Toronto, Ontario April, 1999 National Library Bibliothlzque nationale du Canada Acquisitions and Acquisitions et 8ibliographc Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON KIA ON4 Ottawa ON K1A ON4 Canada Canada The author has granted a non- L'auteur a accorde me licence non exclusive licence allowing the exclusive pennettant a la National Library of Canada to Bibliotheque nationale du Canada de reproduce, loan, distribute or sell reproduire, preter, distribuer ou copies of this thesis in microfom, vendre des copies de cette these sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur fonnat electronique. The author retains ownership of the L'auteur conserve la propriete du copyright in this thesis. Neither the droit d'auteur qui protege cette these. thesis nor substantial extracts fiom it Ni la these ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent &re imprimes reproduced without the author's ou autrement reproduits sans son permission. autorisation. Some Optimal Control Problems in Mathematical Finance by Chaayang Guo a dissertation submitted to the Faculty of Graduate Studies of York University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 0 1999 Permission has been granted to the LIBRARY OF YORK UNIVERSITY to lend or sell copies of this dissertation, to the NATIONAL LIBRARY OF CANADA to microfilm this dissertation and to lend or sell copies of the film, and to UNIVERSITY MICROFILMS to publish an abstract of this dissertation. The author reserves other publication rights, and neither the dissertation nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission. Abstract We study several optimal control problems arising in Mathemati- cal Finance. The first problem is related to models of Term Structure of Interest Rates, which is central to economic and financial theory and provides the basis for pricing interest rate derivatives. Within the framework of optimal control for infinite dimension systems, we de- velop new approaches to estimate the risk neutral drift term for both the one and the multi-factor models. Second, in pricing derivatives, a key and often difficult issue is how to deal with stochastic volatility. We show that our new techniques can also be employed to address this problem. Third, we investigate the problem of building envelopes for the values of convertible bonds under stochastic interest rates. Fi- nally, using our techniques, we examine the bond pricing problem with the presence of default risk, a topic that has received much attention in recent years. The related optimal control problems are analyzed mathematically and conditions for optimality are obtained via a vari- ational approach. Also, numerical algorithms are proposed and tested, and computational results are obtained. Acknowledgment It is a real pleasure to express my sincere thanks to my supervisor, Pro- fessor L.S. Hou for his patient guidance and encouragement with respect to this work. I have tremendously benefited from our stimulating discussions and his generous support. I would also like to thank the supelvising committee members, Professors M.E. Muldoon and Joseph Liu, and Professor M.W. Wong for their invalu- able support and suggestions. Thanks also go to the staff of the Department of Mathematics and Statis- tics, particularly Graduate Program Director, Professor J. Wu and Assistant Primrose Miranda. I would also like to acknowledge some help that I received from B. Luo. Finally, the love and support of my wife, Xiaqing and my lovely daughter, I-Ianzhen have always been the source of my strength to see this work through. Contents 1 Some Background 1 1.1 Some Background in Mathematics ................ 2 1.2 Some background in Mathematical Finance ........... 5 2 One Factor Case 12 2.1 Statement of the problem .................... 13 2.2 Optimal Control (I): Existence .................. 21 2.3 Optimal Control (11): Condition for Optimality ........ 29 2.4 Numerical Analysis ........................ 33 2.5 Comments on related Option Pricing .............. 38 3 Two Factor Case 39 3.1 The factors and the model .................... 40 3.2 Optimal Control ......................... 46 3.3 Algorithm and Numerical results ................ 54 3.4 Stochastic Volatility and Option Pricing ............ 62 4 Convertible Bond 66 4.1 The Model ............................. 67 4.2 Building the Envelope and the Optimal Control ........ 71 4.3 Numerical Results ......................... 77 5 Some Perspectives 85 5.1 Credit Risk and Bond Pricing .................. 86 5.2 Time dependence ......................... 91 Reference Preface Mathematical Finance is a rapidly growing field. With the introduction of more sophisticated mathematical techniques, such as PDE & numerical so- lution, Stochastic Analysis and Optimal Control, both the scope and depth of the field have increased dramatically. It includes the hpics of pricing, hedging, optimization, term structure of interest rates and portfolio opti- mization, etc. For instance, in the daily practice of pricing derivative products, people have to construct various mathematical models to meet different situations. These models, in turn, lead to many kinds of PDEs. The PDE involved are mostly of parabolic type, for which a closed form solution is sometimes obtainable but not too often. Thus a numerical procedure has to be called in, as Wihott et al. (1995) pointed out "we would rather have an accurate numerical solution of the correct model than an explicit solution of the wrong model". In this thesis, we attempt to deal from a different point of view with several classic issues appearing quite often in the practice of Mathematical Finance. More precisely, we try to address several classical problems such as the term structure of interest rates and bond pricing from a new angle. The techniques employed here come mainly from the field of optimal control of systems governed by partial differential equation, which have proved so powerful in several applied fields. It should be stressed that our emphasis here is the use of optimal control to build a rather flexible scheme for some classic problems, so the routine statistical task of parameter estimation is not in our scope. We will recom- mend some references for that. We study several classical problems: term structure of interest rates, bond pricing and option pricing under stochastic volatility. The study of term structure of interest rates is central to economic and financial theory and practice. It provides the basis for pricing fixed income securities and for the evaluation of contingent claims as well as futures con- vii tracts. The literature on the term structure of interest rates is closely related to bond valuation. In fact, many of the term strtlcture models are stated in terms of yield on default free, zero coupon bonds. Note, in particular, that the general scope for the study of term structure of interest rates is large, and we have no intention of covering the whole is- sue here. Instead we confine ourself on one aspect of the subject, for which optimal control theory can be successfully used to attack the problems. The dissertation is organized as follows: In Chapter One, we provide some background in Optimal Control and Sobo'lev Spaces, which will be used extensively. We also indude some rna- terials on Mathematical Finance, which are closely related to the topics dis- cussed here. In Chapter Two, we concentrate on the one-factor model, where the short rate is the only factor driven by a diffusion process. We construct a proper model for optimal control and make a detailed analysis of this model. The analysis made here will not only serve this chapter but also will set a tone for the whole thesis. In Chapter Three, we deal with the two-factor model, where the short rate and the volatility of the short rate are chosen as two factors following two different diffusion processes. Note that the models we construct in both Chapter 1 and Chapter 2 are already beyond the scope of traditional affine models for the term structure of interest rate. In the end, we move to the classical option pricing problem and show how our framework can be used to price the general option under stochastic volatility structure, a problem that receives more attention now. In Chapter Four, the convertible bond under stochastic interest rates is addressed. Our focus is to build an envelope within its lifetime. The re- sulting optimal control problem is a bit different from those in the previous chapters, as we need to obtain the control over the entire time horizon. viii Finally, in Chapter Five we discuss the issue of default risk and its im- pact on bond pricing, which has become increasingly important in modern financial practice in the late 90's. Some other aspects of Optimal Control are also discussed. It should be mentioned that this thesis represents the first attempt to utilize the theory of optimal control for systems governed by partial differ- ential equations in the investigation of some typical problems encountered in modern Mathematical Finance. It is our hope that more work will follow, and we believe firmly that this branch of Mathematics wiII be useful in the area of Mathematical Finance. 1 Some Background In this chapter, we intend to provide some background materials which are closely related the topics addressed here. The chapter is divided into two parts: Mathematical Finance and Optimal Control. The criteria of selecting these materials is that we prefer to choose very basic, relatively simple prin- ciples rather than complicated and longer materials. Based on this principle, the traditional Black-Scholes framework is quickly reviewed.
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