Essays in Financial Mathematics

Essays in Financial Mathematics

Essays in Financial Mathematics Kristoffer Lindensjö

Dissertation for the Degree of Doctor of Philosophy, Ph.D. Stockholm School of Economics, 2013. Dissertation title: Essays in Financial Mathematics c SSE and Kristoffer Lindensjö, 2013 ISBN 978-91-7258-905-6 (printed) ISBN 978-91-7258-906-3 (pdf) Printed by: Ineko, Göteborg, 2013 Keywords: Partial information, Utility maximization, Optimal investment and consumption, Stochastic control, Filtering theory, Portfolio theory, Derivative pricing, Option pricing, Jump-diffusion, Instantaneous forward rate curve, Futures contract, End of the month option, Timing option, Embedded options, Optimal stopping

To you

Foreword This volume is the result of a research project carried out at the Department of Finance at the Stockholm School of Economics (SSE). This volume is submitted as a doctoral thesis at SSE. In keeping with the policies of SSE, the author has been entirely free to conduct and present his research in the manner of his choosing as an expression of his own ideas. SSE is grateful for the financial support provided by the Jan Wallander and Tom Hedelius Foundation and the fund Carl Silfvén which has made it possible to fulfill the project. Göran Lindqvist Director of Research Stockholm School of Economics Clas Bergström Professor and Head of the Department of Finance Stockholm School of Economics

Acknowledgments First and foremost, I am grateful for the support and the advice I have received from my supervisor, Professor Tomas Björk. I am forever indebted for the time and the energy he has invested into this project. Parts of this thesis were written while I was visiting the Department of Mathematics at ETH Zurich. Other parts were written during my visit at the Department of Finance at Copenhagen Business School. I thank both departments and especially Professor Josef Teichmann (ETH Zurich) and Professor Agatha Murgoci (Copenhagen Business School) for their hospitality. I am grateful to a large number of friends, colleagues and others in Stockholm, Zurich and elsewhere, especially the past and present Ph.D. students at the Department of Finance at the Stockholm School of Economics. My time as a Ph.D. student was highly enriching, enjoyable and memorable and this, in many ways, is thanks to all of you. I am also thankful for the stimulating research environment provided by the faculty and the administrative staff at the Department of Finance at the Stockholm School of Economics. This thesis was made possible by the financial support provided by the Jan Wallander and Tom Hedelius Foundation, the Stockholm School of Economics and the fund Carl Silfvén. I am eternally grateful to these generous benefactors. Finally, I thank my parents Sonja and Bo, and the rest of my family for their encouragement and support. Copenhagen, August 2013 Kristoffer Lindensjö vii

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Contents Introduction 1 1 Optimal investment and consumption under partial information 5 1 Introduction............................. 6 2 Mathematical foundation...................... 9 2.1 Strong solutions of stochastic differential equations... 9 2.2 Martingale representation results............. 12 3 The financial market and problem formulation.......... 16 4 The full information case...................... 19 4.1 Power utility........................ 24 4.2 Logarithmic utility..................... 28 4.3 Exponential utility..................... 30 5 The partial information case.................... 34 5.1 Power utility........................ 38 5.2 Logarithmic utility..................... 40 5.3 Exponential utility..................... 40 References 42 2 Option pricing under jump-diffusion dynamics for asset prices and interest rates 45 1 Introduction............................. 46 ix

2 The derivative payoff........................ 48 2.1 Examples of common options we can price using pricing formulas for our derivative................. 49 3 Setup and problem formulation.................. 50 4 An explicit pricing formula..................... 52 5 The instantaneous forward rate curve and bond dynamics.... 60 6 The Greeks............................. 63 7 A numerical example........................ 67 8 Concluding remarks......................... 68 9 Appendix.............................. 69 9.1 The Girsanov theorem, Itô s formula and applications.. 69 9.2 Marked point processes with deterministic intensity... 71 9.3 Differentiation of integrals................. 76 10 Graphs of call option prices.................... 78 References 81 3 The end of the month option and other embedded options in futures contracts 85 1 Introduction............................. 86 2 Setup................................. 87 3 The final futures price F 0...................... 89 3.1 A formula for F 0...................... 90 3.2 Solutions for F 0 in special cases.............. 91 3.3 Bounds for F 0........................ 95 4 The futures price process with several embedded options present 96 4.1 The quality option..................... 103 5 Approximation methods for F 0................... 103 5.1 Differentiability of optimal stopping value functions... 104 5.2 A simple approximation method for F 0.......... 109 x

6 Appendix.............................. 113 References 114 xi

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Introduction This doctoral thesis consists of three independent papers in financial mathematics. Paper 1: Optimal investment and consumption under partial information We present a unified approach for partial information optimal investment and consumption problems in a general non-markovian Itô process market. The main assumption is that volatility is a nonanticipative functional of the asset price trajectory. The local mean rate of return process and the Wiener process cannot be observed by the agent, whereas the asset price volatility, the stochastic interest rate and the asset prices can be observed. The market is shown to be complete in the sense that any contingent claim adapted to the observable filtration generated by the asset prices can be replicated. Utility functions are general. We tackle this stochastic optimal control problem under partial information in two steps. First, we solve the corresponding full information problem. The market is shown to be complete using a nonstandard martingale representation result, for which we need the volatility to be a nonanticipative functional. We then use the martingale approach to study the optimal investment and consumption problem. We characterize the agent s optimal consumption process and optimal portfolio weights process, as well as the resulting value function and wealth process. We also study some common utility function specifications and relate the solutions to the price processes of certain financial derivatives. Second, we translate the original partial information problem into a corresponding full information problem using filtering theory. Using the solutions of the full information problem, we then derive solutions to the original partial information problem. The main contribution of the paper is the unified approach for partial information optimal investment and consumption problems in complete non-markovian Itô process markets with stochastic interest rate and stochastic volatility.

2 Introduction Paper 2: Option pricing under jump-diffusion dynamics for asset prices and interest rates In this paper we study option pricing when stock prices and an instantaneous forward rate curve, with a corresponding continuum of bonds, follow certain jump-diffusions with random jump sizes. Given specifications of these processes, we derive reasonably explicit formulas for the price and the Greeks of a specific derivative payoff. The payoff of the derivative may depend on two assets which can be either bonds or stocks. We show that many of the option payoffs studied in the literature can be represented as linear combinations of our derivative payoff and hence our pricing formula can be used to price these options. Examples of such options are the European call, put, min, max, exchange and digital options on either stocks or bonds. The main contribution of the paper is to provide a reasonably explicit pricing formula for the above mentioned options in a setting where both the price of the underlying asset and the instantaneous forward rate curve follow jump-diffusions with random jump sizes. Paper 3: The end of the month option and other embedded options in futures contracts The end of the month option allows the holder of the short end of a futures contract to deliver the underlying at any time during the last week of the contract period at a fixed price determined at the start of the last week. We derive a formula for this price within a general and incomplete market framework. An approximation method to calculate this price and some special cases in which explicit solutions for the price exist are also presented. We also study the futures price process of a futures contract with a quality option that first has no option active, then it has an active timing option and lastly it has an active end of the month option. This is the natural setup for embedded options in futures contracts. We show that the futures price process in our setting is dominated by a standard futures price process with maturity at the time of the activation of the end of the month option. Moreover, we show that the futures price process during the time when the timing option is active coincides with the futures price process of a certain other futures contract that only has a timing option. The main contribution of the paper is to properly define the end of the month option and to derive the futures price process of a futures contract with the combination of options described above.