UNIVERSITY OF CALGARY Essays on Regime Switching Models in Finance by Hong Miao A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY HASKAYNE SCHOOL OF BUSINESS CALGARY, ALBERTA April 2008 © Hong Miao 2008       ISBN: 978-0-494-41142-1      UNIVERSITY OF CALGARY FACULTY OF GRADUATE STUDIES The undersigned certify that they have read, and recommend to the Faculty of Graduate Studies for acceptance, a dissertation entitled “Essays on Regime Switching Models in Finance” submitted by Hong Miao in partial fulfillment of the requirements for the degree of Doctor of Philosophy. ________________________________________________ Supervisor, Dr. Robert J. Elliott, Haskayne School of Business ________________________________________________ Dr. Anatoliy Swishchuk, Department of Mathematics & Statistics ________________________________________________ Dr. Alfred Lehar, Haskayne School of Business ________________________________________________ Dr. Tony Ware, Department of Mathematics & Statistics _________________________________________________ External Examiner, Dr. Aditya Kaul, University of Alberta ______________________ Date ii Abstract This dissertation consists of four essays focusing on applications of regime switching models in finance. The first essay discusses the investment timing problem in a regime switching framework. We consider a firm facing a future real investment opportunity whose investment cost depends on economic situations. Our approach considers the investment timing as a perpetual American option when the strike price switches between two possible values depending on the economic situations. It gives better optimal investment policy than the widely used standard real option method. The second essay extends the general equilibrium pricing model into an economy with two “states”. Based on assumptions of a CRRA utility function, we have derived a partial differential equation satisfied by the representative agent's cost function. A form of the solution of the partial differential equation has been given in general equilibrium with intermediate consumption. In the case when the representative agent does not have intermediate consumption, we have found an explicit solution of the cost function. A closed-form expression for the riskless rate has been derived. We have also provided a partial differential equation satisfied by any contingent claim written on any risky asset in the market. The stochastic discount factor has been investigated in our framework. Based on the stochastic discount factor, we have suggested an explanation for the equity premium puzzle. The third essay studies a stochastic volatility model in a regime switching world. We have derived closed form expressions for all the parameters by using the filtering iii techniques. The fourth essay provides a new approach for computing value at risk and expected shortfall in a regime switching economy. Based on the Student-t distribution assumption, we have suggested an approach to evaluate value at risk and expected shortfall. We use the Student-t distribution to capture the fat-tail phenomenon and regime switching to model the volatility clustering. Closed form expressions for computing value at risk and expected shortfall for both a single asset and a portfolio are proposed. The approach is easy to apply. iv DEDICATIONS Shanshan Hong My parents, Huanlian Dong and Shuhua Miao My supervisor, Dr. Robert J. Elliott and his wife, Ann Elliott v Acknowledgements I would like to thank my supervisor, Dr. Robert J. Elliott, for his unceasing support and guidance. I also thank the other two professors on the supervisory committee, Dr. Anatoliy Swishchuk and Dr. Alfred Lehar, for their comments and suggestions which have improved this dissertation. Thanks also go to Jin Yu at Vienna School of Finance for his cooperation. vi TABLE OF CONTENTS Approval Page……………………………………………………………………… ii Abstract……………………………………………………………………………… iii Dedication…………………………………………………………………………… v Acknowledgements…………………………………………………………………. vi Table of Contents……………………………………………………………………. vii CHAPTER ONE: INTRODUCTION……………………………………………… 1 CHAPTER TWO: ESSAY 1: INVESTMENT TIMING UNDER REGIME SWITCHING……………………………………………………………………… 4 2.1 Introduction…………………………………………………………………… 4 2.2 Optimal Investment Models………………………………………………… 7 2.2.1 Model Setting……………………………… ….……… …………… 10 2.2.2 The Benchmark Case………………………… ….….………………… 12 2.2.3 Standard Real-options Model………………… ….…………………… 17 2.2.4 Full Information Case……………………… …….…………………… 18 2.3 A Numerical Example ……………… …………… ……………………… 22 2.3.1 Expected Actual Payoff in the Benchmark Case … ….……………… 24 2.3.2 Expected Actual Payoffs in the Standard Real-Option Case…………… 25 2.4 A Numerical Example……………………………………….……………… 26 2.5 Conclusions……………………………………………….………………… 29 CHAPTER THREE: ESSAY 2: GENERAL EQULIBRIUM MODEL WITH REGIME SWITCHING………………………………………… ………………… 30 3.1 Introduction……………….………………………………………………… 30 vii 3.2 The Basic Model……………………………………………………………… 33 3.3 General Equilibrium Model with Representative Agent…… … ………… 36 3.3.1 General Equilibrium with Intermediate Consumption…………………… 36 3.3.2 General Equilibrium without Intermediate Consumption……… ……… 44 3.4 A Partial Differential Equation for Contingent Claims ……………………… 47 3.5 Stochastic Discount Factor and Market Price of Risk ………………………. 51 3.6 Concluding Remarks………………………………………………………… 55 CHAPTER FOUR: ESSAY 3: STOCHASTIC VOLATILITY MODEL WITH FILTEING…………………………………………………………… …………. 57 4.1 Introduction…………………………………………………………………… 57 4.2 Assumptions………… …………………….………………………………. 58 4.3 Change of Measure ………………………………………………………… 61 4.4 Parameter Estimates……………………… …………………………………. 69 4.4.1 Expectation Maximization……………………………………………… 70 4.4.2 Specific Processes ………………………………………………………. 70 4.4.3 Estimate of p ji ……………….……………………………………………. 80 4.4.4 Estimate of α, β, θ and μ….………………………… ………………… 84 4.5 Conclusions…………………………………………………… …………… 86 CHAPTER FIVE: ESSAY 4: VaR and EXPECTED SHORTFALL: A NON-NORMAL REGIME SWICHING FRAMEWORK…………… ………… 87 5.1 Introduction…………………………………………………………………… 87 5.2 Definitions and Model Setup………………….………………………………. 90 5.2.1 Definitions and Features of VaR ……………………………………… 90 5.2.2 Definitions and Features of ES ………………………………………… 91 viii 5.3 Model Setting…………………………………………………………………. 92 5.4 Why a Student-t Distribution? …………………………………………… 95 5.5 Parameter Estimation …………………… …………………………………. 97 5.5.1 Change of Measure…… ……………………………………………… 97 5.5.2 Parameter Estimation….…………………………………………………. 102 5.6 Closed-form VaR and ES for a Single Asset……….………… …………… 105 5.7 Closed-form VaR and ES for a Portfolio…………………………………… 106 5.8 Conclusion…………………………………………………………………… 108 CHAPTER SIX: SUMMARIES AND FUTURE RESEARCH……… ………… 109 REFERENCES…………….………………………………………………………… 111 APPENDIX ……………………………………………………………………… 118 ix [...].. .Essays on Regime Switching Models in Finance Hong Miao Abstract This dissertation consists of four essays focusing on applications of regime switching models in …nance The …rst essay addresses the investment timing problem in a regime switching framework The second essay discusses the general equilibrium pricing model when the economy has two states The third essay studies a stochastic model using... values, meaning we should invest in the project if S Si depending on the state of the world at time zero Condition (2:11) is the value-matching condition at Si : Equation (2:12) is the smooth pasting condition Finally, equation (2:13) is included to rule out a speculative bubble Solving equation (2:10) ; with boundary conditions (2:11) (2:13) ; we have: Proposition 2.3 Under Assumption 2.5, if the economy... …rst essay elaborates regime switching and optimal investment timing in a real option framework We …rst consider an irreversible investment timing decision by adding a hidden Markov process to model the state of the economy in continuous time The cost of the investment is driven the Markov chain Therefore, the investment cost is either K1 or K2 depending on whether the economy is in a ‘ cost’or ‘ low... Veronesi (1999) derives a rational expectations equilibrium model of asset values in the present of regime shifts David and Veronesi (2000), and Bu¢ ngton and Elliott (2002) value contingent claims on an underlying with uncertain parameters This paper is intended to elaborate regime switching and optimal investment timing in a real option framework The paper di¤ers from the existing literature in a... exchange rate modelling, and interest rate modelling Empirically, economic situations impact …nancial decisions and price movements of …nancial instruments The purpose of this study is to extend the applications of regime switching models to investment timing, general equilibrium pricing models, volatility models and risk management This dissertation includes four essays with consolidated references... elements in the Q-matrix, it is possible to have our stochastic investment cost return to its equilibrium level in the long run Hence, our assumption of regime switching investment cost is less misspeci…ed than ones with constant or stochastic costs driven by Brownian motions The paper is organized as follows In Section II, real option models are developed to investigate the investment timing problem... parameters used in the model We model the state of the economy by a Markov chain with a …nite state space Markov regime switching models have been used in electrical engineering since the 1960s Hamilton (1989) proposed the a 5 regime switching model to study postwar US real GNP associated with business cycle See Hamilton (2005) for a recent survey The modelling of asset price processes with regime shifts... in a signi…cant way In this paper we …rst consider an irreversible investment timing decision by adding a hidden Markov process to model the state of the economy in continuous time The cost of the investment is driven by the Markov chain Therefore, the investment cost is either K1 or K2 depending on whether the economy is in a ‘ low cost’ or ‘ high cost’ state K1 and K2 can be considered as strike... volatility clustering phenomenon and variance independent variation in the higher moments by assuming the returns follow Student’ t-distributions s The dissertation is organized as follows The …rst essay on investment timing is presented in Chapter 2, the second essay on general equilibrium model in Chapter 3, the third essay on stochastic volatility in Chapter 4, and the fourth essay on value at risk... constant investment cost identical to the ratinal expectation of the investment cost at time zero The rational investment cost should be the expectation of hK; Xt i : We …rst estimate the investment costs K1 and K2 by computing the expectation of hK; Xt i based on observing the original state of the chain Thus, we shall …rst calculate the expectation of the Markov chain at time t, i.e E[Xt ]: Solving . ……………………………………………………………………… 118 ix Essays on Regime Switching Models in Finance Hong Miao Abstract This dissertation consists of four essays focusing on applications of regime switching models in …nance. The …rst. dissertation consists of four essays focusing on applications of regime switching models in finance. The first essay discusses the investment timing problem in a regime switching framework. We consider. applications of regime switching models to invest- ment timing, general equilibrium pricing models, volatility models and risk management. This dissertation includes four essays with consolidated