Option Pricing Under Regime Switching Models

Option pricing is one of the core issues in mathematical finance. The traditional Black-Scholes option pricing formula has been widely used for pricing option and hedging in finance industry, but there are a large number of empirical results indicate that the asset price follows the geometric Brownian motion is not realistic. Over the past three decades, many different option valuation models have been proposed. Some of these include jump-diffusion models, Levy processes, stochastic volatility models, GARCH model and others. Recently, there has been considerable interest in applications of a regime switching model which is modulated by a continuous time Markov chain to option pricing problem. The states of the continuous time Markov chain can be interpreted as the states of the economy. The transitions of the states of the economy may be attributed to structural changes of the economy and business cycles. Based on the previous research result, this thesis discuss the option pricing under regime switching models and a new two-state regime switching model is provided. Moreover, because the market described by the Markov-modulated regime switching model is incomplete, a locally risk minimizing hedging strategy and the minimal martingale measure are also obtained under a regime switching model. The main contents of this thesis are listed in the following:1. In the first chapter, the origin and development of mathematical finance are first introduced. Then we give a brief description of the option pricing in an incomplete market and the extension of option valuation models. In addition, we also introduce the regime switching model and present research at home and abroad. Finally, some preliminaries and the main results of this thesis are provided.2. In the second chapter, the pricing problem of vulnerable European options is con-sidered under a regime switching model. We suppose that the market interest rate, the appreciation rate and the volatility rate of the risky asset depend on the states of the economy which are modeled by a continuous time Markov chain. Since the market is incomplete, we adopt the regime switching Esscher transform to deter-mine an equivalent martingale measure and provide analytical pricing formulas of vulnerable European options when the dynamics of the risky asset is governed by a Markov-modulated geometric Brownian motion or a Markov-modulated jump dif-fusion process. 3. In the third chapter, a two-state regime switching model is provided. We consider a market which has two states, a stable state and a high volatility state. The dynamic of the risky asset price follows different stochastic processes in different states of the market, the risky asset price is driven by a geometric Brownian motion or a Markov-modulated geometric Brownian motion when the market is stable, but the risky asset price follows a jump diffusion process or a Markov-modulated jump diffusion process if the market state is high volatility. In addition, the pricing problem of several options is considered under this two-state regime switching model.4. In the fourth chapter, the value of catastrophe put option is studied under a regime switching model. Jaimungal and Wang(2006) suppose that the catastrophe risk is unsystem risk which is not priced. In this chapter, we improve their model and assume that the market interest rate, the appreciation rate and the volatility rate of the risky asset all are depend on the states of the economy. In particular, the catastrophe risk is priced and the regime switching Esscher transform is adopted to obtain the simple return and the compound return regime switching Esscher transform martingale measures for the valuation problem in the incomplete market setting. In addition, we also discuss the existence of these two equivalent martin-gale measures and prove that the simple return regime switching Esscher transform martingale measure is the minimal entropy martingale measure when the risky as-set follows a Markov-modulated geometric Levy process. In the end, the explicit analytical formulas of catastrophe put option are derived under these equivalent martingale measures.5. In the fifth chapter, option pricing and hedging is analyzed under a regime switching model. We suppose that the risky asset follows a Markov-modulated geometric Levy process. The market described by a Markov-modulated geometric Levy process is incomplete, it means that contingent claims can not be hedged perfectly by self-financing strategy. The locally risky minimal hedging strategy and the minimal martingale measure are provided under a regime switching model. Moreover, the pricing problem of European call option, European put option and forward starting call option is considered under a regime switching model.In brief, this thesis discuss the option pricing under regime switching models. The pricing formulas of vulnerable European options and catastrophe put option are derived, a new two-state regime switching model is proposed and the valuation of some kinds of option is considered under this option valuation model. Moreover, the market described by a Markov-modulated regime switching model is incomplete, the contingent claim can not be replicated by self-financing strategy, this thesis consider the hedging of contingent claims under a regime switching model and provide the locally risk minimizing hedging strategy and the minimal martingale measure. These results are not only useful in theory but also significant in practice for pricing option and hedging in the financial markets.