A Study On The Pricing Of Several Options In The Incomplete Market

Recently, with the development of financial derivatives market, the pricing of deriva-tives, risk management and others have become more and more important. In order to describe the changing economic environment, more complex models are proposed. By the risk neutral pricing theorem, we investigate the pricing of several options in the incomplete market. The main contents of this thesis are listed in the following:1. In the first chapter, the concept of options and the assets pricing are at first in-troduced. Then, we introduce the pricing of options in the incomplete market. In addition, we give a brief description of the classification and the research of the credit risk model. Finally, some basic definitions, theorems and the main results of this thesis are provided.2. In the second chapter, we consider the pricing of the power options. In order to reflect the price process of the risky assets in the market, we first assume that the price process of the risky assets follow the jump diffusion model and then assume that the parameters in the market vary as the transition of the state of a continuous time Makov chain. In the first case, we suppose that the interest rate follow the Vasicek model, and the risky assets are correlated with the interest rate. By introducing the risk neutral measure, we obtain the pricing formula of the power options. In the second case, we assume that the interest rate, the expected return rate and the volatility are related to the state of the economy which is described by the continuous time Markov chain. Since the market under consideration is incomplete, we give an equivalent martingale measure by the regime switching Esscher transform. We get the price of the power options under the Markov-modulated geometric Brownian motion.3. In the third chapter, we discuss the pricing of several options in a reduced form model. Since the intensity of default may fluctuate severely as unanticipated events happen, we assume that the default intensity is governed by a jump-diffusion process. In addition, we assume that the writer of the option may default and the recovery rate is a constant. In the reduced form model, we give the price of the European option, the power option and the exchange option with credit risk. 4. In the fourth chapter, the value of the guaranteed annuity options is studied with the stochastic mortality intensity. In order to conform to the actual situation, we add the "jump" to the mortality intensity. We assume that the mortality intensity follow the jump diffusion model, underlying assets follow the stochastic volatility model, interest rate is the Vasicek model which is correlated with each other. We obtain the price of the guaranteed annuity options.5. In the fifth chapter, we investigate the hedging strategies and the minimal mar-tingale measure. When the price process of the underlying assets follow the jump diffusion model or the Markov-modulates models, the market is incomplete and the contingent claims can't be hedged by self-financing strategies. We give the lo-cally risk minimizing strategies under the jump diffusion models and the Markov modulated jump diffusion models with stochastic volatility.In brief, this thesis discuss the pricing of several options when the underlying assets follow different models. We get the pricing of the power options under the jump diffusion process and the regime switching model, and the valuation of the guaranteed annuity options with stochastic mortality. The pricing of the options with credit risk in a reduced form model is considered. Moreover, when the market is incomplete, we consider the risk minimizing strategies and the minimal martingale measure.