Option Pricing For Stochastic Volatility Models With Jumps

Financial asset pricing composes a pillar of modern financial theories, and is also one of the most substantial and fundamental areas in mathematical finance research. As a typical class of financial derivatives, option has drawn much attention that makes it the central research topic in financial derivatives pricing field. Since the invention of Black-Scholes option pricing formula, which initiated the second revolution on Wall Street, the pricing theories on all kinds of options such as European options, barrier options, Asian options, American options, as well as a variety of financial innovations have maintained rapid development. However, based on the complete market hypothesis, as well as the presumption of constant volatility, the Black-Scholes model is found to be not appropriate for changes of the modern financial market. Therefore, in order to describe the fluctuating characters of the underlying asset prices better, display heavy tailed distributions, reflect quasi long-range dependence of their stochastic volatility and the leverage effect between the underlying asset returns and their changes of volatility. Barndorff-Nielsen and Shephard proposed a new stochastic volatility model of the Ornstein-Uhlenbeck type, from now on abbreviated as BNS. Moreover, recent empirical works indicate that BNS model possesses good mathematical properties, and has been proven to be able to capture some stylized features of financial time series. As a consequence, this thesis is devoted to the option pricing problems in BNS model.We assume that the price process of the underlying assets follows the BNS model, i.e. the stochastic variance of log-returns is constructed via a mean-reverting stationary process of the Ornstein-Uhlenbeck type driven by a subordinator. First we consider the pricing of the power options. By introducing the risk neutral measure, we obtain the pricing formula of the power options by applying Fourier transforms and martingale methods. In addition, we derive closed-form formulas for the Greeks of options. Secondly, we discuss the pricing of forward starting options in BNS models with stochastic interest rates. We take Hull-White model as an example of stochastic interest rate. Using the stochastic analysis and the character of forward start, we obtain the general pricing formula for the forward starting options. Thirdly, we consider the problem of pricing Asian options in BNS model. By using the risk hedge method and perform- ing a change of numeraire we derive a partial integro-differential equation for Asian options. As we can hardly obtain an explicit solution of the partial integro-differential equation, the key part of our problem becomes the efficiency and accuracy of numerical algorithm. Therefore, using an operator splitting technique, the problem is reduced to a problem of two simple equa-tions. Then, a finite difference discretizations is proposed for dealing with the terms containing the partial derivatives and a simple trapezoidal rule is used for the integral term due to jumps. Numerical experiments confirm that the developed methods are very efficient.Furthemore, the underlying asset return follows a generalized BNS model, i.e. the s-tochastic variance of asset returns is constructed by volatility modulated non-Gaussian Ornstein-Uhlenbeck processes. We consider geometric Asian options with both fixed and floating strike. Guided by Laplace transforms and a change of numeraire, we gain a valuation formula for Asian options.