Option pricing with stochastic volatility models

Despite the success and the user-friendly features of Black-Scholes (BS) pricing, many empirical results in the option pricing literature have shown the departures from the BS model. The motivation of this dissertation starts from these departures. In the first part of dissertation, we take the popular approach of stochastic volatility and jump models that are known to give good explanations to the empirical phenomenon. In order to keep analytic tractability, we derive the Generalized Black-Scholes (GBS) formula by a proper conditioning in a general mixture framework. By taking advantage of this new version of option pricing formula, we propose an approximation scheme that is well suited for the conditional Monte Carlo method. The simulation study and Markov Chain Monte Carlo (MCMC) algorithm give an evidence of a huge computational time reduction without much loss of accuracy. In the second part, we provide a new prospective on the forecasting ability and information content of the BS implied volatility in the presence of nonzero leverage effect. The leverage effect, which is the correlation between the return and volatility process, is introduced to model the observed Black-Scholes implied volatility (BSIV) smile and its skewness. We provide a simple theoretical framework that explains and justifies the use of BSIV from at-the-money option for the volatility forecast. Based on this and simulation study, which show the sensitivity of the concavity of option price with respect to the underlying stock price (the gamma effect), we propose a new approach to improve option pricing accuracy by a proper account for the gamma effect