Abstract
This thesis consists of two parts. In the first part, we propose some computational methods to improve the basic Monte Carlo simulations when estimating option prices because of its slow convergence rate. We also prove that our importance sampling method is efficient theoretically and numeri- cally. And our importance sampling methods is not only suited for European options with deterministic volatility and interest rate, but also applied to the one with stochastic volatility and interest rate. In the latter case, we propose an approximate probability measure to avoid difficulty of compu- tation. Numerical proof is shown for these two cases as well. In the second part, the simulation methods are applied to model cali- bration to option market prices. Since the calibrating procedure is time- consuming, the efficient importance sampling method developed in the pre- vious stage becomes essential. We use Fourier series method, which is fully model-free and nonparametric, to estimate the time series volatility of any stochastic underlying process. By calibrating to the market prices, we can observe some parameters in the volatility process; speed of mean-reversion, volatility of volatility and long-run mean, etc.