This thesis consists of four articles. Two of them focus on asset pricing; option pricing in stochastic volatility odel and its model estimation of unknown parameters are the themes of research. The other two take risk management as their theme, and discussing discuss the importance sampling of Value at Risk under different models.
The first article derives, under Markov Switching model, the closed form formula of the European call option, utilizing Monte Carlo method and Markovian tree to compare the differences among closed form solutions of pricing formula. The second article discusses whether including the materials of option price contribute to estimating the unknown parameters and unobservable states in stochastic volatility model. We consider two kinds of Kalman filter and particle filter. To study whether Heston''s (1993) model should include the materials of option price, we perform a comprehensive comparison and discussion by using a simulation method. The result shows that including the materials of option price and adopting particle filter is a better way to use the model to estimate unknown parameters and unobservable states.
The third article proposes combining the bootstrap method and the Laplace method to improve the efficiency of the importance sampling of VaR in Glasserman et al. (2000, 2002) that assume that portfolios follow multivariate normal distribution and multivariate t distribution. The fourth article studies the efficiency of the importance sampling of the VaR of investment combination of the multivariate jump diffusion model. It proposes the comparative static behavior of relevant parameters and analyzes performing algorithms. The numerical simulation results prove that the efficiency is greatly improved in large deviation. 本篇論文包含四篇文章，其中兩篇以資產定價為主軸，分別以隨機波幅模型的選擇權定價及模型參數估計問題為研究主題，另外兩篇以風險管理為主軸，分別針對不同模型下風險值的重點抽樣法，作一深入探討。
第一篇文章是研究在 Markov Switching 模型下，推導出歐氏買權具封閉解的評價公式，並利用蒙地卡羅法和馬可夫樹來模擬及比較封閉解的評價公式的差異性。第二篇文章是考慮在引進選擇權價格資料是否有助於隨機波幅模型的模型估計問題，文中考慮兩種 Kalman filter 及 particle filter ，針對 Heston (1993) 模型是否引進選擇權價格資料，以模擬方法作一全面性的比較及探討。其結果發現引進選擇權價格資料並採用 particle filter 是有助於模型估計。
第三篇文章提出用拔靴法及 Laplace 法，來改進 Glasserman et al. (2000, 2002) 對多元常態分配, 多元 t 分配的投資組合所構成之風險值的重點抽樣的效率性。第四篇文章是針對多重跳躍擴散模型的投資組合之風險值的重點抽樣的效率性，提出其演算法及相關參數的比較靜態分析，並由數值模擬結果驗證其效率性有大幅改進。
