風險值大多是在假設資產報酬為常態分配下計算而得的，但是這個假設與實際的資產報酬分配不一致，因為很多研究者都發現實際的資產報酬分配都有厚尾的現象，也就是極端事件的發生機率遠比常態假設要來的高，因此利用常態假設來計算風險值對於真實損失的衡量不是很恰當。 針對這個問題，本論文以歷史模擬法、變異數-共變異數法、混合常態模型來模擬報酬率的分配，並依給定的信賴水準估算出風險值，其中混合常態模型的參數是利用準貝式最大概似估計法及EM演算法來估計；然後利用三種風險值的評量方法：回溯測試、前向測試與二項檢定，來評判三種估算風險值方法的優劣。 經由實證結果發現： 1.報酬率分配在左尾臨界機率1％有較明顯厚尾的現象。 2.利用混合常態分配來模擬報酬率分配會比另外兩種方法更能準確的捕捉到左尾臨界機率1％的厚尾。 3.混合常態模型的峰態係數值接近於真實報酬率分配的峰態係數值，因此我們可以確認混合常態模型可以捕捉高峰的現象。 關鍵字：風險值、厚尾、歷史模擬法、變異數-共變異教法、混合常態模型、準貝式最大概似估計法、EM演算法、回溯測試、前向測試、高峰 / Initially, Value at Risk (VaR) is calculated by assuming that the underline asset return is normal distribution, but this assumption sometimes does not consist with the actual distribution of asset return. Many researchers have found that the actual distribution of the underline asset return have Fat-Tail, extreme value events, character. So under normal distribution assumption, the VaR value is improper compared with the actual losses. The paper discuss three methods. Historical Simulated method - Variance-Covariance method and Mixture Normal .simulating those asset, return and VaR by given proper confidence level. About the Mixture Normal Distribution, we use both EM algorithm and Quasi-Bayesian MLE calculating its parameters. Finally, we use tree VaR testing methods, Back test、Forward tes and Binomial test -----comparing its VaR loss probability We find the following results: 1.Under 1% left-tail critical probability, asset return distribution has significant Fat-tail character. 2.Using Mixture Normal distribution we can catch more Fat-tail character precisely than the other two methods. 3.The kurtosis of Mixture Normal is close to the actual kurtosis, this means that the Mixture Normal distribution can catch the Leptokurtosis phenomenon. Key words: Value at Risk、VaR、Fat tail、Historical simulation method、 Variance-Covariance method、Mixture Normal distribution、Quasi-Bayesian MLE、EM algorithm、Back test、 Forward test、 Leptokurtosis

Chen, Chang, Wen and Lin (2013)提出馬可夫調控跳躍過程模型(MMJDM)描述股價指數報酬率，布朗運動項、跳躍項之頻率與市場狀態有關。然而，利率並非常數，本論文以狀態轉換模型配適零息債劵之動態過程，提出狀態轉換下的利率與具跳躍風險的股票報酬之二維模型(MMJDMSI)，並以1999年至2013年的道瓊工業指數與S&P 500指數和同期間之一年期美國國庫劵價格為實證資料，採用EM演算法取得參數估計值。經由概似比檢定結果顯示無論道瓊工業指數還是S&P 500指數，狀態轉換下利率與跳躍風險之股票報酬二維模型更適合描述報酬率。接著，利用Esscher轉換法推導出各模型下的股價指數之歐式買權定價公式，再對MMJDMSI模型進行敏感度分析以評估模型參數發生變動時對於定價公式的影響。最後，以實證資料對各模型進行模型校準及計算隱含波動度，結果顯示MMJDMSI在價內及價外時定價誤差為最小或次小，且此模型亦能呈現出波動度微笑曲線之現象。 / To model asset return, Chen, Chang, Wen and Lin (2013) proposed Markov-Modulated Jump Diffusion Model (MMJDM) assuming that the Brownian motion term and jump frequency are all related to market states. In fact, the interest rate is not constant, Regime-Switching Model is taken to fit the process of the zero-coupon bond price, and a bivariate model for interest rate and stock index return with regime-switching and dependent jump risks (MMJDMSI) is proposed. The empirical data are Dow Jones Industrial Average and S&P 500 Index from 1999 to 2013, together with US 1-Year Treasury Bond over the same period. Model parameters are estimated by the Expectation-Maximization (EM) algorithm. The likelihood ratio test (LRT) is performed to compare nested models, and MMJDMSI is better than the others. Then, European call option pricing formula under each model is derived via Esscher transformation, and sensitivity analysis is conducted to evaluate changes resulted from different parameter values under the MMJDMSI pricing formula. Finally, model calibrations are performed and implied volatilities are computed under each model empirically. In cases of in-the-money and out-the-money, MMJDMSI has either the smallest or the second smallest pricing error. Also, the implied volatilities from MMJDMSI display a volatility smile curve.