Hamilton (1989)發展出馬可夫轉換模型，假設模型母體參數會隨某一無法觀察得到的狀態變數變動而改變，並用馬可夫鏈的機制來掌控狀態間切換，可適當掌握金融與經濟變數所面臨的結構改變，因此是一個十分重要的財務模型。Schwert (1989)觀察股價波動狀況，發現經濟衰退期的股價波動比經濟擴張期大，因此認為Hamilton (1989)所提出的馬可夫轉換模型亦可應用於股票市場。然而，發現當市場上有重大訊息來臨時，大部分標的資產報酬率會產生跳躍現象，因此汪昱頡 (2008)提出跳躍風險下馬可夫轉換模型，以改善馬可夫模型所無法反映之股價不正常跳躍現象。在探討股價指數報酬率之敘述統計量與動態圖後，本文認為跳躍幅度也會受狀態影響，因此進一步拓展周家伃 (2010)跳躍獨立風險下狀態轉換模型，期望對股市報酬率動態過程提供更佳的分析。實證部分使用1999到2010年的國際股價指數之S&P500、道瓊工業指數與日經225三檔作為研究資料，來說明股價指數具有狀態轉換及跳躍的現象，並利用EM(Expectation Maximization)演算法來估計模型的參數，以SEM(Supplemented Expectation Maximization )演算法估計參數的標準差，且使用概似比(Likelihood ratio)檢定結果顯示跳躍相關風險下狀態轉換模型比跳躍獨立風險下狀態轉換模型更適合描述股價指數報酬率。最後，驗證跳躍相關風險下狀態轉換模型能捕捉其報酬率不對稱、高狹峰與波動聚集之特性。 / Hamilton (1989) proposed Markov switching models to suppose the model parameters change with unobserved state variables which control the switch between states by Markov chain. It can be appropriate to grasp the financial and economic variables which facing structural changes, so it’s a very important financial model. Schwert (1989) observed stock prices, and discovered that the volatilities of recession are higher than the volatilities of expansion. Hence, Schwert (1989) suggested to apply the Markov switching models to stock market. However, most of underlying asset return have jump phenomenon when abnormal events occur to financial market. Wong (2008) proposed Markov switching models with jump risks to improve Markov switching models which can not capture the jump risk of asset price. According to stock index return’s descriptive statistics and dynamic graph, we argue that states will impact jump sizes. In this paper, we extend the regime-switching model with independent jump risks (Chou, 2010) to provide better analysis for the dynamic of return. This paper use stock indices of the study period from 1999 to 2010 to estimate the parameters of the model and variance of parameter estimators by Expectation-Maximization (EM) algorithm and SEM(Supplemented Expectation Maximization ) , respectively. And use the likelihood ratio statistics to test which model is appropriate.Finally, the empirical results show that regime-switching model with jump sizes dependency risk can capture leptokurtic feature of the asset return distribution and volatility clustering phenomenon.

本文使用Esscher轉換法推導狀態轉換模型、跳躍獨立風險下狀狀態轉換模型及跳躍相關風險下狀態轉換模型的選擇權定價公式。藉由1999年至2011年道瓊工業指數真實市場資料使用EM演算法估計模型參數並使用概似比檢定得到跳躍相關風險下狀態轉換模型最適合描述報酬率資料。接著進行敏感度分析得知，高波動狀態的機率、報酬率的整體波動度及跳躍頻率三者與買權呈現正相關。最後由市場驗證可知，跳躍相關風險下狀態轉換模型在價平及價外的定價誤差皆是最小，在價平的定價誤差則略高於跳躍獨立風險下狀態轉換模型。 / In this paper, we derive regime switching model, regime switching model with independent jump and regime switching model with dependent jump by Esscher transformation. We use the data from 1999 to 2011 Dow-Jones industrial average index market price to estimate the parameter by EM algorithm. Then we use likelihood ratio test to obtain that regime switching model with dependent jump is the best model to depict return data. Moreover, we do sensitivity analysis and find the result that the probability of the higher volatility state , the overall volatility of rate of return , and the jump frequency are positively correlated with call option value. Finally, we enhance the empirical value of regime switching model with dependent jump by means of calculating the price error.