本論文以修正 Black & Scholes ( B-S model )定價模型為研究方向，主要探討的面向有二， 分別為 B-S 模型資產變動常態假設修正以及修正波動率恆定的假設。本文首先利用台股指數選擇權日資料驗證 B-S 模型的假設在台股指數市場的缺陷，再將 B-S 模型的隨機誤差項由常態分配假設修改為學生 t 分配，在學生 t 分配的基礎之下建立新的定價模型稱為 TDB-S 模型；接著修正波動率的恆定值假設，考慮波動率為有兩種隱藏狀態的馬可夫鍊，利用 Baum-Welch 法 (1970, Ann. Math. Statist) 迭代出隱藏馬可夫鏈的各項係數後，結合佔據時間的機率分配與 TDB-S 模型的架構建構出另一個新的模型 ODMMTD 模型。
文末分析台股指數選擇權 2019/4/1 至 2020/4/1 的收盤價日資料，分別利用本文建構的模型以及 B-S 模型四者不同模型搭配歷史估計量、 動差估計法(MME) 與 最大概似估計法 ( MLE ) 三種不同的參數估計方法進行選擇權定價，並比較在深度價外、價外、價平、價內、深度價內與交割時間距離長短來做績效評估。而績效評估的標準分別使用平均絕對誤差、平均比例誤差與均方根誤差三指標來比較模型的優劣。比較結果顯示：當權證處於價平與價外時，利用 TDB-S 搭配 MLE 估計量的定價績效最佳；而當權證處於深度價外、價內與深度價內時，利用 ODMMTD 模型搭配波動率的 MME 估計量的定價績效最佳。當權證距離交割日的交易日小於 20 天時，由學生 t 分配配飾隨機誤差項的模型並沒有顯著的績效提升，然而，當距離交割日大於 20 天時，學生 t 分配配飾的模型優點便明顯的展現出來，即 TDB-S 模型明顯優於 B-S 模型，且 ODMMTD 模型明顯優於 ODMM 模型。

The main purpose of this article is to modify the Black & Scholes model (B-S)(1973). The assumptions of B-S model are constant volatility and Gaussian distribution of asset returns which had been shown that violate the market phenomenon. Thus we try to use Student-t distribution to replace the error term of B-S model to capture the fat tail of distribution of asset returns and assume that the market volatility is a hidden Markov chain to grasp the market trend. Then we construct two new option pricing model which called TDB-S model and ODMMTD model based on the new assumptions. After that, we find the parameters of HMM such as transition matrix, initial states probability matrix by Baum-Welch method.
At the end of this paper, we compared B-S model, ODMM model, TDB-S model and ODMMTD model these four models with TXO market data from 2019/04/01 to 2020/04/01. The result is that the advantage of student-t distribution is weak if the maturity date is less than 20 days. However, when maturity date is over 20 days, TDB-S model and ODMMTD model have significant improvement on their pricing performance. Moreover, ODMMTD give the most accurate price when option is deep-in-price or deep-out-price and TDB-S model have minimal model error when pricing in-price, at-price and out-price options. Our research also found that over three method of estimating parameters (maximum likelihood estimate, method of moment estimate and historical estimate) (MLE, MME, HE), TDB-S model have the best performance when using MLEs and ODMMTD model match MMEs have the best performance.

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