一、研究動機

　　股票選擇權價格包含了預測股價的波動度的訊息，所以我想研究利用美國銀行 ( Bank of America , BAC ) 的市場股票選擇權價格去檢視是否市場上股價的未來實際波動度( Future Realized Volatility , FRV ) 可以被混合對數常態分配 ( MLN ) 和混合對數常態分配加入破產機率( MLNbk )的無模型隱含波動度Model-free implied volatility ( MFIV ) 解釋，並比較以上兩種模型的解釋力之好壞，本研究檢視是否加入破產機率之模型是比未考慮破產機率的模型具有較多的解釋力，以上兩種模型算出的波動度都是在 Q-measure 下的波動度，我們額外把實際波動度也考慮進來去解釋未來實際波動度，實際波動度是在 P-measure 下的波動度，將以上三者當作解釋變數去解釋未來實際波動度(被解釋變數)。

二、研究方法

依市場資料作分析去推算理論選擇權價格還有市場無風險利率和股利率。
以最小平均誤差平方法及市場資料估計破產機率模型，和混合對數常態的分配模型，去估計各個模型參數。
模型參數記錄下來代入 MFIV 公式中，得到分別配適上述兩種模型之 MFIV。
用線性迴歸模型去驗證這兩種計算之 MFIV 何者對於市場上的實際波動度最有解釋力，其中線性迴歸式為:

FRV_t=α+β_1 MLN+β_2 MLNbk+β_3 RV_t+β_4 GJR+ε

三、研究結果

研究結果顯示出，MLNbk 模型所計算之 MFIV 較 MLN 模型所計算之 MFIV 對實際波動度較有解釋力，其 P 值比較小，也比實際市場波動度還顯著，可以拒絕虛無假設，此意涵在計算 MFIV 時，考量公司破產機率較能預測未來 實際波動度。
 

Stock option prices contain forecasting information about stock price volatility and potentially , the probability of default. So , we use BAC stock option market data to research whether future realized volatility is explained by two types of model-free implied volatility ( MFIV ) . Here are our two MFIV estimations. We adopt a risk-neutral density ( RND ) model consisting of a mixture of lognormal densities with probability of bankruptcy term ( MLNbk2 ) , and a mixture of lognormal ( MLN ) to estimate model-free implied volatility , and the realized volatility ( RV ) we want to know whether we add in the probability of bankruptcy term , it can have significant effect to estimate realized volatility.

After our experiment , we can conclude that MLNbk2 is better than MLN , we can say for forcasting realized volatility , add bankruptcy rate is better than not adding.

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