無模型設定隱含波動率-S&P500指數期貨選擇權的隱含波動率之實證研究

Abstract

Britten-Jones and Neuberger (2000)導出了在標的資產的價格服從擴散過程(diffusion process)的假設之下的無模型設定隱含波動率(the model-free implied volatility)。而Jiang and Tian (2005)把上述之無模型設定的隱含波動率推廣到標的資產的價格服從跳躍-擴散隨機過程(jump-diffusion process)，並且發展了簡單的方式把公式轉換成可以使用市場上的歐式選擇權價格來當作計算的工具。而在計算無模型設定隱含波動率的過程中，需要藉助Black-Scholes模型來當作轉換隱含波動率的橋樑，因為Black-Scholes模型有許多不合理的假設，在此本文加入了另一個轉換隱含波動率的模型來作比較，就是Chen, Palmon and Wald (2003)文中所提到的靜態實證模型(The Static Empirical Model)，他們放寬了幾個Black-Scholes模型中不合理的假設而推導出靜態實證模型。本文的實證研究中，想直接使用S&P 500指數期貨選擇權的資料算出的無模型設定隱含波動率來測試選擇權市場的效率性，不過本文所使用的資料型態是美式選擇權，在此使用了Barone-Adesi and Whaley (1987)的方法，把美式選擇權價格轉換成歐式選擇權的價格之後，再套入本文的模型來加以比較。最後發現，由靜態實證模型轉換出的隱含波動率所算出來的無模型設定隱含波動率對於預測未來的以實現波動率確實會比Black-Scholes模型所轉換出的隱含波動率還要來的有效率。而且由較多選擇權資料所算出的無模型設定隱含波動率也的確比只使用單一一個選擇權的隱含波動率(Black-Scholes模型隱含波動率與靜態實證模型隱含波動率)的預測能力要來的高。

Britten-Jones and Neuberger (2000) derived the model-free implied volatility under the assumption that the price of underlying asset follows diffusion process. Jiang and Tian (2005) further introduce that the price of underlying asset follows jump-diffusion process using the above model-free implied volatility, and build a simple way to transfer the formula to a computing instrument using European option prices on the market. In the process of computing model-free implied volatility, we need to use the Black-Scholes model for the bridge of transferring implied volatility. However, there are many unreasonable assumptions in the Black-Scholes model, we use another bridge to transfer implied volatility in this paper for comparison. That is The Static Empirical Model introduced by Chen, Palmon and Wald (2003). They relax some unreasonable assumptions in the Black-Scholes model to derive The Static Empirical Model. The empirical study in this paper use the data of S&P 500 index futures options to compute the model-free implied volatility to test the efficiency in the option market. But we use the American option data in this paper. So we also use the method introduced by Barone-Adesi and Whaley (1987) to transfer the American option price to European option price to fit our model. At last, we find that when predicting the realized volatility in the future, the model-free implied volatility computed by the implied volatility transferred by The Static Empirical Model is more efficient than the implied volatility transferred by Black-Scholes model. And the prediction ability of model-free implied volatility from more option data is better than the implied volatility of one option (the implied volatility of Black-Scholes model and the implied volatility of The Static Empirical Model).