在Black-Scholes評價模型下台指選擇權最適波動性估計方法之研究

The Mothod of the Volatility Estimator in TXO Under the Black-Scholes Model

倪衍森(Yen-Sen Ni) ； 吳曼華(Man-Hwa Wu) ； 鄭亦妏(Yi Wen Jeng)

《管理科學研究》 2卷1期 (2005/03) Pp. 93-109

https://doi.org/10.6873/MSR.200503_2(1).0006

摘要

在以往的實證研究中，對於波動性的估計模型主要可以區分為兩大類：一類是藉由過去某段時間中的價格走勢做為波動性衡量的依據，稱之為歷史資料波動性，另一類是隱含波動性，乃利用選擇權的市價來代入評價模型中，以反推該價格所隱含的波動性。實證結果顯示兩者在波動性的估計效度上各有其擅長，且實務界常常藉由兩者相互比較的應用，來探討目前市場是否有高、低估的現象，做為投資者進行選擇權買賣操作上的依據。因此雖然相關研究很多，但是在波動性適用問題的探討上，多年以來仍是各界所著墨的焦點。本研究以2001年12月24日上市的台指選擇權（簡稱TXO）做為研究標的，目的在於了解何種波動性模型可以使所估算出之理論價格最貼近於市價，亦即何種波動性模型最能解釋台指選擇權所面臨之波動。吾人得到以下幾點重要發現：1.歷史波動性及GARCH（1，1）波動性對於TXO的估計能力顯著優於隱含波動性模型及隱含波動性與GARCH（1，1）的綜合模型。2.一般公認估計能力較佳的隱含波動性模型中，以考量到選擇權標的資產價格波動彈性的vega加權隱含波動性之價格誤差較小。3.隱含波動性缺乏估計效力代表台指選擇權市場之成熟度不足，但這極可能是在本文的研究期間中，其尚處於新興市場之狀態。本研究認為隱含波動性不指選擇權最適波動性估計模型的原因在於：計算過程的問題、市場無效率與市場制度設計上的問題。4.理論上來說，衍生性商品市場代表的是投資人對於未來現貨市場走勢之預期，應當是由衍生性金融市場來領導現貨市場，然而本文的實證結果卻是以現貨市場的價格變動對於台指選擇權有較高的影響度，可見投資人在選擇權商品的知識及買賣策略的運用上仍未成熟。

關鍵字

股票選擇權 ； 隱含波動性 ； Black-Sholes模型

並列摘要

The underlying asset price, exercise price, risk-free interest rate, duration and volatility are the endogenous variables in the Black-Scholes option pricing model, and we can obtain the theoretical price of an option contract through the model. Five variables except the volatility variable could be collected form market or the option contract, but the volatility variable is hard to be defined. It can be sorted into two categories. One is counted by historical underlying asset price data, and the other is implied volatility (IV). The empirical results about the suitability between volatilities calculated by the Black-Sholes model and real volatilities the option contract faces are not identical. Some of empirical results in this study imply that IV model is the best estimator for calculating the volatility for TAIEX stock index option (TXO), because it represents the true situation the option lies in. But other results show the opposite comments, i.e. IV are not the suitable estimator for pricing TXO, it will lead to estimating errors. The target of the study is a new product (TXO) in Taiwan, and this financial product is available since Dec.24, 2001. This research uses several methods for estimating the volatility for TXO, tries to find the optimal model to estimate the volatility, and search the causes of variation between theoretical value and market value for TXO. The findings are shown as follows. 1. To TXO, the historical standard deviation and GARCH(1,1) model are the best estimators, and IV models are the worst. 2. Among the models of weighted implied volatility, the mis-price of the vega weighted implied volatility model is smallest. 3. The result which IV model is not optimal volatility estimator implies that the TXO market is not mature. 4. The results are quit different between TXO and warrants: TXO call options are over-priced, and warrants are under-priced in Taiwan markets. We try to explain the differentiation in supply aspect, and make reasonable inferences.

並列關鍵字

TXO ； Stock Index Option ； Volatility ； Implied Volatility ； Black-Scholes Model