在經歷了2007年次級房貸的風暴後，人們再度聚焦於極端事件發生的機率。實證結果顯示金融市場存在著厚尾現象，其意指極端事件發生的次數較某些財務理論(例如: 對數常態分配模型)更為頻繁。此論文應用一般化極值(GEV)模型於風險中立分配(RND)，並以此推導出歐式買權與賣權之封閉解。
此模型提供標的物分配更為彈性的峰態與偏態係數，並且不論在in-sample或是out-of-sample中，GEV模型在選擇權評價上，其均方根誤差(RMSE)的表現皆優Black-Scholes 模型。我們同時也呈現GEV模型的三個參數在不同到期期間下的變化，而其中的shape參數決定了分配尾端的厚度。此外我們也推導出在GEV模型下的Delta hedge ratio，並於不同moneyness下，呈現出與Black-Scholes模型避險比率之差異。

Having recently experienced a crisis event -- the 2007 subprime crisis, people refocus on the extreme events which occur with small probability. The empirical results show that the fat tail exist on financial markets, which means the extreme event occurs more often than in some financial theory such as lognormal model. This paper indicates that the Risk Neutral Density (RND) derived from Generalized Extreme Value (GEV) to price option provides a flexible model that captures negative skewness and excess kurtosis of returns.
Either in in-sample or out-of-sample data this paper represents a better performance in the three-parameters GEV model which includes tail shape parameter、location parameter and scale parameter. Of these parameters, we show that tail shape parameter plays an important role to control the size of tail and how the implied tail index to facilitate the fitness of fat tail. We also derive the delta-hedge formula to survey the difference delta between the Black-Scholes model and GEV model.

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