波動率(volatility)在金融市場當中具有不可或缺的角色，它不但會影響到金融商品的評價，也會影響投資者的風險控管策略。Black和Schole定義股票價格變化為一隨機過程，以往業界衍生性金融商品的定價皆以此為基準。

然而，如此的定價卻會造成實際價格和定價之間的明顯差距。透過實證研究發現，實際上會存在「波動率微笑(volatility smile)」的情況，且波動率變化對買權價格的影響相當敏感，波動率為常數的假設明顯和事實不符。另外，在不完全市場的假設下，投資者對單一金融商品的價格看法會有所不同，選取的等價鞅測度(EMM,equivalently martingale measure)並不唯一，導致決定出的價格亦非唯一。

本篇論文首先探討在風險中立機率測度下，如何利用「最適q測度」機率測度轉換，決定出不同機率測度 下的定價模型，接著利用決定出的模型，計算歐式選擇權(European option)以及變異數交換(variance swap)的價格，並且討論在不同的機率測度下，對應到的最適避險策略。最後，利用程式模擬，比較不同機率測度下的歐式選擇權及變異數交換價格，以及各種避險策略下各自的避險效果優劣。

由最終的結果，我們可以觀察到歐式買權及變異數交換中的合理變異數值(fair variance)，在不同的機率測度下確實有些許的不同，且當q值越小，得到的買權價格及合理變異數值也會越小。而在不同的機率測度下對應的避險策略，透過模擬比較，亦可發現其避險效果的確優delta避險。

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