亞式選擇權的可析逼近

Abstract

本篇論文在Black – Scholes (1973) 的模型假設下，以解析方法估計浮動履約的亞式選擇權價格，提出兩個解析公式。主要參考文獻以二階泰勒展開式的方法逼近，並假設此隨機變數為常態分配或卡方分配。本篇論文第一個公式僅修正文獻中展開式的精確度，第二個公式則提出甚至不需要使用泰勒展開式。然而數值結果顯示，在本篇所假設的參數下，第二個公式估計最好。 另外本篇論文對部份文獻所提出的公式提出誤差上界，並以蒙地卡羅模擬直方圖觀察常態分配與卡方分配假設的合理性。最後，我們將本篇論文所需的繁雜計算過程留在附錄。

In this thesis we derive two analytic approximation formulas for floating strike Asian options and assume that assumptions underlying Black-Scholes (1973) model hold. In the literature, some researches use the second-order Taylor expansion to approximate the price of Asian options and they assume the Quadratic approximation is normally or chi-square distributed. In this thesis, we first derive the formula by improving the precision of the Taylor expansions. Next, we suggest that we even do not need to truncate the random variable via Taylor expansions, and the numerical results witness that our second formula is the most accurate approximation for selected values of the underlying parameters. Additionally, we derive the error upper bound for some formulas, and we also observe the rationality of all the assumptions by using the histogram Monte Carlo simulation. Finally, we leave all the troublesome calculations in Appendix.