使用二元樹評價亞式一籃子選擇權

Abstract

亞式一籃子選擇權同時具備亞式選擇權跟一籃子選擇權的特性，故難以找到選擇權價格的封閉解。在這篇論文中，我們使用平移對數常態分配 (shifted lognormal)以及負平移對數常態分配(negative shifted lognormal)搭配動差擬合(moment matching)找出三個參數(shape, scale and shift)來近似一籃子資產的價格。之後，我們利用這三個參數觀察到的性質建構一個可以近似一籃子資產價值的二元樹。最後搭配Hull-White methodology找出美式跟歐式的亞式一籃子選擇權的價格。數值實驗的結果顯示我們的方法所找出來的歐式選擇權價格與蒙地卡羅方法找出來的價格十分接近，但是美式選擇權價格與最小平方蒙地卡羅法找出來的價格相比，我們的方法明顯地高估。

Asian basket option is hard to price. This thesis presents a new approach to price European-style and American-style Asian basket options. First, we use approximation and moment-matching techniques to find the random variable following the shifted lognormal distribution to approximate the basket value. Second, we use the random variable to build a binomial tree and combine it with the Hull-White methodology for pricing path-dependent options to price Asian basket options. Finally, we compare our numerical results with Monte Carlo simulation for European-style Asian basket options and with the least-squares Monte Carlo for American-style ones. They show that the European-style Asian basket option prices obtained by our approach are accurate and the American-style ones are overpriced by our approach.