回顧式選擇權評價-數值方法與程式模擬

Abstract

本論文提出一套直覺且有效率的數值方法來評價回顧式選擇權，特別是針對美式的回顧選擇權，我們先驗證與實作Longstaff和Schwartz [2001]所提出的蒙地卡羅最小平方法(The Least-Squares Monte Carlo Approach)在評價一般美式選擇權的演算法，並延伸應用此評價方法，依回顧選擇權的特性修改最小平方法的模型及參數設定。 由於回顧式選擇權具有路徑相依的性質，所以使用蒙地卡羅法來評價是最直接的方式，然而傳統蒙地卡羅法存在難以判斷美式選擇權提前履約與否的缺陷，所以我們利用LSM方法以價內的路徑資料做最小平方法的迴歸方程用以逼近選擇權持續持有的期望函數，藉此就可以在每一個可執行時間點與提前執行的價值做比較以得到最佳化的執行策略以逼近美式選擇權的價值。 我們以C++程式語言實作評價模型並修改參數設定以求取最適合評價回顧式選擇權的迴圈數、樣本產生頻率、及基底函數與項數之選擇。並驗證標的資產價格的波動度、存續期間長度與可執行履約天數對於美式回顧選擇權價格的影響，最後與Conze and Viswanathan [1991]推導的歐式回顧選擇權之封閉解做比較並推論出回顧式選擇權之提前履約價值比例上略低於一般美式選擇權的可能原因，再附予實證資料證明我們的數值方法之實用性與精確度。

This thesis concentrates on the pricing problem of lookback options whose values depend on the extreme price of the underlying asset. We provided a straightforward numerical method which extended from the least-square Monte Carlo simulation (LSM) approach derived by Longstaff and Schwartz [2001] to overcome the difficulty of pricing problem result from the path-dependent property and achieve the optimal early exercising strategic for an American lookback option. With Lagendre polynomials for regressing to approximate the expectation value of continuation, we can easily estimate the holding value of the option and decide whether to early exercise or not at any exercisable date before the expiration. By setting the optimal parameter, our pricing algorithm can accurately produce the approximation value of a lookback option with considering the computational efficiency. According to the simulation result, we also verify the impact of several factors of the option, which likes the volatility, the exercisable times, and the duration, on the value of the lookback option and the numerical result is in accordance with our expectation. Taking the closed-form solution of European lookback options derived by Conze and Viswanathan [1991], we also find out the difference between the early-exercise premium of lookback options and ordinary options.