控制變數技術(control variate technique)是利用兩個相似的選擇權，一個無封閉解之選擇權當作被評價之選擇權，而另一個具有封閉解之選擇權來當控制變數，並利用此兩個選擇權在相同的數值方法下估計誤差會有具相關性，以增加估計無封閉解選擇權的效率。本論文主要是研究利用控制變數技術在蒙地卡羅模擬法評價無封閉解選擇權時，如何選擇最適的控制變數並檢驗蒙地卡羅模擬法是否可分辨出控制變數的優劣。 選擇好的控制變數可依循兩個原則:首先是找尋一選擇權與欲評價之選擇權滿足相同之偏微分方程式;其次是讓控制變數選擇權的邊界條件與欲評價之選擇權越相似越好。 文中將檢視不同型態之選擇權當作欲評價之選擇權包括:美式選擇權,障礙式選擇權,亞式選擇權,價差式選擇權。結果顯示在每一個例子中,選擇較佳的控制變數可以使控制變數法更加強蒙地卡羅模擬法的估計效率,此外也驗證出蒙地卡羅模擬法可以正確的區分出控制變數的優劣。 For many complex options, analytical solutions are not available. In these cases a Monte Carlo simulation is an important numerical method. In its basic form, however, the Monte Carlo simulation is computationally inefficient, the control variate technique can be used to improve the efficiency of a Monte Carlo simulation. This paper presents a principle for finding better control variates when considering an option. A good control variate has to satisfy two conditions: The first is that a good control variate satisfies the same PDE satisfied by the target option. The second is that the boundary condition for the control variate is similar to the boundary condition for the target option. Options under consideration in this paper include American put options, barrier options, Asian options, and spread options. The result shows that a good control variate can improve the efficiency of the simulation dramatically and a good control variate can be differentiated from a bad control variate in a Monte Carlo simulation.
