本篇論文包含兩大研究主題，第一部份是單資產雙障礙選擇權評價，第二部份是雙資產雙障礙選擇權評價。由於評價障礙選擇權是Black-Scholes偏微分方程式（partial differential equation，PDE）的邊界值問題，本研究採用最有效的PDE方法來探討雙障礙選擇權評價問題。

因Black-Scholes PDE可經由一系列變數轉換成線性齊次熱傳導方程式（linear homogeneous heat equation），故使用解決線性熱傳導方程式最有效的邊界元素法（boundary element method，BEM）來處理單資產障礙選擇權評價問題。再者，由於評價雙資產障礙選擇權為2-D邊界值問題，計算較為複雜，故引用Chen and Chen（1988a,b）提出之hybrid method，可有效降低評價維度，提昇計算效率。其後，我們舉一些數值例來探討評價方法的效率與準確性，並與其他評價方法做比較。數值結果顯示我們的方法可穩定且精確的處理障礙選擇權評價問題。

This thesis has two main parts. The first focus on the valuation of one-asset double barrier options, and the other discusses the pricing of two-asset double barrier options. Since the valuation of barrier options is a boundary value problem of the Black-Scholes partial differential equation (PDE), we use efficient PDE approaches to price double barrier options in this thesis.

The Black-Scholes PDE can be transformed to a linear homogeneous heat equation via a set of variable transformations. A boundary element method (BEM) can be used to solve the boundary value problem of the heat equation powerfully in one-dimensional case. Moreover, since the valuation of two-asset double barrier options is a 2-D boundary value problem of PDE, it is difficult to solve the problem using traditional methods. The usage of the hybrid method provided by Chen and Chen (1988a,b) can reduce the dimensions of the valuation problem to enhance the efficiency of calculation. Then we give some numerical examples to demonstrate the flexibility and accuracy of our methods and to compare with other pricing approaches. The numerical results reveal that our methods are useful to price double barrier options stably and precisely.

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