障礙選擇權自1967年即在市場上被廣泛交易之新奇選擇權。部份障礙選擇權為障礙選擇權之延伸，其差別之處在於部份障礙選擇權的觀察期間只涵蓋選擇權存續期間之一部份，故被強迫履約之風險較低，價格較障礙選擇權高。由於部份障礙選擇權之觀察期間時點可以變動，交易者可針對某特定期間來做避險，鎖定標的資產價格於特定期間之波動，為一具有彈性之投資避險工具。

　　Heynen and Kat (1994, 1996)、Carr (1995)與Armstrong (2001)提出部份障礙選擇權之封閉解，但當考慮變動的償金、多資產或障礙變動時等因素，將很難找出該選擇權之封閉解。Boyle and Lau (1994)、Ritchken (1995)與Hull (2002)之樹狀評價模型與Figlewski and Gao (1999)之Adaptive Mesh Model雖可解決評價障礙選擇權barrier too close之問題。但部份障礙選擇權障礙價格之觀察點為一段期間，故以樹狀評價模型或AMM來評價部份障礙選擇權時，模型可能無法求解或產生計算過程繁複等問題。

　　部份障礙選擇權之評價乃為初始值問題與邊界值問題，故本研究擬以求解帶有初始條件與邊界條件之偏微分方程式予以評價部份障礙選擇權，只要偏微分方程式存在一格林函數為其基本解，即可透過回溯積分法求解此偏微分方程式，此法不僅節省運算時間並可得到較精確之數值解。

　Barrier options are the most popular exotic options and have been extensively traded in markets since 1967. Partial barrier options are the extension of barrier options, and the major difference is that partial barrier options assume the barrier just prevails for some fraction of the option’s life time. Hence, partial barrier options are more expensive than standard barrier options due to the lower risk to be knocked, and the adaptable monitoring period that provides traders the full flexibility to lock volatility risks during a specific time period.

　Cox, Ross, and Rubinstein’s binomial method (1979) can be used to price the barrier options. However, the convergence of the lattice models is slow, and the results tend to have a large bias when the asset price is close to the barrier. Boyle and Lau (1994), Ritchken (1995), and Hull (2002) propose a modified technique to place the true barrier at node point, thus, an accurate result can be obtained. Figlewski and Gao’s Adaptive Mesh Model (1999) provides a highly flexible approach that greatly increases the efficiency in lattice models. However, the partial barrier options are monitored only at some specific time period, thus, it creates additional difficulty for modified lattice models to price partial barrier options.

　The closed-form solution exists for only a small subset of all exotic options. Although Heynen and Kat (1994, 1996), Carr (1995) and Armstrong (2001) derive the closed-form solutions for the price of various types of barrier options, it will be more difficult to find a closed-form solution when more complex features, such as varying rebates, a changing barrier, double barriers, or multiple assets, is incorporated into the barrier option specifications.

　Since the original Balck-Scholes PDE can be transformed into a linear homogeneous equation, the integral method has been proven proved to be a highly efficient method for finding a precise numerical solution for the PDE. The valuation of partial barrier options, regardless of varying rebates, changing barrier, or double barriers, is still a combination of initial value problem and boundary value problem from the aspect of the PDE approach. The integral method introduced in this project will be a highly efficient method to find a generally accurate numerical solution for partial barrier options.

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