使用Proxy降低樹狀評價模型的評價誤差

Abstract

許多衍生性金融商品因其合約規格或標的物價格隨機過程過於複雜，無法使用封閉公式解評價，而必須使用數值方法計算該選擇權的價格。其中樹狀結構是一個常見的數值評價方法。該方法將評價日到到期日切割成許多期n，並模擬每一期的標的物價格，而評價結果隨期數增加而收斂到理論值。 然而切割的期數增加會提高評價程式計算所需耗費的時間，因此本論文希望針對樹狀結構評價方法提出一種提高評價收斂速度的改良方法，本文稱之為Proxy方法。該方法將欲評價的選擇權價格拆成兩個部分，第一部分的值可用公式求解而且其量值近似選擇權的價格，本文稱之為Proxy part，第二部分為選擇權價格扣除proxy之後殘餘的部分，本文稱之為 Residual part，該部分則用樹狀結構求值。此外本演算法會配合數學優化降低每次的運算量，使得本演算法能夠較一般樹狀結構演算法有效率並準確。

Many financial derivatives cannot be priced with analytical formulas due to complicated contract provisions or complex stochastic process for modeling the underlying asset. These derivatives must be priced by numerical methods. The tree model is one of the popular numerical models. This model divides the time span from evaluation date to maturity date into n equal time steps and simulates the price of the underlying asset at each time step. The pricing results converge to the theoretical value with the increment of n. However, the increment of n would also increase the running time. This thesis proposes a method, called the proxy method, to improve the convergence speed. This method divides the derivative price into two parts: the proxy part and the residual part. The proxy part approximates the derivative’s price and can be analytically priced by the closed form. The residual part is defined as the difference between the derivative’s value and the proxy part, and it can be evaluated by the tree method. Besides, this thesis uses some mathematical optimizations algorithm to improve the efficiency of the proxy method. Experimental results show that the proxy method is more efficient and accurate than traditional tree methods.