一個創新的減小誤差演算法：針對快速傅立葉變換選擇權定價法

Abstract

應用快速傅立葉變換來評價衍生性金融商品是一項廣為人知且重要的評價方法。其中一種普遍且被引用很多次的以快速傅立葉變換為基礎的歐式選擇權評價演算法是由Carr和Madan所提出的。 他們利用了簡潔的特徵方程式來轉換成選擇權價格。然而，Carr和Madan所提出演算法的收斂速度很慢；在評價深度價外選擇權的時候，也會產生負值。 本文提出了一種創新的方法來改善上述兩個Carr和Madan演算法的問題。我將選擇權價格分成一個逼近部分和一個剩餘部分。逼近部分用來逼近目標分配假設下的選擇權價格，並且可以被解析的求出來。 因此，逼近部分並不會產生數值誤差；這就是為什麼我的演算法產生較少的數值誤差。剩餘部分數值的估計選擇權的理論價格和逼近部分的差額。 數值結果顯示，我提出的演算法有效地減少了評價的數值誤差，且減輕了深度價外選擇權的負價格現象。

Applying the fast Fourier transform (FFT) for pricing derivatives is one of the popular and important evaluation methodologies. A general, and highly cited FFT-based approach proposed by Carr and Madan can efficiently price vanilla options given that the characteristic function of the underlying asset's return is analytically known. However, their pricing results converge slowly and even are negative for deep-out-of-the-money options. This thesis proposes a novel approach to address these problems. My approach decomposes the option value into the proxy and residual terms: The proxy term approximates the theoretical option value and can be analytically evaluated without generating numerical error; that is why my approach can generate less pricing error. The residual term numerically estimates the difference between the theoretical option value and the proxy term. Numerical experiments suggest that my superior approach efficiently reduces the pricing error and alleviates the negative price problem for evaluating deep-out-of-the-money options.