以快速傅利葉轉換為基礎的選擇權定價演算法及改進

Abstract

本篇論文研究了一種日益受重視的定價演算法，可廣泛地用在各種衍生性商品的定價，且標的物的報酬過程不受傳統的布朗運動假設所限制，而是可假設其為更廣義的 Levy process。此演算法大量的應用了褶積，而褶積可很容易的被改寫為高階收斂，計算速度更快，結果更精準。此種演算法可約略分為向後推導法和向前推導法。本文報告了向後推導法的基本原理及其一些應用，向前推導法的部份，所有討論專注在亞式選擇權的定價上。本文並且提出一種加速既有亞式選擇權定價演算法的方法，數值結果顯示效果相當成功。

This thesis analyzes and improves FFT-based pricing algorithms that can be applied to price a wide class of derivatives under Levy processes. Convolutions and interpolations are widely used in these algorithms. Our paper constructs faster error convergence rate algorithms by improving error convergence rate of these convolutions and interpolations. Both backward and forward induction methods can be applied in our algorithms. Our paper derives the fundamental properties of FFT-based algorithms with backward induction method and the related applications, like the pricing algorithms for Bermudan options, barrier options and lookback options. We also price Asian options with FFT and forward induction method. Our paper also derives a control variates methods on (probability) distribution that can significantly improve the convergence rate for pricing Asian options.