萊維過程方差伽瑪選擇權定價模型以台指選為例

Abstract

本論文旨在探索萊維過程及其在真實世界中的應用，重點著重於金融領域。論文的第一部分深入探討了萊維過程的基本數學理論，包括萊維-伊托分解。第二部分涵蓋了對於半鞅的隨機積分定義，以及伊托公式。第三部分介紹了基本選擇權定價定理，使數學模型貼近實際應用，探討了隨機積分在金融領域中的應用。最後，第四部分呈現了前幾節所討論理論的實作，具體研究了幾何布朗運動和幾何變異伽瑪過程下歐式購買期權的定價公式。為了估算公式中的參數，分析了台灣加權股價指數期權的價格資料。

The aim of this thesis is to explore Levy processes and their applications in the real world, with a focus on the field of finance. The first part of the thesis delves into the fundamental mathematical theory behind Levy processes, including the Levy-Ito decomposition. The second part contains the definition of stochastic integral with respect to semi-martingales, and the Ito Formula. The third part introduces the fundamental pricing theorem to bridge the gap between mathematical models and real-world applications and the application of stochastic integrals in finance. Finally, the fourth part presents an implementation of the theory discussed in the earlier sections, specifically investigating the pricing formula of European call option for Geometric Brownian motion and Geometric Variance Gamma Process. To estimate the parameters in the formula, the prices of options on TAIEX were analyzed.