Option Pricing Based on the Alternating Direction Implicit Finite Difference Method

Abstract

本文建立在方向互換有限差分法則之下的斷續性選擇權評價模型。本文提供單一資產之歐式及雙資產美式極大或極小選擇權之方向互換有限差分選擇權評價模型。我們延伸Stulz（1982）對於極大或極小選擇權的評價，相對於其所探討的問題是為歐式且不考慮股利發放，本文探究美式問題，並於模型中允許股利發放的考量。本文所建立之模型可應用於具多重標的之選擇權的評價。我們並對文章中所建立之斷續性選擇權評價模型作出數值分析，提供理論證明本文所建立之方向互換有限差分選擇權評價模型於數值求解上具無條件穩定性。

In this paper, we establish the way in which the alternating direction implicit (ADI) finite different method can be applied to option pricing problems. We develop ADI schemes for both European call option values written on a single underlying asset and American call option values on the maximum or minimum of two underlying assets. While Stulz (1982) assumes no dividend streams for the underlying assets, here we extend his model to American-type and allows for continuous dividend yields for each underlying asset. We address the problem of option pricing on multiple underlying assets, and provide theoretical justifications for the numerical stability of the ADI schemes that we develops in this paper. Our ADI schemes are shown to be unconditionally stable.