在隨機波動率下之雙變數樹評價模型

Abstract

Hilliard和Schwartz（1996）提出一個雙變數二元樹模型，其模型可以在隨機波動率下評價選擇權，並允許股價與波動率有相關性。本論文旨在探討 Hilliard和 Schwartz（1996）的雙變數二元樹模型演算法之缺點，以及提供一個部分修正方案。論文針對在 Hilliard和Schwartz（1996）模型中會發生錯誤機率的股價二元樹，改用平均數追蹤法建構的三元樹，而波動率還是維持二元樹，稱此架構為雙變數2/3-元樹模型，最後用此模型去評價選擇權。

The bivariate binomial framework of Hilliard and Schwartz (1996) allows non-zero correlation between the stochastic volatility and the underlying process. It can also be used to value American options. This thesis points out the problems with their bivariate binomial model. It also provides a partial solution to deal with those problems. When pricing options with the Hilliard-Schwartz model, it is easy to demonstrate that incorrect probabilities will occur in some situations. We use the mean-tracking method to construct trinomial trees instead of binomial trees for one dimension. Nevertheless, the stochastic volatility dimension still adopts the binomial tree as Hilliard and Schwartz (1996). The framework will be called the bivariate bino-trinomial model, and it is used to evaluate options.