隨機波動度 Heston 模型下之效率樹狀模型

Abstract

Heston 模型是最廣為人知的隨機波動度模型。然而 Heston 模型的 樹狀評價方法卻相對稀少。本研究使用 Nawalka-Beliaeva 樹狀模型離 散化 Heston 模型中的變異數隨機過程。我們正交化股價隨機過程與變 異數隨機過程後,建立一個五元樹與一個六元樹來評價選擇權的價格。 用我們提出的樹狀模型計算歐式選擇權的價格,並將之與其他數值方 法做比較,結果顯示我們的六元樹可以精確且有效率的算出選擇權的 價格。除此之外,我們用六元樹計算在 Heston 模型下,歐式選擇權的 風險值 (value-at-risk),並與在 Black-Scholes 模型下所得之值做比較。 比較結果發現兩者的風險值有相當程度的差異,此差異可以作為交易 時的一些參考。

Heston’s model ranks among the most popular stochastic-volatility mod- els. However, trees for the Heston model are few. In this thesis, we use the Nawalkha-Beliaeva tree to discretize the variance process of the Heston model. After decorrelating the stock price process and the variance process, a pentanomial and a hexanomial trees are built. Numerical results for European options are presented and analyzed. Comparisons are made with competing numerical methods. Our hexanomial tree is found to be both accurate and ef- ficient. The value-at-risk numbers calculated by our tree for the Heston model are compared with those under the Black-Scholes model. The results show that they are significantly different, which suggests trading opportunities.