以隱含波動樹評價選擇權之另一方法

Abstract

本文提出了一個固定機率-隨機波動度的隱含波動二元樹的建構方法。此方法改善了先前其他學者曾提出方法的缺點。相較於Derman-Kani 隱含波動二元樹與Li 隱含波動二元樹，以此方法建構隱含波動樹時，具有相當的穩定性。在Derman-Kani 隱含波動二元樹中有不良機率的問題，亦即在二元樹建構的同時，會出現機率大於1 或小於0 的狀況；在Li 隱含波動二元樹中，雖改良了不良機率發生的情形，但當隱含波動微笑曲線陡峭時，在建構樹的過程中，股價仍會發生違反無套利原則的狀況。然而，本文所提出的新方法，不僅改善了上述二者的缺點，在二元樹的建構概念上相當的簡單易懂，選擇權評價的結果也相當穩定。

This thesis proposes a constant probability-stochastic volatility implied binomial tree. Our method improves upon some weaknesses of previous works. Compared with the Derman-Kani tree (1994) and the Li tree (2000), our method is considerably more stable. In our method, neither the nvalid transition probability problem occurs, like in the Derman-Kani tree, nor the results of option pricing diverge when the slope of volatility with respect to the strike price is steep, as in the Li tree. Incorporating the known local volatility function, our method constructs the implied binomial tree directly by forward induction. The option value is calculated from the stock prices in the terminal nodes of the tree backward. As a whole, for the proposed constant probability-stochastic volatility implied binomial tree, its construction is direct, and its implementation is straightforward.