三元樹模型奇異點法評價美式的亞式選擇權

Abstract

亞式選擇權 (Asian option) 是路徑相依選擇權的一種，其報酬取決於選擇權於存續期間標的資產價格的平均值。大多數美式路徑相依選擇權 (path-dependent option) 都沒有封閉解 (closed-form)，導致評價的困難，所以實務上多以數值方法來近似選擇權的理論價格。在2010年，Gaudenzi, Zanette and Lepellere 提出一種基於二元樹模型所建構出的數值方法可以用來有效率地評價美式路徑相依選擇權，並稱之為二元樹奇異點法 (singular-points binomial method)。本篇論文針對美式的亞式選擇權做評價，並以樹狀模型為基礎，建立一個三元樹的價格模型，每一期除了股價的上漲與下跌外，較二元樹模型多考慮了股價持平的情形，並將奇異點應用在三元樹模型中，主要想法是用奇異點來描述三元樹中任意節點的價格平均及其報酬間的關係。此方法較過去評價美式的亞式選擇權的方法，在效率上來的佳，且在收斂行為的表現上也優於 Gaudenzi, Zanette and Lepellere (2010) 所提出的方法。

Asian option is a path-dependent option whose payoff is based on the arithmetic averaging of the underlying asset price over the life of the option. Most American path-dependent options do not admit of closed-form analytical formulas for their option values, or, if they do, the formulas are complex. In practice, numerical methods are used to approximate the option value. In 2010, Gaudenzi, Zanette and Lepellere proposed a new method called the singular-points binomial method to price American path-dependent options efficiently. It is based on the binomial tree. In this thesis, we focus on pricing American Asian options. We establish a trinomial tree that adds a flat move to the binomial tree in every time step. Moreover, we apply the singular-points method into trinomial tree. The idea is to use the singular points to depict the relation between the average price and payoff of any node in the tree. The method is more efficient than existing methods. The convergence behavior is also better than the method of Gaudenzi, Zanette and Lepellere (2010).