二項樹的平滑收斂

Abstract

近年來衍生性金融商品發展相當迅速，亦發展出許多訂價商品的方法，例如二項樹、偏微分方程、平賭過程及蒙地卡羅法等等。其中二項樹模型是最簡單也廣泛的為人所用，但是其收斂並非平滑收斂，有許多學者為改善收斂行為，提出控制二項樹傾斜的參數，其中Flexible model, WAND model, Joshi model, center binomial model有著重要的地位。 在本篇文章中，我們利用彭栢堅教授以及張洛賓先生所提出的重要定理證明在WAND model中其作者發現但並未詳細說明的收斂行為，並且利用一些性質提出改善的模型以減少所需的計算量，最後我們觀察數值結果並與上樹的模型作比較。

The products of derivative develop rapidly in recent years. There are many methods to price derivatives including using binomial tree, partial differential equations, martingale methods, and Monte Carlo simulation, etc. In these methods, binomial tree model is the simplest method that is used widely. The binomial model of Cox, Ross, and Rubinstein, CRR model, is well known. But CRR model converge to correct option price oscillatory and non-monotonic. Some models use a 'tilt' parameter that alters the shape and span of the binomial tree to improve the behavior of convergence. In these models, Tian's flexible model, Widdicks, Andricopoulos, Newton, and Duck's WAND model, Joshi's model, and Chang and Palmer's center binomial model are significant. In this article, we use the main theorem of Chang-Palmer to prove the convergence rate that is not unspecitied in their paper of WAND model, and we use some relation to estimate the implied n of WAND model to save the computation of using Newton-Raphson iteration. Finally, we compared with the numerical results of these models.