各期取樣點數量自適應的Willow Tree Models

Abstract

Willow tree是一個很有效率的定價模型，它有效率的原因在於每期都採用固定數量的點來近似布朗運動(Brownian motion)。不同於二項期權定價模型(binomial option pricing model)或者三項期權定價模型(trinomial option pricing model)之類的方法，willow tree隨切期數增加，點增加的速度是線性的，而二項及三項期權定價模型之類的做法點的數量增加的速度是平方成長，效率上來講willow tree 會有一定的優勢。然而，價格的布朗運動，會使價格隨著時間的增加，分布範圍更廣。 所以，如果要維持同樣的精準度，取樣的點的數量也要增加才行。而在原本Curran (1998)提出來的willow tree中，每個時間點使用的點的數量都是固定的，其中勢必是可以刪去一些點而不影響結果。這篇論文會提供三個方法自適應(adaptive)決定每個時間點應該使用的點的數量，減少不影響精準度的點。

Willow tree is an efficient option pricing model presented by Curran (1998). Willow tree use constant nodes for each time step to calculate the price and has two main advantage: linear time complexity and linear space complexity. But there is a flaw in Curran’s willow tree: the time’s growth make the price distribute wider, but Curran’s willow tree use constant nodes for each time, that cause the calculation error grow with time’s growth. This thesis present a concept that we can calculate the error at each time for specific number of nodes by using the error function we designed. And adaptively define the number of nodes at each time to control the error. We can trade the increasing of preprocessing time for the decreasing of primary processing time.