在二項式模型中的高階誤差分析

Analysis of Higher Order Error in the Binomial Model

黃冠閔(Kuan-Min Huang)

指導教授 ： 彭柏堅

國立臺灣大學/理學院/數學研究所/碩士(2011年)

https://doi.org/10.6342/NTU.2011.00849

摘要

本文主要在探討在二元樹模型中的歐式買權價格收斂到Black-Scholes(BS)模型中價格的速度，當每一分割期間的長度愈縮小。在在二元樹模型中，選擇權的價格是由股價的未來變動百分比u和d及風險中立機率(Risk-Neutral Probability)來決定。文獻一(Chang-Palmer)給出在誤差項中1/n的確切係數。在這篇論文中，我們考慮更一般化的u和d來證明我們的主要定理，應用主要定理加強文獻一所提出的結果，將誤差項提高項並給出高項的確切係數。我們也利用加強的結果在Joshi模型中來說明二元樹模型中的價格與BS模型中的價格兩者的誤差。我們也應用主要定理在Leisen-Reimer模型中得到一個收斂定理，在Tian模型中得到一個新定理。

關鍵字

歐式買權價格 ； 二元樹模型

並列摘要

In this paper, we study the rate of convergence of the European call option price by the binomial model to the Black-Scholes price as the number of period n tends to
infinity. The binomial option pricing is determined by the jump sizes u and d and the risk-neutral probability p. Chang and Palmer [1] gives an explicit formula for the coefficient of 1/n in the expansion of the error. This paper discusses the higher order in the expansion of the error. We consider more general u and d to prove
the Main Theorem and apply it to strengthen the Chang-Palmer result, expanding up to the higher term in the expansion of the error and also giving an explicit formula for the coefficient of the higher term. We use the strengthened Chang-Palmer result to prove the error between the binomial price and the Black-Scholes price in Joshi's model [4]. We also use the Main Theorem to obtain a proof of the convergence rate in Leiser-Reimer's model [5] and a new theorem in Tian's model [7].

並列關鍵字

binomial model ； option value ； Black-Scholes price ； digital call option

參考文獻

[1] Chang, L. and Palmer K.: Smooth convergence in the binomial model. Finance

[2] Cox, J., Ross, S.A., Rubinstein, M.: Option pricing: a simplified approach.

Journal of Financial Economics 7 (1979), 229-263

[4] Joshi, M.: Achieving higher order convergence for the prices of European options

in binomail trees. Mathematical Finance. 20 (2010), 89-103