障碍期权定价问题的有限元解的收敛性  

Convergence of Finite Element Methods for Barrier Option Pricing

作　　者：武亚男[1] WU Ya-nan(Department of Mathematics and Artificial Intelligence,Lyuliang University,Lvliang 033000,China)

机构地区：[1]吕梁学院数学与人工智能系,山西吕梁033000

出　　处：《数学的实践与认识》2024年第7期237-246,共10页Mathematics in Practice and Theory

摘　　要：障碍期权是与路径有关的期权,它的定价比较复杂。对于期权定价最常用的数值方法有二叉树方法,有限差分方法,有限元方法.文章以欧式下降敲出看涨期权为例,给出了障碍期权的有限差分格式,讨论了这三种数值方法的相互关系,并且利用这些关系给出了随着时间步数趋于无穷大,欧式看涨期权的有限元解收敛于Black-Scholes解,且收敛速度是1/√n,在误差展开式中给出了1/√n和1/n的系数表达公式此外,给出了数值算例,用以验证文章的结论.The barrier option is related to paths,and its pricing is more complicated.To option pricing,the most commonly numerical methods are binomial tree method,finite difference method and finite element method.In this paper,we take the European down-and-out call option as an example.In particular,we show the explicit finite difference form,and discuss the interrelationship of these three numerical methods.Furthermore,we show that,as the number of periods n tends to infinity,in the case of the European barrier call option,the finite element price converges to the Black-Scholes price at the rate of 1/√n.At the same time,we give formulas for the coefficients of 1/√n and 1/n in the expansion of the error.In addition,one numerical example is provided to validate the theoretical results.

关 键 词：障碍期权定价 有限元方法 二叉树方法 有限差分法 BLACK-SCHOLES模型 

分 类 号：F830.9[经济管理—金融学] O241.82[理学—计算数学]