自适应隐式有限差分方法求解美式期权定价问题  

ADAPTIVE IMPLICIT FINITE DIFFERENCE FOR AMERICAN OPTIONS

作　　者：解雯佳 黄忠亿[1] Xie Wenjia;Huang Zhongyi(Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China)

机构地区：[1]清华大学数学科学系,北京100084

出　　处：《计算数学》2023年第3期284-298,共15页Mathematica Numerica Sinica

基　　金：国家自然科学基金(12025104,11871298)资助。

摘　　要：本文针对美式期权的定价问题设计了基于有限差分方法的预估-校正数值算法.该算法采用显式离散格式先对自由边界条件进行预估,再对经过变量替换后的关于期权价格的偏微分方程采用隐式格式离散,并用Fourier方法分析了此离散格式的稳定性.接下来,引入基于Richardson外推法的后验误差指示子.这个后验误差指示子能够在给定的误差阈值范围内,针对期权价格和自由边界找到合适的网格划分.最后,通过设计多组数值实验并与Fazio[1]采用显式离散格式算得的数值结果相比较,验证了所提算法的有效性,稳定性和收敛性.In this paper,we design a numerical prediction-correction algorithm based on the finite difference method for the pricing of American options.The algorithm first predicts the free boundary condition using an explicit discretization formula,then discretizes the partial differential equation for the option price after variable substitution in an implicit formula.And we analyze the stability of this discretization formula using the Fourier method.Next,a posteriori error estimator based on the Richardson extrapolation method is introduced.This error estimator allows us to find a suitable grid where the computed solution,both the option price variable and the free boundary,verify a prefixed error tolerance.Finally,the validity,stability and convergence of the proposed algorithm are verified by designing several sets of numerical experiments and comparing them with the numerical results obtained by Fazio[1]using an explicit discrete formula.

关 键 词：美式期权 自由边界问题 有限差分方法 RICHARDSON外推法 后验误差估计子 

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