中文摘要：

本文研究与永久美式经理股票期权（ESOs）有关的一个抛物型变分不等式的自由边界。该变分不等式是退化的，且障碍条件中含有未知函数的偏导数。采用切片法来逼近原问题，将偏微分方程的变分不等式转化为常微分方程的变分不等式，并且分析了该逼近问题解的误差估计及其收敛性。利用迭代法得到逼近问题的数值算法。在给定参数的条件下，对自由边界的性质进行了数值分析，并解释了其金融意义。

英文摘要：

This paper is concerned with the free boundary of a parabolic variational inequality related to the perpetual American executive stock options （ESOs）. It is a degenerate variational inequality and its obstacle condition contains the partial derivative of an unknown function. The microtomy method is used to approximate the problem, and the variational inequality of a partial differential equation can be transformed into that of an ordinary differential equation. The error estimates of the approximate solution and its convergence are analyzed. By using the iteration method, a numerical algorithm for the approximation problem is obtained. Under given parameters, numerical analysis of the properties of the free boundary and interpretation of its financial meaning are provided.