An Implicit Scheme for American Put Options

In this paper, an implicit scheme is proposed to solve a parabolic variational inequality arising from the American put options. The discretization leads to a class of discrete elliptic variational inequalities. Well-posedness, including existence, uniqueness, comparison principle, and stability of the discrete elliptic variational inequality is established. A simple and efficient algorithm to solve the implicit discretized variational inequality is discovered. The novelty here is an explicit formula for the optimal exercise boundary. An improved algorithm is also presented to eliminate the singularity near the time to expiry. Numerical examples are carried out to show the accuracy and efficiency of the proposed algorithms.