Characterization of the American put option using convexity

Understanding the behaviour of the American put option is one of the classic problems in mathematical finance. Considerable efforts have been made to understand the asymptotic expansion of the optimal early exercise boundary for small time near expiry. Here we focus on the large-time expansion of the boundary. Based on a recent development of the convexity property, we are able to establish two integral identities pertaining to the boundary, from which the upper bound of its large-time expansion is derived. The bound includes parameter dependence in the exponential decay to its limiting value. In addition, these time explicit identities provide very efficient numerical approximations to the true solution to the problem.