An explicit series approximation to the optimal exercise boundary of American put options

This paper derives an explicit series approximation solution for the optimal exercise boundary of an American put option by means of a new analytical method for strongly nonlinear problems, namely the homotopy analysis method (HAM). The Black–Sholes equation subject to the moving boundary conditions for an American put option is transferred into an infinite number of linear sub-problems in a fixed domain through the deformation equations. Different from perturbation/asymptotic approximations, the HAM approximation can be applicable for options with much longer expiry. Accuracy tests are made in comparison with numerical solutions. It is found that the current approximation is as accurate as many numerical methods. Considering its explicit form of expression, it can bring great convenience to the market practitioners.