Approximations for the values of american options

The solution of the American option valuation problem is the solution of a parabolic partial differential equation satisfying free boundary conditions. The free boundary represents the critical price, at which the option should be exercised. In this paper the free boundary is determined by an algebraic relation and an approximate solution derived. A suitable modification of the approximate solution gives the exact solution. The uniqueness of the free boundary implies the expression determined by the algebraic relation is the true critical price