Before revision:  Convergence of BTM and explicit difference schemes for the variational inequality model of American put options to viscosity solution is proved .
After revision:  Convergence of BTM and explicit difference schemes for the variational inequality model of American put options to viscosity solution the decreasing property of the price of American put options and increasing property of the optimal exercise boundary on time variable are proved. First, put options are considered. Then the linear homogeneity and call-put symmetry of the price functions in the BTM and the EDS for the variational inequality model of American options with time dependent coefficients are studied and using them call options are studied .