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Binomial tree methods (BTM) and explicit difference schemes (EDS) for thevariational inequality model of American options with time dependentcoefficients are studied. When volatility is time dependent, it is notreasonable to assume that the dynamics of the underlying asset's price forms abinomial tree if a partition of time interval with equal parts is used. A timeinterval partition method that allows binomial tree dynamics of the underlyingasset's price is provided. Conditions under which the prices of American optionby BTM and EDS have the monotonic property on time variable are found. Usingconvergence of EDS for variational inequality model of American options toviscosity solution the decreasing property of the price of American put optionsand increasing property of the optimal exercise boundary on time variable areproved. First, put options are considered. Then the linear homogeneity andcall-put symmetry of the price functions in the BTM and the EDS for thevariational inequality model of American options with time dependentcoefficients are studied and using them call options are studied.

The Binomial Tree Method and Explicit Difference Schemes for American Options with Time Dependent Coefficients