This paper investigates analytical solutions of American option pricing based on the non-sensitivity principle of homotopy.Specifically,American option price is decomposed into a sum of the European price,early exercise premium and probabilistic interpretation.By transforming topological space,non-linear PDE is converted into an infinite sum of many linear subproblems and an exact and explicit solution for Black Scholes American options on a dividend-paying stock is derived by homotopy perturbation method.Besides,discrete approximations to the continuous process underlying the Black-Scholes model provides opportunities to check for early exercise and the Greeks for American options.