Abstract
This article discusses some features of calculating the option price V(t,x), stock price x(t) and the optimal stop (execution) moment \tau (=t) on finite and infinite time intervals. Then we consider the problem of finding a rational price for American-type options for the optimal stopping moment on diffusion stock (B, S) markets. Then we consider the problem of finding a rational price for European-type options. First the option is considered from the buyer's point of view - the buyer's option. Then the seller's option is considered. All problems under consideration are solved exactly if the optimal stopping moment is found in advance, or numerically by the sweep method and finite element methods, by reducing them to the Stefan's problem regarding Y*(t,x) is a rational option value, \tauT* is rational execution and x*(t) is a rational stock price.