The valuation of American options is an optimal stopping time problem which typically leads to a free boundary problem. We introduce here the randomization of the exercisability of the option. This method considerably simplifies the problematic by transforming the free boundary problem into an evolution equation. This evolution equation can be transformed in a way that decomposes the value of the randomized option into a European option and the present value of continuously paid benefits. This yields a new binomial approximation for American options. We prove that the method is accurate and numerical results illustrate that it is computationally efficient.