  The existence and uniqueness of probabilistic solutions of variational
inequalities for the general American options are proved under the hypothesis
of hypoellipticity of the infinitesimal generator of the underlying diffusion
process which represents the risky assets of the stock market with which the
option is created. The main tool is an extension of the It\^o formula which is
valid for the tempered distributions on $\R^d$ and for nondegenerate It\^o
processes in the sense of the Malliavin calculus.
