  We give an analytical characterization of the price function of an American
option in Heston-type models. Our approach is based on variational inequalities
and extends recent results of Daskalopoulos and Feehan (2011). We study the
existence and uniqueness of a weak solution of the associated degenerate
parabolic obstacle problem. Then, we use suitable estimates on the joint
distribution of the log-price process and the volatility process in order to
characterize the analytical weak solution as the solution to the optimal
stopping problem. We also rely on semi-group techniques and on the affine
property of the model.
