  The aim of this study was to develop methods for evaluating the
American-style option prices when the volatility of the underlying asset is
described by a stochastic process. As part of this problem were developed
techniques for modeling the early exercise surface of the American option.
These methods of present work are compared to the complexity of modeling and
computation speed. The paper presents the semi-analytic expression for the
price of American options with stochastic volatility. The results of numerical
computations and their calibration are also presented. The obtained results
were compared with results excluding the effect of volatility smile.
