This paper studies the valuation of American options for a general one-factor stochastic volatility model.  Using the local time-space calculus on surfaces we derive an early exercise premium representation for the option price, parametrized by the optimal exercise surface. The exercise surface is the unique solution to an integral equation of Volterra type. The paper proposes a new numerical scheme to solve the integral equation based on the Picard iterations method. The method is flexible and can handle a wide class of non-affine models. Performance is illustrated for the Black-Scholes and Heston models. The approach provides fast convergence, simple implementation and good runtime/RMSE tradeoff and can be extended for other multi-dimensional stopping problems.