  We study some properties of the American option price in the stochastic
volatility Heston model. We first prove that, if the payoff function is convex
and satisfies some regularity assumptions, then the option value function is
increasing with respect to the volatility variable. Then, we focus on the
standard put option and we extend to the Heston model some well known results
in the Black and Scholes world, most by using probabilistic techniques. In
particular, we study the exercise boundary, we prove the strict convexity of
the value function in the continuation region, we extend to this model the
early exercise premium formula and we prove a weak form of the smooth fit
property.
