American options are classical financial derivative contracts which lead to free boundary problems. The objective of this article is to give some qualitative properties of the exercise region of American options by means of analytic techniques. We prove that the price of an American option is the unique viscosity solution of the obstacle problem. We also prove comparison principles and strict comparison principles. These results enable us to localize the exercise region and to prove the propagation of convexity for American options. As a result, we study the influence of the volatility parameter on the price of American options.