A semigroup approach to American options

In this paper we give new conditions for the existence and uniqueness of the value of an American option by using semigroup theory. Some open questions given by Gozzi et al. and Kholodnyi are solved and improved, specifically, that the Black-Scholes operator is degenerate and that it is sectorial. Moreover, our results are extended to the original American problem. Finally, we obtain an integral equation which gives the value of an American option, providing an alternative method to calculate the value of American options.