In this paper, we consider an option pricing problem in a pure jump model where the process X(t) models the logarithm of the stock price. By the Schauder fixed point theorem, we show the existence and uniqueness of the solutions in Hlder spaces for the European and American option pricing problems respectively. Due to the estimates of fractional heat kernel, we give the regularity of the value functions u(E)(t, x) and u(A)(t,x) of the European option and the American option respectively.