An option is a financial contract whose value is based on some underlying stochastic process. After recalling the basic ingredients of the Black Scholes model, I will give a characterization of the value function of the Integral option, which is of American type. This means that the holder can decide when to exercise. Then I will move on to the Israeli Integral option. Israeli (or game) options give both the writer and the holder the right to exercise at any time. I will show that finding the value function of Israeli Integral option is equivalent to solving a specific free boundary value problem.