I am trying to get a grip on the difference in value between American and European options. I have read up on and understand that the value of an American call option is equal to that of a European call, and the value of an American put is greater than that of a European put.
My problem is with determining whether the value of an American option is equal to that of a European option in general.
Take, as a simple example, an American option with payoff E - S(t) and compare this to the corresponding European option with maturity date T. My intuition tells me that the American option is more valuable, but I can't seem to justify this rigorously, because when I exercise at t < T, I don't know what will happen at T, or if E - S(t) > E - S(T). I also understand that if I exercise early I can earn interest on my profit but again am struggling to formulate a rigorous argument to this effect.
Any explanation/guidance would be greatly appreciated.