# Difference in Value between General American & European Options

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.

• Hint: you only exercise early if you are in the money, otherwise you wait and see if the price brings you into the money. – MD-Tech Dec 9 '16 at 16:03
• – MD-Tech Dec 9 '16 at 16:05
• Thank you for your contribution @MD_Tech. I have read your link and think I have a slightly better understanding now. Am I right in saying, then, that because I can exercise on any given day if the American option is in the money, or else wait it out and get the same payoff as the European option at the end, that the American option is worth more? – user234 Dec 10 '16 at 8:58