Why are American-style options worth more than European-style options?
I realize I can exercise American-style options anytime before the expire, but I can only exercise European-style options during their "exercise period" (usually right when they expire, but no earlier).
So it makes sense that an American option is worth at least as much as a European option.
But why is it worth more? If I exercise my American option early, I may make more money than waiting until expiration, but I also may make less. Mathematically, there's no advantage, since I'm equally likely to make as much money by waiting for expiration.
Mathematically speaking, is there ever a good reason to exercise an American option early?
I realize people don't always behave logically, but even the formulas that valuate options show that American options are worth more.
EDIT (answering comments):
Many of the comments people made would apply to lookback options (http://en.wikipedia.org/wiki/Lookback_option), but not to American options.
American options can do everything European options can and more. I understand that, and that means American options can't be worth any LESS than European options, but I'm still not clear on how you would mathematically calculate this extra value.
@Aaronaught: "The difference between an American and European option is the difference between getting N chances to get it right (N being the number of days 'til expiration) and getting just one chance. It should be easy to see why you're more likely to profit with the former, even if you can't accurately predict price movement."
REPLY: I don't think you really get N chances. Once you exercise the option, that's it, no more chances. And, if you decide not to exercise the option today, and the underlying's price falls, you can't go back in time and exercise it yesterday.
- @jdsweet: Think of it this way, if you traveled back through time one month - with perfect knowledge of AAPL's stock price over that period - which happens to peak viciously then return to its old price at the end of the period - wouldn't you pay more for an American option?
REPLY: Well, no. I'd simply buy a cheaper European option that expires right at the spike. Or, I'd buy a European option that expires later, and sell it when the spike occurs. American options don't give you precognition, so you're still guessing as to when to sell/exercise.
- @jdsweet: "one of the more common reasons people exercise (as opposed to sell) an American option before expiration is if an unexpected dividend (larger than remaining time value of the option) was just announced that's going to be paid before the option contract expires"
REPLY: This makes sense. I'm actually trading FOREX options, so there is no dividend. In that scenario, are European and American options worth the same?