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Why are American-style options worth more than European-style options?
If you're into math, do this thought experiment: Consider the outcome X of a random walk process (a stock doesn't behave this way, but for understanding the question you asked, this is useful): On the first day, X=some integer X1. On each subsequent day, X goes up or down by 1 with probability 1/2. Let's think of buying a call option on X. A European option with a strike price of S that expires on day N, if held until that day and then exercised if profitable, would yield a value Y = min(X[N]-S, 0). This has an expected value E[Y] that you could actually calculate. (should be related to the binomial distribution, but my probability & statistics hat isn't working too well today) The market value V[k] of that option on day #k, where 1 < k < N, should be V[k] = E[Y]|X[k], which you can also actually calculate. On day #N, V[N] = Y. (the value is known) An American option, if held until day #k and then exercised if profitable, would yield a value Y[k] = min(X[k]-S, 0).  For the moment, forget about selling the option on the market. (so, the choices are either exercise it on some day #k, or letting it expire) Let's say it's day k=N-1.  If X[N-1] >= S+1 (in the money), then you have two choices: exercise today, or exercise tomorrow if profitable. The expected value is the same. (Both are equal to X[N-1]-S). So you might as well exercise it and make use of your money elsewhere. If X[N-1] <= S-1 (out of the money), the expected value is 0, whether you exercise today, when you know it's worthless, or if you wait until tomorrow, when the best case is if X[N-1]=S-1 and X[N] goes up to S, so the option is still worthless. But if X[N-1] = S (at the money), here's where it gets interesting. If you exercise today, it's worth 0. If wait until tomorrow, there's a 1/2 chance it's worth 0 (X[N]=S-1), and a 1/2 chance it's worth 1 (X[N]=S+1). Aha! So the expected value is 1/2. Therefore you should wait until tomorrow. Now let's say it's day k=N-2.  Similar situation, but more choices: If X[N-2] >= S+2, you can either sell it today, in which case you know the value = X[N-2]-S, or you can wait until tomorrow, when the expected value is also X[N-2]-S. Again, you might as well exercise it now. If X[N-2] <= S-2, you know the option is worthless. If X[N-2] = S-1, it's worth 0 today, whereas if you wait until tomorrow, it's either worth an expected value of 1/2 if it goes up (X[N-1]=S), or 0 if it goes down, for a net expected value of 1/4, so you should wait. If X[N-2] = S, it's worth 0 today, whereas tomorrow it's either worth an expected value of 1 if it goes up, or 0 if it goes down -> net expected value of 1/2, so you should wait. If X[N-2] = S+1, it's worth 1 today, whereas tomorrow it's either worth an expected value of 2 if it goes up, or 1/2 if it goes down (X[N-1]=S) -> net expected value of 1.25, so you should wait. If it's day k=N-3, and X[N-3] >= S+3 then E[Y] = X[N-3]-S and you should exercise it now; or if X[N-3] <= S-3 then E[Y]=0.  But if X[N-3] = S+2 then there's an expected value E[Y] of (3+1.25)/2 = 2.125 if you wait until tomorrow, vs. exercising it now with a value of 2; if X[N-3] = S+1 then E[Y] = (2+0.5)/2 = 1.25, vs. exercise value of 1; if X[N-3] = S then E[Y] = (1+0.5)/2 = 0.75 vs. exercise value of 0; if X[N-3] = S-1 then E[Y] = (0.5 + 0)/2 = 0.25, vs. exercise value of 0; if X[N-3] = S-2 then E[Y] = (0.25 + 0)/2 = 0.125, vs. exercise value of 0. (In all 5 cases, wait until tomorrow.) You can keep this up; the recursion formula is E[Y]|X[k]=S+d = {(E[Y]|X[k+1]=S+d+1)/2 + (E[Y]|X[k+1]=S+d-1) for N-k > d > -(N-k), when you should wait and see} or {0 for d <= -(N-k), when it doesn't matter and the option is worthless} or {d for d >= N-k, when you should exercise the option now}. The market value of the option on day #k should be the same as the expected value to someone who can either exercise it or wait. It should be possible to show that the expected value of an American option on X is greater than the expected value of a European option on X. The intuitive reason is that if the option is in the money by a large enough amount that it is not possible to be out of the money, the option should be exercised early  (or sold), something a European option doesn't allow, whereas if it is nearly at the money, the option should be held, whereas if it is out of the money by a large enough amount that it is not possible to be in the money, the option is definitely worthless. As far as real securities go, they're not random walks (or at least, the probabilities are time-varying and more complex), but there should be analogous situations. And if there's ever a high probability a stock will go down, it's time to exercise/sell an in-the-money American option, whereas you can't do that with a European option. edit: ...what do you know: the computation I gave above for the random walk isn't too different conceptually from the Binomial options pricing model.