# Can American options with no dividends and zero risk-free rate be treated as European?

Let's say you've got American options on a future of a stock index. There are no dividends, and no risk-free rate either (assume $r=0$). Can these options then be treated as European from the perspective of using Black-Scholes to price them and calculate the Greeks?

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All this assumes the absence of arbitrage:

As you probably know without dividends it's is never optimal to early exercise a call option on a non dividend paying stock because then the time value is lost, if $r$ is non-negative.

Early exercise of an American put option can be optimal if the option is sufficiently far in the money and $r > 0$. Then you can earn interest on the money gained.

So yes, see also Hull, 7nd edition, chapter 9 'Properties of Stock Options', section 9.6 in particular.

In the rare case that $r < 0$ it can be call options can be optimal to exercise call options. However $r$ rarely if ever goes sufficiently negative on its own.

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Technically, call options can be optimal to exercise early if $r<0$. $r$ rarely if ever goes sufficiently negative on its own, though. –  Brian B May 7 '12 at 12:31
Great comment! I added it to the answer since I feel an answer should stand on its own. –  Bob Jansen May 7 '12 at 18:37