Unless things have changed, Black-Scholes-Merton does a pretty good job on American options as well.

To remind the reader: a European option can only be exercised at expiration, whereas an American option can be exercised at any time; the intrinsic value of an option is the amount you’d make if you exercised the option right now, and the time value is the option value minus the intrinsic value — the option premium, in other words. Finally, the cost of carry is how much it costs to finance your position, or conversely, the interest rate you’d get on holding cash.

The only reason to exercise an American option early is if the cost of carrying the intrinsic value is greater than the time value — in other words, if you’d make more money by exercising the option today and keeping the cash than delta-hedging to capture the time value.

Given today’s low interest rates and thus low cost of carry, this means that the only reason to execute an American option is if it’s deep in the money — which only happens a pretty small fraction of the time, particularly since traders tend to unwind their contracts pretty fast when options get into the money (as most of the play is in options that are at the money, near their strike price).

This means that you can use a European option model like BSM to value American options and then correct for that one case — and for many options, that “correction” is effectively 0.