[Rotating Rabbit]

You're mistaken.

Sure, an american option cannot be expressed analytically (thats what discrete approximation models are for) but an american call is the same as a european call under geometric brownian motion, the assumption for BS. Its never preferable to exercise an american call before its expiry.

e.g. S(t) = price of security at time t. If an american call (Price C,Strike K,Expiry T) is in the money, at time t1 say, t1<T, you can exercise and realise a time-T gain of:

[S(t1)-K]e^r(T-t1)

If instead you sell short at T1, stick the funds S(t1) into a bank paying continuously compounded interest at rate r (as above), then at time T buy the stock at the minimum of K and S(T) with the bank money you have a time-T gain of:

S(t1)e^(T-t1) - min {K, S(T)}

which is greater. So its never preferable to exercise early, thus its the same expected value as its european sibling, under Black-Scholes and its assumptions.