Does the Black-Scholes Model apply to American Style options?

Yes, your understanding is correct. Strictly speaking, the Black-Scholes model is used to price European options. However, the payoff (price) of European and American options are close enough and can be used as an approximation if no dividends are paid on the underlying, and liquidity cost is close to zero (e.g. in a very low-interest rate scenario).

As of now, there are no closed-form methods to price American options. At least none that I know of. You should rely on lattices for multi-period binomial pricing, which is mostly recursive.

Just a few observations within the Black-Scholes framework:

• American calls have the same price as European calls on non-dividend paying assets.
• The Black-Scholes formula is applicable only to European options (and, by the above, to American calls on non-dividend paying assets).
• By the call-put parity, if you have European call prices for some expiry dates and strikes, you also have the European put prices for those expiry dates and strikes.
• If you have European call prices for a given expiry date T for all strikes, you can easily compute the price of any "European" payoff for that expiry (for example, a digital call V = 1_{S>K}, or a parabola V = S^2, or whatever). Conceptually, you form butterfly spreads __/_ for a series of increasing strikes, and they give you the "risk-neutral" probability that you end up there, and then you just integrate over your payoff.

Next, you can now use the Black-Scholes framework (stock price is a Geometric Brownian Motion, no transaction costs, single interest rate, etc. etc.) and numerical methods (such as a PDE solver) to price American style options numerically, but not with a simple closed form formula (though there are closed-form approximations).

as no advantage from exerting American call option early,we can use Black schole formula to evaluate the option.However, American put option is more likely to be exercised early which mean Black schole does not apply for this style of option

The difference between an American and European option is that the American option can be exercised at any time, whereas the European option can be liquidated only on the settlement date. The American option is "continuous time" instrument, while the European option is a "point in time" instrument. Black Scholes applies to the latter, European, option. Under "certain" (but by no means all) circumstances, the two are close enough to be regarded as substitutes.

One of their disciples, Robert Merton, "tweaked" it to describe American options. There are debates about this, and other tweaks, years later.

Black-Scholes is "close enough" for American options since there aren't usually reasons to exercise early, so the ability to do so doesn't matter. Which is good since it's tough to model mathematically, I've read.

Early exercise would usually be caused by a weird mispricing for some technical / market-action reason where the theoretical option valuations are messed up. If you sell a call that's far in the money and don't get any time value (after the spread), for example, you probably sold the call to an arbitrageur who's just going to exercise it. But unusual stuff like this doesn't change the big picture much.

A minor tangent. One can claim the S&P has a mean return of say 10%, and standard deviation of say 14% or so, but when you run with that, you find that the actual returns aren't such a great fit to the standard bell curve. Market anomalies producing the "100-year flood" far more often than predicted over even a 20 year period. This just means that the model doesn't reflect reality at the tails, even if the +/- 2 standard deviations look pretty.

This goes for the Black-Sholes (I almost abbreviated it to initials, then thought better, I actually like the model) as well. The distinction between American and European is small enough that the precision of the model is wider than the difference of these two option styles. I believe if you look at the model and actual pricing, you can determine the volatility of a given stock by using prices around the strike price, but when you then model the well out of money options, you often find the market creating its own valuation.