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As we seen, European call and puts are highly liquid derivatives, so their pricesis dicided by the market and they can be used to calibrate some models.Question, how about American option? Are they always priced by quants? In other words, since they are not highly liquid, they do not have a equilibrium market price? If I am wrong and they do have marketprice, what is their market price useful?Thanks.

It depends on what kind of calls and puts you're talking about. If you're talking about equity options, then I dare you to find a stock with an actively traded European option chain! Most of the equity options are American style.In the fixed income world the situation is the opposite. European, say, swaptions are much more common than American swaptions. It all depends on the market.

American options on equity (stocks) are usually liquid of they are on stocks that have a high volume and/or are at the money with relatively medium-short expirations. I suggest you go to yahoo finance and type in several tickers and look at their option chains. Most small companies will barely have volume on their options of they have options. For medium sized companies it depends.

In addition to what has been said below, many liquid american options are exchange traded (e.g. ED and ER futures options) but they are treated like Europeans, since you get paid daily interest on your margins. No need to exercise them early.