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I've noticed that in the literature, whenever European vanilla options are to be priced, the classical approach is to price a European call.

I guess it doesn't matter because we have put-call parity.

However, almost every time I see somebody mention the pricing of American options, the standard approach is to consider an American PUT. We don't have a put-call parity, so considering American Call options seems equally relevant, and yet this doesn't happen.

Is there a reason for this?

EXAMPLE: Shiryev and Peskir's monograph on optimal stopping problems consider pricing American puts and perpetual puts... they don't even mention a call option.

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That’s because in the case of a non dividend paying asset (the usual studied case), an American call is worth the same as a European call. Conversely for a non dividend paying asset the American put is different from the European put, so the American put needs special methods.

Nonetheless, once you study a pricing algorithm on an American put such as a tree, finite difference or a Monte Carlo method, you can generalise it to the case of an American call on a dividend paying asset

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