Generally, you back out local vols (as a function of S, t) of the BS vols (as a function of K, T) by a process described first by Dupire, and then you price Ame

Generally, you back out local vols (as a function of S, t) of the BS vols (as a function of K, T) by a process described first by Dupire, and then you price American options (and other products that are not sensitive to vol of vol) with that using a numerical PDE solver. https://en.wikipedia.org/wiki/Local_volatility https://en.wikipedia.org/wiki/Local_volatility