In contrast to the holder of a European option, the holder of an American option is allowed to exercise his right to buy (or sell) the underlying asset at any time before or at the expiry date. This special feature of American-style options – and more generally of American claims – makes the arbitrage pricing of American options much more involved than the valuation of standard European claims. We know already that arbitrage valuation of American claims is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, under risk-neutral probability). Thus, the uniqueness of the arbitrage price of an American claim holds.
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