Consider an American option that pays G(X^*_t) when exercised at time t,
where G is a positive increasing function, X^*_t := \sup_{s\le t}X_s, and X_s
is the price of the underlying security at time s. Assuming zero interest
rates, we show that the seller of this option can hedge his position by trading
in the underlying security if he begins with initial capital
X_0\int_{X_0}^{\infty}G(x)x^{-2}dx (and this is the smallest initial capital
that allows him to hedge his position).