  In this paper, we investigate the generalization of the Call-Put duality
equality obtained in [1] for perpetual American options when the Call-Put
payoff $(y-x)^+$ is replaced by $\phi(x,y)$. It turns out that the duality
still holds under monotonicity and concavity assumptions on $\phi$. The
specific analytical form of the Call-Put payoff only makes calculations easier
but is not crucial unlike in the derivation of the Call-Put duality equality
for European options. Last, we give some examples for which the optimal
strategy is known explicitly.
