General Duality for Perpetual American Options

Abstract

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 $ϕ(x,y)$. It turns out that the duality still holds under monotonicity and concavity assumptions on $ϕ$. 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.