We investigate an American exchange option (AEO) pricing problem. Under the perfect market assumption, an AEO pricing problem can be modeled as a free boundary problem (FBP). The FBP is converted into an integral equation by using the Green's function. When the expiration date tends to infinity, we obtain a time-invariant constant of the exercise boundary. Moreover, we provide a pricing formula for valuating the early exercise premium of the perpetual AEO.