Put-Call Parity for European Exotic Options

I propose a simple generalization of put-call parity that holds for a large class of exotic European options. The result rests on a reasonable generalization of the concepts of put and call. The proof is based on the fundamental theorem of arbitrage pricing and elementary properties of real numbers. I also propose a generalization of the notion of intrinsic value and volatility smile. Here I leverage on the relationship between put-call parity and the smile/smirk in the vanilla case.