Static replications with traffic light options

It is well known that any sufficiently regular one-dimensional payoff function has an explicit static hedge by bonds, forward contracts and lots of vanilla options. We show that the natural extension of the corresponding representation leads to a static hedge based on the same instruments along with traffic light options, which have recently been introduced in the market. One big advantage of these replication strategies is the easy structure of the hedge. Hence, traffic light options are particularly powerful building blocks for more complicated bivariate options. While it is well known that the second strike derivative of non-discounted prices of vanilla options are related to the risk-neutral density of the underlying asset price in the corresponding absolutely continuous settings, similar statements hold for traffic light options in sufficiently regular bivariate settings.