Quadratic hedging is a practically appealing approach for approximately replicating the random payoff of European options in incomplete markets. This work extends this methodology to the case of Bermudan options, for which an exercise policy needs to be optimized. This extension relies on constructing date specific approximate replicating portfolios. This modeling idea leads to a decomposition finding that enables adapting known quadratic hedging results to the date specific cash flows and formulating an option exercise policy optimization model. Optimal option exercise policies are in general time inconsistent. Time consistent option exercise policies can be optimized recursively.