Quadratic Hedging and Optimization of Option Exercise Policies

Abstract

Quadratic hedging of option payoffs generates the variance optimal martingale measure. When an option features an exercise policy and its cash flows are hedged according to this approach, it may be tempting to optimize such a policy under this measure. Because the variance optimal martingale measure may not be an equivalent probability measure, focusing on American options we show that the resulting exercise policy may be unappealing. This drawback can sometimes be remedied by imposing time consistency on exercise policies, but in general persists even in this case, which compounds the familiar issue that valuing an option using this measure may not result in an arbitrage free value. An alternative and known approach bypasses both of these pitfalls by optimizing option exercise policies under any given equivalent martingale measure and anchoring quadratic hedging to the resulting value of this policy. Additional research may assess on realistic applications the magnitude of the limitations associated with optimizing option exercise policies based on the variance optimal martingale measure.