  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.
