On q-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Lévy models. As a consequence we find that, in the onedimensional case, the q-optimal equivalent martingale measures may exist only, if the tails for upward jumps are extraordinarily light. Moreover, we derive convergence of the q-optimal signed, resp. absolutely continuous, martingale measures to the entropy minimal martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.