Exponential hedging and entropic penalties

We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X. We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q-price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk-averse asymptotics. KEY WORDS: hedging, exponential utility, relative entropy, duality, minimal martingale measure, minimal entropy martingale measure, reverse Hölder inequalities The research of FD is sponsored by Crédit Suisse. TR gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft through the Graduiertenkolleg ‘‘Stochastische Prozesse und probabilistische Analysis’ ’ at the Technical University of Berlin. DS thanks S. R. S. Varadhan for helpful conversations. MS thanks Dirk Becherer for useful discussions and gratefully acknowledges financial support by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich 373 ‘‘Quantifikation und Simulation Ökonomischer Prozesse’ ’ at Humboldt University, Berlin. CS thanks Marc Yor for helpful conversations. We are all grateful t