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.

[1]  I. Csiszár.  $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[2]  C. Doléans-Dade,et al.  Inegalites de normes avec poids , 1979 .

[3]  M. Émery.  Une Topologie sur l’Espace des Semimartingales , 1979 .

[4]  N. Kazamaki.  A note on two criteria for the uniform integrability of exponential martingales , 1983 .

[5]  A. Barron.  THE STRONG ERGODIC THEOREM FOR DENSITIES: GENERALIZED SHANNON-MCMILLAN-BREIMAN THEOREM' , 1985 .

[6]  Aharon Ben-Tal,et al.  The Entropic Penalty Approach to Stochastic Programming , 1985, Math. Oper. Res..

[7]  Stanley R. Pliska,et al.  A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios , 1986, Math. Oper. Res..

[8]  N. Kazamaki.  Continuous Exponential Martingales and Bmo , 1994 .

[9]  Christophe Stricker,et al.  Couverture des actifs contingents et prix maximum , 1994 .

[10]  Sid Browne,et al.  Optimal Investment Policies for a Firm With a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin , 1995, Math. Oper. Res..

[11]  M. Schweizer.  On the Minimal Martingale Measure and the Foellmer- Schweizer Decomposition , 1995 .

[12]  Yoshio Miyahara,et al.  CANONICAL MARTINGALE MEASURES OF INCOMPLETE ASSETS MARKETS , 1996 .

[13]  Walter Schachermayer,et al.  Weighted norm inequalities and hedging in incomplete markets , 1997, Finance Stochastics.

[14]  H. Pham,et al.  Mean‐Variance Hedging and Numéraire , 1998 .

[15]  Closedness of some spaces of stochastic integrals , 1998 .

[16]  Dominick Samperi.  Inverse problems, model selection and entropy in derivative security pricing , 1998 .

[17]  Huyên Pham,et al.  Dynamic programming and mean-variance hedging , 1999, Finance Stochastics.

[18]  W. Schachermayer,et al.  The asymptotic elasticity of utility functions and optimal investment in incomplete markets , 1999 .

[19]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .

[20]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[21]  Nicole El Karoui,et al.  Pricing Via Utility Maximization and Entropy , 2000 .

[22]  Marco Frittelli,et al.  Introduction to a theory of value coherent with the no-arbitrage principle , 2000, Finance Stochastics.

[23]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[24]  W. Schachermayer.  Optimal investment in incomplete markets when wealth may become negative , 2001 .

[25]  Hui Wang,et al.  Utility maximization in incomplete markets with random endowment , 2001, Finance Stochastics.

[26]  Ludger Rüschendorf,et al.  Minimax and minimal distance martingale measures and their relationship to portfolio optimization , 2001, Finance Stochastics.

[27]  Yuri Kabanov,et al.  On equivalent martingale measures with bounded densities , 2001 .

[28]  Doktor der Naturwissenschaften,et al.  Rational Hedging and Valuation with Utility-Based Preferences , 2001 .

[29]  Marco Frittelli,et al.  On the Existence of Minimax Martingale Measures , 2002 .

[30]  Y. Kabanov,et al.  On the optimal portfolio for the exponential utility maximization: remarks to the six‐author paper , 2002 .

[31]  Walter Schachermayer,et al.  How Potential Investments may Change the Optimal Portfolio for the Exponential Utility , 2002 .

[32]  Peter Grandits,et al.  On the minimal entropy martingale measure , 2002 .