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Optimal Strategies and Utility-Based Prices Converge When Agents’ Preferences Do

Laboratoire de Probabilités et Modèles Aléeatoires, Université Denis Diderot, Paris 7, 16 Rue Clisson, 75013 Paris, France
Computer and Automation Institute of the Hungarian Academy of Sciences, P.O. Box 63, 1518 Budapest, Hungary
carassus{at}math.jussieu.fr
rasonyi{at}sztaki.hu

A discrete-time financial market model is considered with a sequence of investors whose preferences are described by their utility functions U_n, defined on the whole real line and assumed to be strictly concave and increasing. Under suitable hypotheses, it is shown that whenever U_n tends to another utility function U, the respective optimal strategies converge, too. Under additional assumptions the rate of convergence is estimated. We also establish the continuity of the fair price of Davis [Davis, M. H. A. 1997. Option pricing in incomplete markets. M. A. H. Dempster, S. R. Pliska, eds. Mathematics of Derivative Securities. Cambridge University Press, pp. 216–226] and the utility indifference price of Hodges and Neuberger [Hodges, R., K. Neuberger. 1989. Optimal replication of contingent claims under transaction costs. Rev. Futures Markets 8 222–239] with respect to changes in agents’ preferences.

Key Words: utility maximization; optimal strategies; convergence rate; utility indifference price; fair price
History: Received: June 22, 2005; revision received: March 23, 2006;