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Modern Stochastics: Theory and Applications


Exponential utility maximization in small/large financial markets
Volume 12, Issue 2 (2025), pp. 225–250
Pub. online: 23 January 2025      Type: Research Article      Open accessOpen Access
Received
13 June 2024
Revised
15 October 2024
Accepted
23 December 2024
Published
23 January 2025

Abstract

Obtaining a utility-maximizing optimal portfolio in a closed form is a challenging issue when the return vector follows a more general distribution than the normal one. In this paper, for markets based on finitely many assets, a closed-form expression is given for optimal portfolios that maximize an exponential utility function when the return vector follows normal mean-variance mixture models. Especially, the used approach expresses the closed-form solution in terms of the Laplace transformation of the mixing distribution of the normal mean-variance mixture model and no distributional assumptions on the mixing distribution are made.
Also considered are large financial markets based on normal mean-variance mixture models, and it is shown that the optimal exponential utilities in small markets converge to the optimal exponential utility in the large financial market. This shows, in particular, that to reach the best utility level investors need to diversify their investments to include infinitely many assets into their portfolio, and with portfolios based on only finitely many assets they will never be able to reach the optimum level of utility.

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Keywords
Expected utility mean-variance mixtures Hara utility functions large financial markets martingale measures

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