Exponential utility maximization in small/large financial markets

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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.