Optimal positioning in financial derivatives under mixture distributions

In this paper, we study and extend the optimal portfolio positioning problem introduced by Brennan and Solanki (1981) and by Leland (1980). For that purpose, we introduce mixtures of probability distributions to model the log returns of financial assets. In this framework, we provide and analyze the general solution for log return with mixture distributions, in particular for the mixture Gaussian case. Our solution is characterized for arbitrary utility functions and for any risk neutral probability. Moreover, we illustrate the solution for a CRRA utility and for the minimal risk-neutral probability. Lastly, we compare our solution with the optimal portfolio within ambiguity. Our results have significant implications to improve the management of structured financial portfolios.