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Optimal portfolio positioning within generalized Johnson distributions. (English) Zbl 1402.91721
Summary: Many empirical studies have shown that financial asset returns do not always exhibit Gaussian distributions, for example hedge fund returns. The introduction of the family of Johnson distributions allows a better fit to empirical financial data. Additionally, this class can be extended to a quite general family of distributions by considering all possible regular transformations of the standard Gaussian distribution. In this framework, we consider the portfolio optimal positioning problem, which has been first addressed by M. J. Brennan and R. Solanki [“Optimal portfolio insurance”, J. Financ. Quant. Anal. 16, No. 3, 279–300 (1981; doi:10.2307/2330239)], H. E. Leland [“Who should buy portfolio insurance?”, J. Finance 35, No. 2, 581–594 (1980; doi:10.1111/j.1540-6261.1980.tb02190.x)] and further developed by P. Carr and D. Madan [“Optimal positioning in derivative securities”, Quant. Finance 1, No. 1, 19–37 (2001; doi:10.1080/713665549)] and J.-L. Prigent [“Generalized option based portfolio insurance”, Working Paper, THEMA, University of Cergy-Pontoise (2006)]. As a by-product, we introduce the notion of Johnson stochastic processes. We determine and analyse the optimal portfolio for log return having Johnson distributions. The solution is characterized for arbitrary utility functions and illustrated in particular for a CRRA utility. Our findings show how the profiles of financial structured products must be selected when taking account of non Gaussian log-returns.

MSC:
91G10 Portfolio theory
Software:
AS 99
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