Deformed exponentials and financial markets: applications to portfolio selection and asset pricing

Abstract

In this work, we propose a portfolio selection model based on a mean-divergence criteria, adapted to financial returns distributed according deformed exponential probability densities. Fixed a desired expected return, the method reduces to the minimization of a risk premium defined in terms of a statistical divergence, In the particular case of Gaussian returns, we recover the classical mean-divergence model by H. Markowitz. Next, we reformulate the projection pricing theory by Luenberger in the context of divergences as risk measures. This allowed us to define single factor models, including a variant of the CAPM whose beta coefficients depend on a Fisher metric that plays the role of a generalized covariance matrix. The eigenvalues of this matrix are used to define an extended notion of principal curves that adapts the work by Hastie and Stuetzle to the case of deformed exponentials and their correspondent Bregman divergences.