A projection pricing model for non-Gaussian financial returns

Abstract

Stephen LeRoy, Jan Werner and David Luenberger have developed a geometric approach to the capital asset pricing model (CAPM) in terms of projections in a Hilbert space onto a mean–variance efficient frontier. Using this projection method, they were able to elegantly deduce a geometric interpretation of CAPM and factor asset pricing models. In this paper we extend their geometric methods to non-Euclidean divergence geometries. This extension has relevant consequences. First, it permits to deal with higher order moments of the probability distributions since general statistical divergences could encode global information about these distributions as is the case of the entropy. Secondly, orthogonal Euclidean projections and the corresponding least squares problem give place to Riemannian projections onto a possibly curved efficient frontier. Finally, our method is flexible enough to deal with huge families of probability distributions. In particular, there is no need to assume normality of the returns of the financial assets.