Principal Curves for Statistical Divergences and an Application to Finance

Abstract

This paper proposes a method for the beta pricing model under the consideration of non-Gaussian returns by means of a generalization of the mean-variance model and the use of principal curves to define a divergence model for the optimization of the pricing model. We rely on the q-exponential model so consider the properties of the divergences which are used to describe the statistical model and fully characterize the behavior of the assets. We derive the minimum divergence portfolio, which generalizes the Markowitz's (mean-divergence) approach and relying on the information geometrical aspects of the distributions the Capital Asset Pricing Model (CAPM) is then derived under the geometrical characterization of the distributions which model the data, all by the consideration of principal curves approach. We discuss the possibility of integration of our model into an adaptive procedure that can be used for the search of optimum points on finance applications.

An Application of Maximal Exponential Models to Duality Theory

“…Moreover, the dual results obtained in the present paper could probably be extended in the framework described by [ 18 , 19 ], where a portfolio optimization problem, which involves deformed exponentials, is investigated.…”

Relative Entropy and Minimum-Variance Pricing Kernel in Asset Pricing Model Evaluation

“…Additionally, reference portfolios other than the HJ pricing kernel should be validated. Entropy-based methods, such as those suggested by [ 42 , 43 ], can provide useful insight in this regard. Furthermore, given that the absence of arbitrage opportunities—that is, the fact that non-negative payoffs that are positive with positive probability have positive prices—guarantees the existence of at least a strictly positive pricing kernel [ 14 ] and the fact that intertemporal marginal rates of substitution must be positive, the implications of this constraint for our entropy-based decomposition must be sufficiently addressed.…”