Clustering Financial Return Distributions Using the Fisher Information Metric

Abstract

Information Geometry provides a correspondence between differential geometry and statistics through the Fisher Information matrix. In particular, given two models from the same parametric family of distributions, one can define the distance between these models as the length of the shortest geodesic connecting them in a Riemannian manifold whose metric is given by the model’s Fisher Information matrix. One limitation that had hinder the adoption of this similarity measure in practical applications is that this distance is typically difficult to compute in a robust manner. We review such complications and provide a general form for the distance function for one parameter models. We next focus on two higher dimensional extreme value models including the Generalized Pareto and Generalized Extreme Value distributions that will be used in financial risk applications. Specifically, we first develop a technique to identify the nearest neighbors of a target security in the sense that their best fit model distributions have minimal Fisher distance to that of target. Second, we develop a hierarchical clustering technique that compares Generalized Extreme Value distributions fit to block maxima of a set of equity loss distributions to group together securities whose worst single day yearly loss distributions exhibit commonalities.

Keywords: Information Geometry, Extreme Value Theory, Quantitative Risk