Exchangeable models of financial correlations matrices. Bayesian nonparametric models and network derived measures of financial assets

Abstract

Abstract De Finetti theorem establishes the conceptual basis of Bayesian inference replacing the independent and identically distributed sampling hypothesis prevalent in frequentist statistics with the much easier to justify in practical settings hypothesis of exchangeability. In this paper we make use of the extension of the concept of exchangeability from sequences to arrays arguing that the invariance to ordering is a much more tenable assumption than independent and identically distributed sampling in the financial modeling problems. Making use of the celebrated Aldous-Hoover representation theorem of exchangeable matrix we construct a Bayesian non-parametric model of the financial returns correlation matrices arguing that a Bayesian approach can mitigate many of the known shortcomings of the usual Pearson correlation coefficient. We posit the correlation matrix to be an exchangeable matrix and construct a Bayesian neural network to estimate the functions from the Aldous-Hoover representation theorem. The correlation matrix model is coupled with a Student-t likelihood (accounting for the heavy tails of financial returns). The model is estimated with a Hamiltonian Monte Carlo sampler. The samples are used to construct an ensemble of networks where each edge is weighted by the size of the correlation between two financial instruments. Various centrality measures are being calculated (betweenness, eigenvector) for each network of the ensemble allowing us to obtain a probabilistic view of each financial instrument’s importance. We also construct a minimum spanning tree associated with the mean correlation matrix allowing us to visualize the most important financial instruments from the universe selected.