Non-Commutative Probability Theory and Applications in Finance

Abstract

We introduce non-commutative probability theory as a tool to analyse sample covariance matrices. We develop the theory necessary for derivation of the spectral distribution of covariance matrix estimates of VARMA(p, q) random matrix models and introduce an extension to VARFIMA(p, d, q) random matrix models. The relationship between sample covariance matrices and there population counterparts are investigated. Specifically, we showcase efficient algorithms for calculating various VARMA(p, q) spectral densities.

Both model classes are implemented so that parameter estimation is possible. For a feasible subset of a high-dimensional data set of stock returns we estimate the model parameters for VARMA(1, 1) random matrix models.