Dynamic Covariance Modelling Using Generalised Wishart Processes

Abstract: Modern portfolio theory was pioneered by Markowitz who formulated the mean-variance problem, without which any discussion on quantitative approaches to portfolio selection would be incomplete. The framework boils down to finding the expected return $\mu$ and covariance $\Sigma$, after which the solution is proportional to $\Sigma^{-1}\mu$. Although the problem is simple at heart, finding estimates of the components constitutes an entire field of research. Common estimators are weighted sample means, and there exist various techniques designed to separate information from noise -- the difficulty of which makes matters even worse when inverting the covariance. In this project, we take a more probabilistic route to modelling the covariance and deploy a Markov chain Monte Carlo algorithm to perform Bayesian inference. We extend on existing frameworks by tailoring a Hamiltonian Monte Carlo algorithm to improve sampling efficiency. The model is validated on synthetic datasets and deployed on financial data in the form of future contract return series. Results are on par with benchmark models based on exponentially weighted moving averages, and we notice particular improvement by modelling the precision matrix $\Sigma^{-1}$ directly, thus circumventing the otherwise problematic inversion