Predicting Stock Market Crashes using Urns and the Quantum Lorenz Ordering of Correlation Matrices

Abstract

By sampling financial correlation matrices over sliding windows, it has been shown in recent work that the quantum majorization induced partial ordering on this space of correlation matrices known as the "quantum Lorenz ordering" (QLO) can be used to characterize systemic risk by clustering correlation matrices according to their degree centrality on the associated directed graph called the "quantum majorization graph" (QMG). In this work, clusterings of the QMG are used to construct an online Bayesian nonparametric alarm system for the prediction of stock market crashes via the so-called "reinforced urn process" (RUP). To test the efficacy of this modelling methodology we exploit extreme value theory to systematically define stock market crashes by studying the tail of an appropriately fitted generalized pareto distribution (GPD) for stock market drawdowns. This approach identified 13 extreme drawdowns between 1985-2020, for which the RUP was trained from 1986-2005 to predict the 8 extreme drawdowns from 2005-2020. Of the three correlation metrics used to test this approach, the QLO corresponding to the set of upper Tail-Dependence matrices was shown to outperform the others: Pearson's and the Gini correlation. Tail-Dependence was able to predict all 8 crashes with just 5 false alarms over a 12 month time horizon, all 8 with 7 false alarms over an 8 month time horizon, 7 out of 8 with 9 false alarms over a 4 time horizon, and 7 out of 8 with 17 false alarms over a 2 month time horizon. This approach was then tested against the usage of the Log-Periodic Power Law Singularity (LPPLS) model's confidence indicators with promising results. The quantum Lorenz ordering is meant to rank a set of correlation matrices by the amount of dispersion reflected in their spectra: a true heterogeneity. We consider this dispersion from the standpoint of measurement error as has been in the application of random matrix theory (RMT) to correlation matrices in portfolio risk theory. We provide analytical relations between quantum majorization and random matrix cleaning for a few RMT filtering schemes posing quantum majorization as a desirable condition for RMT filtering. The RUP is tested using these RMT cleaned correlation matrices as well.