Complex market dynamics in the light of random matrix theory

TL;DR

This paper explores how random matrix theory can be applied to analyze complex financial market dynamics, addressing issues in correlation matrix computation and identifying market states and instabilities.

Contribution

It introduces the use of power mapping to improve correlation matrix analysis and compares simulated and empirical data to reveal eigenvalue spectrum properties.

Findings

Power mapping effectively suppresses noise in short epoch correlation matrices.

Eigenvalue spectra reveal interesting properties depending on correlation structures.

Empirical analysis of S&P 500 data identifies market states and precursors to crashes.

Abstract

We present a brief overview of random matrix theory (RMT) with the objectives of highlighting the computational results and applications in financial markets as complex systems. An oft-encountered problem in computational finance is the choice of an appropriate epoch over which the empirical cross-correlation return matrix is computed. A long epoch would smoothen the fluctuations in the return time series and suffers from non-stationarity, whereas a short epoch results in noisy fluctuations in the return time series and the correlation matrices turn out to be highly singular. An effective method to tackle this issue is the use of the power mapping, where a non-linear distortion is applied to a short epoch correlation matrix. The value of distortion parameter controls the noise-suppression. The distortion also removes the degeneracy of zero eigenvalues. Depending on the correlation structures, interesting properties of the eigenvalue spectra are found. We simulate different correlated Wishart matrices to compare the results with empirical return matrices computed using the S&P 500 (USA) market data for the period 1985-2016. We also briefly review two recent applications of RMT in financial stock markets: (i) Identification of "market states" and long-term precursor to a critical state; (ii) Characterization of catastrophic instabilities (market crashes).