Signal inference in financial stock return correlations through phase-ordering kinetics in the quenched regime

TL;DR

The paper tackles the problem of detecting signals in financial stock return correlations when the eigen-spectrum is nearly continuous, where PCA fails. It develops a non-equilibrium statistical-field theory in the eigenbasis of the correlation matrix, yielding a Langevin-type dynamics and a self-consistent, quenched solution that reveals a critical temperature and long memory in the continuous spectrum. Applying the framework to S&P 500 data, the authors show that the largest eigenvalues within the continuous spectrum exhibit dynamics inconsistent with purely random (MP) matrices, indicating detectable signals. This work provides a rigorous, thermodynamics-inspired method to uncover subtle, tail-end correlations in financial markets with potential implications for risk management and portfolio construction.

Abstract

Financial stock return correlations have been analyzed through the lens of random matrix theory to differentiate the underlying signal from spurious correlations. The continuous spectrum of the eigenvalue distribution derived from the stock return correlation matrix typically aligns with a rescaled Marchenko-Pastur distribution, indicating no detectable signal. In this study, we introduce a stochastic field theory model to establish a detection threshold for signals present in the limit where the eigenvalues are within the continuous spectrum, which itself closely resembles that of a random matrix where standard methods such as principal component analysis fail to infer a signal. We then apply our method to Standard & Poor's 500 financial stocks' return correlations, detecting the presence of a signal in the largest eigenvalues within the continuous spectrum.

Figures (7)

• Figure 1: Empirical spectra can exhibit some localized spikes (left) out of the continius spectrum (i.e. bulk noise, in red) made of delocalized eigenvectors (i.e. relevant information, in blue), in which case the cut-off $\Lambda$ provides a clean separation between delocalized eigenvectors and localized ones. For nearly continuous spectra (right), the position of the cut-off $\Lambda$ is more difficult to understand.
• Figure 2: Illustration of the interpolation for different values of the parameter $\beta$ On the left the entries of the correlation matrix and on the right the corresponding eigenvalue distribution with the cut.
• Figure 3: Behavior of the function $a$ (eq. \ref{['refat']})for different values of $a_0:=-h_1/h_0$ and parameter $\beta$, respectively for high temperature (right panels) and low temperature (left panels).
• Figure 4: Short time numerical evolution of the averaged trajectories for different values of $\beta$, different eigenvalues and different temperatures.
• Figure 5: Dependency on the fitting parameters $\alpha$ and $\gamma$ on $\beta$ for different eigenvalues and different temperatures.