The paper introduces a stochastic field theory model to establish a detection threshold for signals present in the limit where the eigenvalues are within the continuous spectrum of financial stock return correlations. The key insights are:
The authors construct a formal solution of the Langevin equation in the quenched regime for a real vector of size N, where disorder is represented by a Wigner matrix. They investigate the self-consistent evolution equation for the effective potential arising from the self-averaging of the square length.
Applying the formalism to study the S&P 500 financial market, the authors vary the signal-to-noise ratio by perturbing the correlations with an appropriate Brownian motion. They find that the evolution of the component corresponding to the largest eigenvalues in the original correlation matrix exhibits different behavior compared to a purely Gaussian matrix, indicating the presence of a signal.
The authors demonstrate that the underlying kinetics at the tail of the spectrum significantly diverges from what is expected for large Wishart random matrices, suggesting that the degrees of freedom near the cut-off imposed by the continuity criterion from the bulk are informative.
The analysis reveals that below a critical temperature, the system never reaches equilibrium (i.e., infinite correlation time), consistent with spin glass dynamics theory.
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by Ixandra Achi... at arxiv.org 10-01-2024
https://arxiv.org/pdf/2409.19711.pdfDeeper Inquiries