Extreme Value Analysis for Finite, Multivariate and Correlated Systems with Finance as an Example

2026-03-05

AI Summary

This study addresses the challenges of modeling extremes in multivariate high-frequency financial time series—namely, cross-sectional dependence, non-stationarity, and discretization effects—by introducing a novel approach that leverages eigen-decomposition of the correlation matrix. The method projects the original series onto an orthogonal eigenbasis to disentangle market-wide, sector-specific, and idiosyncratic components. Within this decorrelated space, peak-over-threshold (POT) extreme value analysis is applied to each component separately. This work represents the first integration of eigenbasis rotation with extreme value theory in a finite-dimensional dependent system, effectively decoupling collective dynamics from individual noise. By explicitly accounting for non-stationarity and intraday seasonality, the framework enables precise quantification and attribution of tail risk arising from distinct sources.

Technology Category

Machine Learning: Multi-class/Multi-label Learning & Extreme ClassificationData Mining & Knowledge Management: Anomaly/Outlier DetectionGame Theory and Economic Paradigms: Other Foundations of Game Theory & Economic Paradigms

Application Category

Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphsEconomics, Online Markets and Human Computation: The sharing economySecurity and Privacy: Large-scale security measurements

Abstract

Extreme values and the tail behavior of probability distributions are essential for quantifying and mitigating risk in complex systems of all kinds. In multivariate settings, accounting for correlations is crucial. Although extreme value analysis for infinite correlated systems remains an open challenge, we propose a practical framework for handling a large but finite number of correlated time series. We develop our approach for finance as a concrete example but emphasize its generality. We study the extremal behavior of high-frequency stock returns after rotating them into the eigenbasis of the correlation matrix. This separates and extracts various collective effects, including information on the correlated market as a whole and on correlated sectoral behavior from idiosyncratic features, while allowing us to use univariate tools of extreme value analysis. This holds even for high-frequency data where discretization effects normally complicate analysis. We employ a peaks-over-threshold approach and thereby fully avoid the analysis of block maxima. We estimate the tail shape of the rotated returns while explicitly accounting for nonstationarity, a key feature in finance and many other complex systems. Our framework facilitates tail risk estimation relative to larger trends and intraday seasonalities at both market and sectoral levels.

Problem

• Extreme Value Analysis
• Multivariate Systems
• Correlated Time Series
• Tail Risk
• Nonstationarity

Innovation

• extreme value analysis
• correlation matrix eigenbasis
• peaks-over-threshold
• nonstationarity
• tail risk estimation