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

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.

Figures (25)

• Figure 1: Probability density function $g^{\text{EV}}_{\gamma, b, a}(x)$ for $\gamma=0$ and special cases $\gamma=-1/2<0, \gamma=1/2>0$.
• Figure 2: Probability density function $g^{\text{GPD}}_{\gamma, \sigma}(x)$ for $\gamma=0$ and special cases $\gamma=-1/2<0, \gamma=1/2>0$.
• Figure 3: Schematic depiction of clustering of peaks over the threshold. Top: Independent, unclustered exceedances. $\hat{\Theta} = 1$. Bottom: Exceedances derived from the time series above, but occurring in clusters of two. $\hat{\Theta} = 0.68$.
• Figure 4: Schematic depiction of the estimation procedure.
• Figure 5: Top left: Spectral density of the financial correlation matrix ($\Delta t = 1$s) of 2014. Top right and bottom: Largest three normalized eigenvectors visibly corresponding to the market, the Energy sector and the Utility sector of the GICS GICS.