arXiv subjects

Anton J. Heckens

Publications and source records attributed to Anton J. Heckens.

8 recordsLinked to original sources

arXiv2026

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

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.

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arXiv2026

Ultrafast Extreme Events: Empirical Analysis of Mechanisms and Recovery in a Historical Perspective

To understand the emergence of Ultrafast Extreme Events (UEEs), the influence of algorithmic trading or high-frequency traders is of major interest as they make it extremely difficult to intervene and to stabilize financial markets. In an empirical analysis, we compare various characteristics of UEEs over different years for the US stock market to assess the possible non-stationarity of the effects. We show that liquidity plays a dominant role in the emergence of UEEs and find a general pattern in their dynamics. We also empirically investigate the after-effects in view of the recovery rate. We find common patterns for different years. We explain changes in the recovery rate by varying market sentiments for the different years. Overall, our results hint at a certain degree of universal behavior.

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arXiv2025

Multivariate Distributions in Non-Stationary Complex Systems II: Empirical Results for Correlated Stock Markets

Multivariate Distributions are needed to capture the correlation structure of complex systems. In previous works, we developed a Random Matrix Model for such correlated multivariate joint probability density functions that accounts for the non-stationarity typically found in complex systems. Here, we apply these results to the returns measured in correlated stock markets. Only the knowledge of the multivariate return distributions allows for a full-fledged risk assessment. We analyze intraday data of 479 US stocks included in the S&P500 index during the trading year of 2014. We focus particularly on the tails which are algebraic and heavy. The non-stationary fluctuations of the correlations make the tails heavier. With the few-parameter formulae of our Random Matrix Model we can describe and quantify how the empirical distributions change for varying time resolution and in the presence of non-stationarity.

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arXiv2025

Multivariate Distributions in Non-Stationary Complex Systems I: Random Matrix Model and Formulae for Data Analysis

Risk assessment for rare events is essential for understanding systemic stability in complex systems. As rare events are typically highly correlated, it is important to study heavy-tailed multivariate distributions of the relevant variables, especially in the presence of non-stationarity. We use a generalized scalar product between correlation matrices to clearly demonstrate this non-stationarity. Further, we present a model that we recently put forward, which captures how the non-stationary fluctuations of correlations make the tails of multivariate distributions heavier. Here, we provide the resulting formulae including Gaussian or Algebraic features. Compared to our previous results, we manage to remove in the Algebraic cases one out of the two, respectively three, fit parameters which considerably facilitates applications. We demonstrate the usefulness of these results by deriving joint distributions for linear combinations of amplitudes and validating them with financial data. Furthermore, we explicitly work out the moments of our model distributions. In a forthcoming paper we apply the model to financial markets.

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arXiv2025

A New Traders' Game? -- Empirical Analysis of Response Functions in a Historical Perspective

Traders on financial markets generate non-Markovian effects in various ways, particularly through their competition with one another which can be interpreted as a game between different (types of) traders. To quantify the market mechanisms, we empirically analyze self-response functions for pairs of different stocks and the corresponding trade sign correlators. While the non-Markovian dynamics in the self-responses is liquidity-driven, it is expectation-driven in the cross-responses which is related to the emergence of correlations. We empirically study the non-stationarity of these responses over time. In our previous data analysis, we only investigated the crisis year 2008. We now considerably extend this by also analyzing the years 2007, 2014 and 2021. To improve statistics, we also work out averaged response functions for the different years. We find significant variations over time revealing changes in the traders' game.

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arXiv2022

New Collectivity Measures for Financial Covariances and Correlations

Complex systems are usually non-stationary and their dynamics is often dominated by collective effects. Collectivity, defined as coherent motion of the whole system or of some of its parts, manifests itself in the time-dependent structures of covariance and correlation matrices. The largest eigenvalue corresponds to the collective motion of the system as a whole, while the other large, isolated, eigenvalues indicate collectivity in parts of the system. In the case of finance, these are industrial sectors. By removing the collective motion of the system as a whole, the latter effects are much better revealed. We measure a remaining collectivity to which we refer as average sector collectivity. We identify collective signals around the Lehman Brothers crash and after the dot-com bubble burst. For the Lehman Brothers crash, we find a potential precursor. We analyze 213 US stocks over a period of more than 30 years from 1990 to 2021. We plot the average sector collectivity versus the collectivity corresponding to the largest eigenvalue to study the whole market trajectory in a two dimensional space spanned by both collectivities. Therefore, we capture the average sector collectivity in a much more precise way. Additionally, we observe that larger values in the average sector collectivity are often accompanied by trend shifts in the mean covariances and mean correlations. As of 2015/2016 the collectivity in the US stock markets changed fundamentally.

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arXiv2022

A New Attempt to Identify Long-term Precursors for Endogenous Financial Crises in the Market Correlation Structures

Prediction of events in financial markets is every investor's dream and, usually, wishful thinking. From a more general, economic and societal viewpoint, the identification of indicators for large events is highly desirable to assess systemic risks. Unfortunately, the very nature of financial markets, particularly the predominantly non-Markovian character as well as non-stationarity, make this challenge a formidable one, leaving little hope for fully fledged answers. Nevertheless, it is called for to collect pieces of evidence in a variety of observables to be assembled like the pieces of a puzzle that eventually might help to catch a glimpse of long-term indicators or precursors for large events - if at all in a statistical sense. Here, we present a new piece for this puzzle. We use the quasi-stationary market states which exist in the time evolution of the correlation structure in financial markets. Recently, we identified such market states relative to the collective motion of the market as a whole. We study their precursor properties in the US stock markets over 16 years, including two endogenous crises, the dot-com bubble burst and the pre-phase of the Lehman Brothers crash. We identify certain interesting features and critically discuss their suitability as indicators.

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arXiv2020

Uncovering the Dynamics of Correlation Structures Relative to the Collective Market Motion

The measured correlations of financial time series in subsequent epochs change considerably as a function of time. When studying the whole correlation matrices, quasi-stationary patterns, referred to as market states, are seen by applying clustering methods. They emerge, disappear or reemerge, but they are dominated by the collective motion of all stocks. In the jargon, one speaks of the market motion, it is always associated with the largest eigenvalue of the correlation matrices. Thus the question arises, if one can extract more refined information on the system by subtracting the dominating market motion in a proper way. To this end we introduce a new approach by clustering reduced-rank correlation matrices which are obtained by subtracting the dyadic matrix belonging to the largest eigenvalue from the standard correlation matrices. We analyze daily data of 262 companies of the S&P 500 index over a period of almost 15 years from 2002 to 2016. The resulting dynamics is remarkably different, and the corresponding market states are quasi-stationary over a long period of time. Our approach adds to the attempts to separate endogenous from exogenous effects.

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