Constructing analytically tractable ensembles of stochastic covariances with an application to financial data

Meudt, Frederik; Theissen, Martin; Schäfer, Rudi; Guhr, Thomas

doi:10.1088/1742-5468/2015/11/p11025

Cited by 10 publications

(13 citation statements)

References 44 publications

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“…As pointed out in Sec. 2.4, the algebraic distribution w A (X|C, D) converges to the Gaussian w G (X|C, D) in the limit L, M → ∞ under the condition (27). Consequently, this limit of Eq.…”

Section: Gaussian-algebraicmentioning

confidence: 89%

“…In Refs. [26,27] we only considered the Markovian case, i.e., D = 1 N , here we go beyond that by allowing arbitrary positive non-Markovian correlation matrices D. To take care of heavy tails in the distribution of the truly existing ensemble, we also choose the algebraic, determinant distribution…”

Section: Choice Of Amplitude and Ensemble Distributionsmentioning

confidence: 99%

“…We considerably extend a random matrix approach to tackle these issues that we presented some years ago [26], see also some modification in Ref. [27]. It also turned out useful in the context of credit risk and its impact on systemic stability [28,29].…”

Section: Introductionmentioning

confidence: 99%

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Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations

Guhr,

Schell

2020

Preprint

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Complex systems are often non-stationary, typical indicators are continuously changing statistical properties of time series.In particular, the correlations between different time series fluctuate. Models that describe the multivariate amplitude distributions of such systems are of considerable interest. Extending previous work, we view a set of measured, non-stationary correlation matrices as an ensemble for which we set up a random matrix model. We use this ensemble to average the stationary multivariate amplitude distributions measured on short time scales and thus obtain for large time scales multivariate amplitude distributions which feature heavy tails. We explicitly work out four cases, combining Gaussian and algebraic distributions. The results are either of closed forms or single integrals. We thus provide, first, explicit multivariate distributions for such nonstationary systems and, second, a tool that quantitatively captures the degree of nonstationarity in the correlations.

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Section: Gaussian-algebraicmentioning

confidence: 89%

Section: Choice Of Amplitude and Ensemble Distributionsmentioning

confidence: 99%

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Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations

Guhr,

Schell

2020

Preprint

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“…In the top part, the tails are heavier because the time interval ∆t is much shorter. To further account for this, we need to modify the Wishart model by using a distribution different from a Gaussian one (Meudt et al 2015). Second, Figure 2 clearly shows that the empirical ensemble of correlation matrices has inner structures, which are also contained in our model, because the mean Σ 0 enters.…”

Section: New Interpretation and Application Of The Wishart Modelmentioning

confidence: 99%

Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations

Mühlbacher

Guhr

2018

Risks

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We review recent progress in modeling credit risk for correlated assets. We employ a new interpretation of the Wishart model for random correlation matrices to model non-stationary effects. We then use the Merton model in which default events and losses are derived from the asset values at maturity. To estimate the time development of the asset values, the stock prices are used, the correlations of which have a strong impact on the loss distribution, particularly on its tails. These correlations are non-stationary, which also influences the tails. We account for the asset fluctuations by averaging over an ensemble of random matrices that models the truly existing set of measured correlation matrices. As a most welcome side effect, this approach drastically reduces the parameter dependence of the loss distribution, allowing us to obtain very explicit results, which show quantitatively that the heavy tails prevail over diversification benefits even for small correlations. We calibrate our random matrix model with market data and show how it is capable of grasping different market situations. Furthermore, we present numerical simulations for concurrent portfolio risks, i.e., for the joint probability densities of losses for two portfolios. For the convenience of the reader, we give an introduction to the Wishart random matrix model.

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“…We mention in passing that Ref. [21] also extends a new interpretation [24,25] of the Wishart model, namely a random matrix model for non-stationarity, conceptually different from the usage in statistical inference. Various multivariate algebraic amplitude distributions were modeled and calculated.…”

Section: Introductionmentioning

confidence: 99%

Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model

Guhr,

Schell

2020

Preprint

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The Wishart model of random covariance or correlation matrices continues to find ever more applications as the wealth of data on complex systems of all types grows. The heavy tails often encountered prompt generalizations of the Wishart model, involving algebraic distributions instead of a Gaussian. The mathematical properties pose new challenges, particularly for the doubly correlated versions. Here we investigate such a doubly correlated algebraic model for real covariance or correlation matrices. We focus on the matrix moments and explicitly calculate the first and the second one, the computation of the latter is non-trivial. We solve the problem by relating it to the Aomoto integral and by extending the recursive technique to calculate Ingham-Siegel integrals. We compare our results with the Gaussian case.

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Constructing analytically tractable ensembles of stochastic covariances with an application to financial data

Cited by 10 publications

References 44 publications

Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations

Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations

Credit Risk Meets Random Matrices: Coping with Non-Stationary Asset Correlations

Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model

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