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Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case


  • Zdzisław Burda
  • Andrzej Jarosz
  • Maciej Nowak
  • Jerzy Jurkiewicz
  • Gabor Papp
  • Ismail Zahed


We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.

Suggested Citation

  • Zdzisław Burda & Andrzej Jarosz & Maciej Nowak & Jerzy Jurkiewicz & Gabor Papp & Ismail Zahed, 2011. "Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1103-1124.
  • Handle: RePEc:taf:quantf:v:11:y:2011:i:7:p:1103-1124
    DOI: 10.1080/14697688.2010.484025

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    References listed on IDEAS

    1. Szilard Pafka & Marc Potters & Imre Kondor, 2004. "Exponential Weighting and Random-Matrix-Theory-Based Filtering of Financial Covariance Matrices for Portfolio Optimization," Papers cond-mat/0402573,
    2. Christoly Biely & Stefan Thurner, 2006. "Random matrix ensembles of time-lagged correlation matrices: Derivation of eigenvalue spectra and analysis of financial time-series," Papers physics/0609053,
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    Cited by:

    1. Desislava Chetalova & Rudi Schafer & Thomas Guhr, 2014. "Zooming into market states," Papers 1406.5386,
    2. Collins, Benoît & Matsumoto, Sho & Saad, Nadia, 2014. "Integration of invariant matrices and moments of inverses of Ginibre and Wishart matrices," Journal of Multivariate Analysis, Elsevier, vol. 126(C), pages 1-13.
    3. Stephan Süss, 2012. "The pricing of idiosyncratic risk: evidence from the implied volatility distribution," Financial Markets and Portfolio Management, Springer;Swiss Society for Financial Market Research, vol. 26(2), pages 247-267, June.


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