Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case |
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| Authors: | Zdzisław Burda Andrzej Jarosz Maciej A. Nowak Jerzy Jurkiewicz Gábor Papp Ismail Zahed |
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| Affiliation: | 1. Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre , Jagiellonian University , Reymonta 4, 30-059 Kraków, Poland zdzislaw.burda@uj.edu.pl;3. Clico Ltd. , Oleandry 2, 30-063 Kraków, Poland;4. Marian Smoluchowski Institute of Physics and Mark Kac Complex Systems Research Centre , Jagiellonian University , Reymonta 4, 30-059 Kraków, Poland;5. Institute for Physics, E?tv?s University , 1518 Budapest, Hungary;6. Department of Physics and Astronomy , SUNY Stony Brook , NY 11794, USA |
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| Abstract: | 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. |
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| Keywords: | Portfolio theory Power laws Statistical physics Risk measures Random walks Options pricing Random matrix theory |
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