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A 2D Levy-flight model for the complex dynamics of real-life financial markets

We report on the emergence of scaling laws in the temporal evolution of the daily closing values of the S\&P 500 index prices and its modeling based on the L\'evy flights in two dimensions (2D). The efficacy of our proposed model is verified and validated by using the extreme value statistics in random matrix theory. We find that the random evolution of each pair of stocks in a 2D price space is a scale-invariant complex trajectory whose tortuosity is governed by a $2/3$ geometric law between the gyration radius $R_g(t)$ and the total length $\ell(t)$ of the path, i.e., $R_g(t)\sim\ell(t)^{2/3}$. We construct a Wishart matrix containing all stocks up to a specific variable period and look at its spectral properties over 30 years. In contrast to the standard random matrix theory, we find that the distribution of eigenvalues has a power-law tail with a decreasing exponent over time -- a quantitative indicator of the temporal correlations. We find that the time evolution of the distance of a 2D L\'evy flights with index $\alpha=3/2$ from origin generates the same empirical spectral properties. The statistics of the largest eigenvalues of the model and the observations are in perfect agreement.

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