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We report on the emergence of scaling laws in the temporal evolution of thedaily closing values of the S\&P 500 index prices and its modeling based on theL\'evy flights in two dimensions (2D). The efficacy of our proposed model isverified and validated by using the extreme value statistics in random matrixtheory. We find that the random evolution of each pair of stocks in a 2D pricespace 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 Wishartmatrix containing all stocks up to a specific variable period and look at itsspectral properties over 30 years. In contrast to the standard random matrixtheory, we find that the distribution of eigenvalues has a power-law tail witha decreasing exponent over time -- a quantitative indicator of the temporalcorrelations. We find that the time evolution of the distance of a 2D L\'evyflights with index $\alpha=3/2$ from origin generates the same empiricalspectral properties. The statistics of the largest eigenvalues of the model andthe observations are in perfect agreement.

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