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

Abstract

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 Lévy flights in two dimensions (2D). The efficacy of our proposed model is verified and validated by using the extreme value statistics in the 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 [Formula: see text] geometric law between the gyration radius [Formula: see text] and the total length [Formula: see text] of the path, i.e., [Formula: see text]. We construct a Wishart matrix containing all stocks up to a specific variable period and look at its spectral properties for 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 2D Lévy flights with index [Formula: see text] from origin generates the same empirical spectral properties. The statistics of the largest eigenvalues of the model and the observations are in perfect agreement.

One-dimensional Lévy quasicrystal

“…It is well established that to understand the traveling behavior of animals [14,15], the complex dynamics of real-life financial markets [18], Lévy description has been extremely useful. If one wants to model a situation to understand how these things can get affected due to correlated random events, our Lévy quasicrystal will be an automatic choice.…”

“…the anomalous scaling of dynamical correlations of conserved quantities in one-dimensional (1D) Hamiltonian systems, the Lévy scaling for the spreading of local energy perturbation has been predicted, as well as diverging thermal conductivity (via GreenKubo formula) [8][9][10][11]. Also, in order to understand the motion of the particle in a rotating flow [12,13] or even the traveling behavior of animals [14][15][16][17], the complex dynamics of real-life financial markets [18], Lévy description has been extremely useful. It has been shown recently in [5], one can discretize space fractional Schrödinger equation, introduce a system which is referred to as Lévy crystal.…”