A 2D Levy-flight model for the complex dynamics of real-life financial markets
A 2D Lévy flight model explains S&P 500 index dynamics, showing scaling laws and power-law eigenvalue distributions.
What it examines
This paper introduces a 2D Lévy flight model to describe the complex dynamics of financial markets, specifically the S&P 500 index. It aims to model the scaling laws and temporal evolution of stock prices using extreme value statistics and random matrix theory.
What it concludes
The research suggests that the 2D Lévy flight model can effectively predict the dynamics of financial markets and assess risks. Potential applications include portfolio management and risk estimation. Future research could explore the model's response to external perturbations and its applicability to other financial indices.
Evidence objects
The research suggests that the 2D Lévy flight model can effectively predict the dynamics of financial markets and assess risks. Potential applications include portfolio management and risk estimation. Future research could explore the model's response to external perturbations and its applicability to other financial indices.
key_findings bullet 1 · key_findings · validation V0
Raw abstract and provenance
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 random matrix theory. We find that the random evolution of each pair of stocks in a 2D price space is a s… ▽ More 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 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 Lévy flights with index $α=3/2$ from origin generates the same empirical spectral properties. The statistics of the largest eigenvalues of the model and the observations are in perfect agreement. △ Less
Source row: 2 · abstract type: unknown