Variable order porous media equations: Application on modeling the S&P500 and Bitcoin price return

TL;DR

This paper introduces variable order fractional Fokker-Planck solutions using VO q-Gaussian functions, enhancing modeling of financial returns like S&P 500 and Bitcoin by capturing long-range memory and autocorrelation.

Contribution

It develops a novel solution framework for variable order fractional equations with VO q-Gaussian functions, applied to financial market modeling.

Findings

VO q-Gaussian functions effectively model price return distributions.

Analysis of anomalous exponents reveals long-range memory in financial data.

The approach improves understanding of autocorrelation in stock and cryptocurrency returns.

Abstract

This article reveals a specific category of solutions for the $1+1$ Variable Order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO $q$-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO $q$-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-range memory in time series data and autocorrelation patterns.