Space-time fractional porous media equation: Application on modeling of S&P500 price return

We present the fractional extensions of the porous media equation (PME) with an emphasis on the applications in stock markets. Three kinds of “fractionalization” are considered: local, where the fractional derivatives for both space and time are local; nonlocal, where both space and time fractional derivatives are nonlocal; and mixed, where one derivative is local, and another is nonlocal. Our study shows that these fractional equations admit solutions in terms of generalized $q$-Gaussian functions. Each solution of these fractional formulations contains a certain number of free parameters that can be fitted with experimental data. Our focus is to analyze stock market data and determine the model that better describes the time evolution of the probability distribution of the price return. We proposed a generalized PME motivated by recent observations showing that $q$-Gaussian distributions can model the evolution of the probability distribution. Various phases (weak, strong super diffusion, and normal diffusion) were observed on the time evolution of the probability distribution of the price return separated by different fitting parameters [Phys. Rev. E 99, 062313 (2019)]. After testing the obtained solutions for the S&P500 price return, we found that the local and nonlocal schemes fit the data better than the classic porous media equation.

Figure 1

Comparison of the probability distribution of the price return of S&P500 for $\mathrm{\Delta }t=1$ min with two different fittings. (a) For the Levy-Stable distribution, the fitting values after applying a least-squares fitting method are $\alpha =0.95$ and $\gamma =0.11$. (b) A second calculation was made to obtain the characteristic exponent from the power law of the tails of the PDF. The value of $\alpha =2.43$ lies outside the Levy regime, ruling out the Levy-stable distribution function. (c) The $q$-Gaussian distribution function (blue). Two $q$-Gaussians capture better the anatomy probability distribution of the price return of S&P500.

Figure 2

(a) $\mathrm{L}q$-Gaussians for different $q$ values as indicated in the plot for $t=1,\xi =1,\gamma =2.$ (b) $\mathrm{L}q$-Gaussians for different $\xi$ values as indicated in the plot for $t=1,q=1.5,\gamma =2.$ (c) $\mathrm{L}q$-Gaussians for different $\gamma$ values as indicated in the plot for $t=1,q=1.5,\xi =1.$ (d) Time evolution of the Green function of Eq. (6). Over time, the peaks of curves decrease and the PDFs lose the behavior of heavier tails; in contrast with (c), by increasing $\gamma ,$ the peaks of curves and behavior of heavier tails decreases. In (a), by increasing $q$, the peaks of curves increase and the distribution becomes narrower (the tails become heavier).

Figure 3

(a) The PDFs of the $\mathrm{LNL}q$-Gaussian distributions for different values of $\gamma$ and $\xi =1$. (b) The PDFs of the $\mathrm{LNL}q$-Gaussian distributions for different values of $\xi$ and $\gamma =0.25$. (c) Time evolution of the Green function of Eq. (18).

Figure 4

(a) The PDFs of the $\mathrm{NL}q$-Gaussian distributions for $\alpha =2$ and indicated values of $q.$ (b) The PDFs of the $\mathrm{NL}q$-Gaussian distributions for $q=1.5$ and indicated value of $\alpha$. (c) The time evolution of PDFs of the $\mathrm{NL}q$-Gaussian distributions for $q=1.5$ and $\alpha =2$. These are the Green functions of Eq. (22). We can see the $\mathrm{NL}q$-Gaussian for a constant value $\alpha$, and different $q$ values show a multiple behavior; for $q=1.1,1.2,1.3$ the peaks of curves and behavior of heavier tails increase, in contrast with $q=1.4,1.5.$ Also, for a constant value $q$ and different $\alpha$ values, by increasing $\alpha ,$ the peaks of curves and behavior of heavier tails increase. In time evolution of $\mathrm{L}q$-Gaussian, over time, the peaks of curves decrease and the PDFs lost behavior of heavier tails.

Figure 5

Time evolution of the PDF of the S&P500 price return. Three different zones were determined based on an abrupt slope change of the fitting parameters $\alpha$ and $q$. The contour plot represents the PDF of the detrended price return. The black circles represent the end points of the strong superdiffusion regime (zone A) from $t=0$ to $t=35$ min. These points are the ends of the bump obtained from the two points at the PDF with an abrupt change of slope (Fig. 1(c) in Ref. [8]). The remaining area during the first 35 min corresponds to a weak superdiffusion regime (zone $B_1$). From 80 min to 10 days, the zone corresponds to a weak superdiffusion regime (zone $B_2$). A normal diffusion process is reached after 30 days. The gray dashed lines represent the transitions between each zone.

Figure 6

This figure shows the collapse of the time evolution of the PDFs for each zone displayed previously in Fig. 5. (a) The collapse of the PDFs of the strong superdiffusion regime is represented in zone A. The collapse in zone A has been fitted with the solution of the C-PME ($\gamma =2,q=1.94$), T-FPME ($\gamma =2,q=2.73,\xi =0.21$), TS-FPME ($\gamma =1.97,q=2.70,\xi =0.21$), and G-PME ($\alpha =2.18,q=3.14,\lambda =-0.52$). (b) The collapse of the PDFs of the weak superdiffusion regime for the first 35 min while the bump remains in the PDFs occurs in zone $B_1$. The collapse in zone $B_1$ has been fitted with the solution of the C-PME ($\gamma =2,q=1.85$), T-FPME ($\gamma =2,q=1.72,\xi =0.82$), TS-FPME ($\gamma =2.09,q=1.91,\xi =0.75$), and G-PME ($\alpha =2.05,q=2.44,\lambda =-0.99$). (c) The collapse of the PDFs during the weak superdiffusion regime after 100 min, when the bump disappears completely, occurs in zone $B_2$. The collapse in zone $B_2$ has been fitted with the solution of the C-PME ($\gamma =2, q=1.81$), T-FPME ($\gamma =2,q=1.42,\xi =0.92$), TS-FPME ($\gamma =1.73,q=1.41,\xi =0.77$), and G-PME ($\alpha =1.56,q=1.19,\lambda =-4.62$). (d) The collapse of the PDFs for the normal diffusion regime occurs in zone C. The collapsed data is fitted by a Gaussian distribution function, which is a concurrent solution for the four previous PDEs when $\gamma =2,q=1.00,\xi =1.00$.