  Recently it was observed that the probability distribution of the price
return in S\&P500 can be modeled by $q$-Gaussian distributions, where various
phases (weak, strong super diffusion and normal diffusion) are separated by
different fitting parameters (Phys Rev. E 99, 062313, 2019). Here we analyze
the fractional extensions of the porous media equation and show that all of
them admit solutions in terms of generalized $q$-Gaussian functions. Three
kinds of "fractionalization" are considered: \textit{local}, referring to the
situation where the fractional derivatives for both space and time are local;
\textit{non-local}, where both space and time fractional derivatives are
non-local; and \textit{mixed}, where one derivative is local, and another is
non-local. Although, for the \textit{local} and \textit{non-local} cases we
find $q$-Gaussian solutions , they differ in the number of free parameters.
This makes differences to the quality of fitting to the real data. We test the
results for the S\&P 500 price return and found that the local and non-local
schemes fit the data better than the classic porous media equation.
