$Q$-Gaussian diffusion in stock markets

We analyze the Standard & Poor's 500 stock market index from the past 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong superdiffusion together with short-time correlations and the second one corresponds to weak superdiffusion with weak time correlations. Both regimes are well described by $q$-Gaussian distributions. The porous media equation—a special case of the Tsallis-Bukman equation—is used to derive the governing equation for these regimes and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.

Figure 1

(a) Time evolution of the PDF of price return. Initially the PDF has a pronounced bump in the center that fully disappears close to 78 min. (b) The time evolution of the height of the PDF. Two well-defined power laws are observed. (c) From (a) for the first hour analysis, time increases from top to bottom. The ends of the bump are obtained from the two points at the PDF with abrupt change of slope. These points correspond to a transition from strong to weak superdiffusion. (d) The circles represent the end points plotted against time that are fitted in (e) by the power law $x=\pm a{(t/t_o)}^{\nu }$, with $a=(3.39\pm 0.01)\times {10}^{-2}$, $t_o=1$ min, and $\nu =0.62\pm 0.05$. This curve and the line $t=35$ min (red dotted line) define the strong superdiffusion regime (zone A). The bump disappears completely at $t=78$ min (blue dashed line). The remaining area corresponds to the weak superdiffusion regime (zone C). The crossover regime (zone B) is limited by the curve and $35\text{min}<t<78$ min. In this regime the bump still has not dissipated but experiences a transition from strong to weak superdiffusion.

Figure 2

Collapse of the PDFs of the weak superdiffusion regime (zone C). The exponent used for the collapse is $\alpha =1.79\pm 0.01$. The collapse data is fitted by a $q$-Gauss (red line) with exponent $q=1.71\pm 0.01$ and $D=0.1118\pm 0.005{\min }^{-1}$. Both exponents are related by Eq. (10).

Figure 3

Collapse of the PDFs of the strong superdiffusion regime (zone A). The exponent used for the collapse is $\alpha =1.26\pm 0.04$ and $D=(4.8\pm 0.2)\times {10}^{-3}{\min }^{-1}$. The collapse data is fitted by a $q$-Gauss (red line) with exponent $q=2.73\pm 0.005$.

Figure 4

Time evolution of the second moment of the full PDF of the S&P500 index. The data is compared to the best fitting and the linear fitting that corresponds to classical diffusion. The zones where the integration is performed are also shown.