Volatility Clustering and Scaling for Financial Time Series due to Attractor Bubbling

A microscopic model of financial markets is considered, consisting of many interacting agents (spins) with global coupling and discrete-time heat bath dynamics, similar to random Ising systems. The interactions between agents change randomly in time. In the thermodynamic limit, the obtained time series of price returns show chaotic bursts resulting from the emergence of attractor bubbling or on-off intermittency, resembling the empirical financial time series with volatility clustering. For a proper choice of the model parameters, the probability distributions of returns exhibit power-law tails with scaling exponents close to the empirical ones.

Figure 1

(a) Time series of returns $x(t)$ from the map (5) with $A=2$, $h={10}^{-2}$. (b) Probability distributions $\rho (x)\propto \rho (G_1)$ of returns from the map (5) and fits to the power-law scaling $\rho (x)\propto x^{-\alpha -1}$ of their tails for $h={10}^{-3}$ and $A=1.6$ ($\circ$, $\alpha =2.67$, ${\alpha }_{an}=2.89$); $h={10}^{-2}$ and $A=1.6$ ($\square$, $\alpha =2.62$, ${\alpha }_{an}=2.89$), $A=2.0$ ($△$, $\alpha =0.98$, ${\alpha }_{an}=1.0$), $A=2.75$ ($♦$, $\alpha =-0.10$, ${\alpha }_{an}=-0.032$); ${\alpha }_{an}$ denotes analytic values of $\alpha$. (c) Cumulative distributions of normalized returns $G_{\Delta t}$ (where the angular brackets denote time average) from the map (5) with $A=1.6$, $h={10}^{-2}$, $\Delta t=1$ ($\circ$), $\Delta t=10$ ($\square$), $\Delta t=100$ ($△$), $\Delta t=1000$ ($♦$), and comparison with the power law $P(G_{\Delta t})\propto G_{\Delta t}^{-\alpha }$ with exponent $\alpha =2.76$. All distributions were obtained from positive returns.

Figure 2

(a) Time series of returns $x(t)$ from the model with many interacting agents with $A=2.0$, $a=4.0$, and, from above, $N=25$ and $N=4000$. (b) Probability distributions $\rho (x)$ of returns and fits to the power-law scaling $\rho (x)\propto x^{-\alpha -1}$ of their tails from the model with $N=1000$, $h=0$, $a=2A$, and $A=1.6$ ($\square$, $\alpha =2.59$), $A=2.0$ ($△$, $\alpha =0.98$), and $A=2.75$ ($♦$, $\alpha =-0.13$). All distributions were obtained from positive returns.