Structural model for fluctuations in financial markets

In this paper we provide a comprehensive analysis of a structural model for the dynamics of prices of assets traded in a market which takes the form of an interacting generalization of the geometric Brownian motion model. It is formally equivalent to a model describing the stochastic dynamics of a system of analog neurons, which is expected to exhibit glassy properties and thus many metastable states in a large portion of its parameter space. We perform a generating functional analysis, introducing a slow driving of the dynamics to mimic the effect of slowly varying macroeconomic conditions. Distributions of asset returns over various time separations are evaluated analytically and are found to be fat-tailed in a manner broadly in line with empirical observations. Our model also allows us to identify collective, interaction-mediated properties of pricing distributions and it predicts pricing distributions which are significantly broader than their noninteracting counterparts, if interactions between prices in the model contain a ferromagnetic bias. Using simulations, we are able to substantiate one of the main hypotheses underlying the original modeling, viz., that the phenomenon of volatility clustering can be rationalized in terms of an interplay between the dynamics within metastable states and the dynamics of occasional transitions between them.

Figure 1

Magnetization as a function of $u_0$, shown for three different values ${\kappa }_0$ of the mean of the kappa distribution. Here, we take $J_0=J=0.5$, with $\alpha =0.5$, while $I_0=0$, ${\sigma }_I^2=0.1$, and $\sigma =0.1$. From top to bottom the curves correspond to ${\kappa }_0=0.2,0.7$, and 1.2, respectively.

Figure 2

Magnetization $m$ (blue full line), time-persistent correlation $q$ (red dashed line), and integrated response $\chi$ (black dotted line) as functions of $J_0$ in the absence of any global symmetry breaking fields, i.e., for $I_0=u_0=0$. Other parameters are ${\sigma }_I^2=0.1$, so there is a local random field, $J=0.5$, $\alpha =0.5$, ${\kappa }_0=0.2$, and $\sigma =0.1$. The figure shows the appearance of a ferromagnetic phase as $J_0$ is increased beyond $J_0^c\simeq 0.75$. For $J_0<J_0^c$ the system is in a frozen “spin-glass” like phase.

Figure 3

Phase boundary separating a spin-glass like phase at small values of $J_0$ from a ferromagnetic phase at larger values of $J_0$ as a function of ${\kappa }_0$ in the absence of global symmetry breaking fields, i.e., for $I_0=u_0=0$, but ${\sigma }_I^2=0.1$. Other parameters are $J=0.5$, $\alpha =0.5$, and $\sigma =0.1$.

Figure 4

Return distribution in the quasistationary regime evaluated for ${\kappa }_0|t-t^{\prime }|=20$ (red dashed line) compared with the analytic prediction for its asymptotic behavior Eq. (48) (blue full line) for an exponential $\kappa$ distribution with $\nu =1$.

Figure 5

Distributions for long time log returns across the ensemble averaged over all market conditions. Again, power law tails are observed and we expect that upon suitable normalization the distributions across timescales should scale very well as is seen in microscopic simulations.

Figure 6

Distribution of equilibrium log-prices ${\overline{u}}_{\vartheta }$ for the noninteracting system (first panel), compared to those of the corresponding interacting system (second panel), for selected values of the mean reversion $\kappa$ and the slow process $u_0$; note the different scales. System parameters are $I_0=0$, ${\sigma }_I^2=0.1$, ${\kappa }_0=0.2$, and $\sigma =0.1$; for the interacting system we chose $J_0=J=0.5$ and $\alpha =0.5$. For the individual curves the parameters are $(\kappa =0.5,u_0=0.1)$ (blue narrow pair of distributions), $(\kappa =0.2,u_0=0.1)$ (black, nearly symmetric, broad pair of distributions), and $(\kappa =0.2,u_0=1)$ (red, nonsymmetric, broad pair of distributions).

Figure 7

Equilibrium distribution of log-prices across the entire ensemble for $u_0=1$, both for the interacting system (full line) and for the noninteracting system (dashed line). Parameters are the same as in Fig 6.

Figure 8

Simulation of a market with of $N=50$ traded assets, exhibiting the relation between volatility and metastable state structures. The upper panel shows overlaps of the system state with three random attractors embedded in the coupling matrix in a Hebbian form as explained in the main text, while the lower panel shows returns on the index as a function of time. The other system parameters are ${\kappa }_0=0.2$, $I_0=0$, ${\sigma }_I^2=0.5$, $\sigma =0.1$, $J_0=J=0.5$, $\alpha =0.5$, and $\gamma ={10}^{-4}$.