Anomalous Impact in Reaction-Diffusion Financial Models

We generalize the reaction-diffusion model $A+B\to \cancel{0}$ in order to study the impact of an excess of $A$ (or $B$) at the reaction front. We provide an exact solution of the model, which shows that the linear response breaks down: the average displacement of the reaction front grows as the square root of the imbalance. We argue that this model provides a highly simplified but generic framework to understand the square-root impact of large orders in financial markets.

Figure 1

Fluctuations in the interface position for a modified model in which the terms $u_A$ and $u_B$ are random variables. In particular we change Eq. (2) by choosing with probability $1-p$ the sign of the reaction ($A+B\to \text{either}$ $A$ or $B$) according to a zero-mean, long-range correlated process with tail exponent $\gamma$. We find that the diffusion properties of the model change even though the impact properties are unaffected. We plot the variance of the interface position for different values of $\gamma$ for the set of parameters $L=400$, $J=D=1$, $\lambda =1000$, and $p=m=0$. The inset shows a perfect data collapse for the impact curves (as a function of $T$) obtained at different $\gamma$.

Figure 2

Main figure: average change in the position of the midpoint $\mathcal{I}=⟨x_t^{*}⟩-L/2$ after a perturbation of duration $T$. We compare the results of simulations of systems of different length (dashed lines) with the analytical prediction valid in the limit $L\to \infty$ (solid line), finding very good agreement. We used the parameters $J=D=1$, $p=0.5$, and $m=0.75$. The limit $\lambda \to \infty$ is enforced by setting $\lambda =10^3$. The crossover of the curve to the linear regime (indicating $Dt/L^2\gtrsim 1$) appears in the curve for $L=50$. Inset: average change in the midpoint position $\mathcal{I}$ against the volume imbalance $Q$ for a simulated system of length $L=100$ (dashed lines). The solid line indicates the mean-field (MF) estimate predicted by Eq. (20). We have chosen the parameters $D=J=1$, $\lambda =1000$, $p=1$, and $m=0.5$.