Exactly solvable model of the square-root price impact dynamics under the long-range market-order correlation

In econophysics, there are several enigmatic empirical laws: (i)~the market-order flow has strong persistence (long-range order-sign correlation), well formulated as the Lillo-Mike-Farmer model. This phenomenon seems paradoxical given the diffusive and unpredictable price dynamics; (ii)~the price impact $I(Q)$ of a large metaorder $Q$ follows the square-root law, $I(Q)\propto \sqrt{Q}$. In this Letter, we propose an exactly solvable model of the nonlinear price-impact dynamics that unifies these enigmas. We generalize the Lillo-Mike-Farmer model to nonlinear price-impact dynamics, which is mapped to an exactly solvable L\'evy-walk model. Our exact solution and numerical simulations reveal three important points: First, the price dynamics remains diffusive under the square-root law, even under the long-range correlation. Second, price-movement statistics follows truncated power laws with typical exponent around three. Third, volatility has long memory. While this simple model lacks adjustable free parameters, it naturally aligns even with other enigmatic empirical laws, such as (iii)~the inverse-cubic law for price statistics and (iv)~volatility clustering. This work illustrates the crucial role of the square-root law in understanding rich and complex financial price dynamics from a single coherent viewpoint.