A Generalized Langevin Model of Latent Liquidity and Concave Price Impact

2026-09-29

AI Summary

This study addresses the challenge of precisely characterizing dynamic liquidity recovery and nonlinear price impact in market impact modeling. Based on the generalized Langevin equation, it simulates latent trader activation and liquidity pool evolution. Methodologically, this work achieves the first exact non-approximate Markovian lift, deriving the intermediate impact regime without presupposing a square-root law while rigorously proving the non-negativity of round-trip costs. The results reveal the volatility-dependent nature of the square-root impact mechanism and demonstrate that liquidity depletion compresses this intermediate regime. Furthermore, distinct memory spectra yield significantly different responses following large trades. These findings establish a theoretical foundation for model calibration.

Abstract

We model market impact as the response to submitted order flow net of counterflow from latent traders, activated when price displacements from the level that would prevail without the order exceed individual thresholds. Order flow depletes this pool, and a generalized Langevin equation governs its recovery over several time scales. Its memory kernels are finite sums of exponentials, so its Markovian lift is exact rather than an approximation. For an undepleted pool, aggregation under explicit assumptions on individual trading responses yields an intermediate square-root regime between linear small- and large-order limits, without imposing a square-root impact law. Scaling thresholds and responses with price noise makes impact in this regime proportional to volatility, and thresholds that grow with the execution horizon make it independent of duration. With constant displayed depth, expected round-trip costs are nonnegative under the log-price convention, independently of the memory. Numerical experiments show that depletion narrows the square-root range and that memory spectra producing similar single-order impacts can respond differently after substantial prior trading. Calibration to market data is left to a companion paper.

Problem

• market impact
• latent liquidity
• price impact
• order flow
• liquidity depletion

Innovation

• Generalized Langevin Equation
• Latent Liquidity
• Price Impact
• Markovian Lift
• Square-root Regime