Latent Liquidity, Memory, and Concave Market Impact

Summary

This model explains market impact through submitted order flow and offsetting activity from latent traders. Those traders enter when price moves beyond individual thresholds, and order flow depletes their available liquidity. A generalized Langevin equation describes recovery over multiple time scales; because its memory kernels are sums of exponentials, the model can be represented exactly as a Markovian system.

Under stated assumptions about trader responses, an undepleted liquidity pool produces linear impact for very small and very large orders, with an intermediate square-root regime emerging from aggregation. The model also links impact in that regime to volatility and, under a threshold-scaling assumption, makes it independent of execution duration. Numerical experiments find that depletion narrows the square-root range and that liquidity-memory spectra with similar isolated-order impact can behave differently after substantial prior trading. Calibration to market data is deferred to a companion paper, so the description provides theoretical and numerical evidence rather than empirical calibration.

Key ideas

• Latent traders provide counterflow after prices cross their individual activation thresholds.
• Order flow depletes latent liquidity, whose recovery is modeled across several time scales.
• Aggregated trading responses can produce an intermediate square-root impact regime without assuming that law directly.
• The model derives volatility scaling and duration independence under specified assumptions.
• Depletion narrows the square-root regime, and market-data calibration is left to later work.

Tags

# A Generalized Langevin Model of Latent Liquidity and Concave Price Impact


# A Generalized Langevin Model of Latent Liquidity and Concave Price Impact









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.