Diffusive in plain sight: An inconspicuous law of market impact

Battle Count: 53

This paper is directly and fundamentally relevant to quantitative trading. It provides a theoretical foundation for the square-root law of market impact, which is central to optimal execution, trading cost estimation, and algorithmic trading. The framework challenges the assumption of exogenous, strategy-agnostic impact kernels used in most execution algorithms (Almgren-Chriss, Gatheral, etc.). Key practical implications include: (1) impact models must adapt to each participant's order-flow statistics, (2) backtests outside the historical statistical envelope may be unreliable, (3) the square-root law is a structural consequence of diffusivity rather than an anomaly, (4) many existing propagator and latent-liquidity models fail a basic diffusivity diagnostic. The paper also provides a rigorous no-arbitrage foundation for positive impact costs.

The theoretical framework involves advanced mathematics: autocorrelation functions, all-pass filter theory, Kolmogorov-Szegő formula, hydrodynamic limits, and Gaussian regression. Implementing the diagnostic (checking whether a model's impact increments are white) is relatively straightforward. However, implementing an adaptive impact model that satisfies the structural constraint while accommodating transient dynamics (the all-pass degree of freedom) is non-trivial. The framework requires careful handling of stationarity assumptions, participation-rate scaling, and the distinction between mechanical and adaptive impact components. No reference implementation is provided.

The paper is purely theoretical with complete mathematical derivations (Eqs. 2.1-7.2). All proofs and algebraic steps are presented in the text. However, no code, simulation scripts, or empirical data are provided. Reproduction requires implementing the theoretical framework and validating against external empirical data. The author references companion papers (Lodhia 2026, Vasaikar 2026) that perform empirical validation. A proof for general LTI kernels is noted as 'to be published elsewhere.'