Mechanical Origin of Market Impact

• Mechanical origin of market impact is defined as the emergence of universal price response functions driven solely by order book mechanics, liquidity consumption, and autocorrelated order flow.
• The study details mathematical models that derive power-law relations and equilibrium properties explaining temporary, transient, and permanent price impacts.
• It underscores the universality of the square-root impact law and its implications for optimal execution strategies, risk management, and market regulation.

The mechanical origin of market impact is the emergence of universal price response functions purely from the interactions, matching rules, and supply/demand frictions of modern electronic markets—independent of informational asymmetry or detection of metaorders. Throughout diverse theoretical and empirical frameworks, market impact laws—particularly the square-root impact law—arise from deterministic liquidity consumption, autocorrelated order flow, optimal supply/demand clearing, and strategic risk transfer among market participants. This article surveys precise mathematical definitions, model derivations, empirical regularities, and equilibrium properties that collectively ground market impact in mechanical microstructure.

1. Key Definitions and Mathematical Formalism

Market impact quantifies the average price change—temporary or permanent—induced by a large order (metaorder) executed through a sequence of smaller child orders. Three principal notions emerge:

• Bare (Instantaneous) Impact Function $f(n)$: The incremental price change from the $n$th trade in a metaorder, measured before any decay or recovery. For a metaorder of $L$ trades, $p(L)=\sum_{k=1}^L f(k)\approx C L^\delta$, with exponent $\delta\in(0,1)$ concave.
• Transient (Temporary) Impact: The time-dependent price path $p_t\approx C L_{t}^\delta$ during execution, typically rising sub-linearly in volume due to diminishing liquidity.
• Permanent Impact: After completion, only a fraction $\alpha_\infty=(\gamma-1)/\gamma$ of peak impact remains. For power-law distributed metaorders, $p_\infty=p_{\max}/(1+\delta)$.

Market clearing, martingale pricing, and fair pricing conditions enforce mechanical constraints on impact. Under perfect competition and perfect information, the zero-profit requirement ensures $E[p_{t+1}|\mathcal{F}_t]=p_t$ even with autocorrelated order flow (Donier, 2012).

2. Mechanical Origin: Order Book Structure and Matching Rules

Central to the mechanical theory is the structure of supply/demand encoded in the limit order book or, more fundamentally, in the "latent" order book of agent intentions.

• Locally Linear Latent Order Book (LLOB) Approximation: The stationary density of supply/demand near the mid-price is linear, $\varphi(y)=-(J/D)y$, where $y$ is distance to the price. A metaorder "eats" into this linear region. The resulting price trajectory for a metaorder of rate $m_t$ is determined by a self-consistent integral equation (Donier et al., 2014):

$$y_t = \frac{1}{\mathcal{L}} \int_0^t \frac{m_s}{\sqrt{4\pi D(t-s)}}\,ds$$

• Concave Impact Law: For constant-rate execution, the impacted price follows $y_T \propto \sqrt{Q}$, directly yielding the square-root impact law, regardless of informational content.
• Immediate Impact Decomposition: At the trade level, mechanical impact is strictly determined by trade size $\omega$, bid-ask spread $S$, standing book volumes $V^A_i, V^B_i$, and price gaps $G^A_i, G^B_i$ (Zhou, 2012), with regression

$$r_{k,t+1} \sim a\,\omega^\alpha + bS + \sum_i c_i (V^A_i)^\beta + \sum_i d_i (V^B_i)^\beta + \sum_i e_i G^A_i + \sum_i f_i G^B_i$$

Concave exponents $\alpha, \beta$ and book-side-specific coefficients capture the nonlinearity observed empirically.

3. Order Flow Autocorrelation, Optimal Execution, and Impact Decay

Metaorders typically exhibit autocorrelated order flow, modeled as random lengths $l$ drawn from $P(l>L)\sim L^{-\gamma}$, yielding sign autocorrelations $\rho(\tau)\sim\tau^{-(\gamma-1)}$ (Donier, 2012). This introduces rich temporal dynamics:

• Decay of Impact: After metaorder completion, the probability of continuation decays as $t^{-\delta}$ ($\delta=\gamma-1$), so the residual price deviation above $p_\infty$ decays as $t^{-(1-\delta)}$.
• Optimal Execution: Splitting execution into slices and waiting for decay leads to cost savings, quantified as $$\Delta_{\mathrm{slice}}\sim g(\delta) C \ell^{\delta+1}/(\delta+1)$$. The mean–variance trade-off produces schedules analogous to Almgren–Chriss but derived from underlying order-flow autocorrelation.
• Transient Impact Emergence: Even with purely permanent impact at the micro-level, equilibrium among multiple strategic agents (directional trader plus arbitrageur) generates time-dependent "transient" impact kernels. Nash equilibrium analysis yields U-shaped trading profiles and implied empirical decay kernels (Cordoni et al., 2022).

4. Universality and Dimensional Analysis

Dimensional invariance arguments enforce mechanical universality:

• Square-Root Law Necessity: If market impact $G$ depends only on metaorder size $Q$, price $P$, turnover $V$, and volatility $\sigma^2$, plus leverage neutrality, dimensional analysis uniquely yields $G \propto \sigma\sqrt{Q/V}$ (Pohl et al., 2017).
• Empirical Robustness: Square-root scaling emerges regardless of trade identification, trader information, or book dynamics—confirmed in full anonymous exchange data and synthetic metaorder reconstruction (Maitrier et al., 22 Feb 2025).
• Mechanical Contradiction to Informational Theories: Kyle-type models predict $G\propto Q$ or other exponents tied to autocorrelation and information, but universal $\delta=1/2$ is robust against $\gamma$ and volume distribution.

5. Equilibrium, Martingale Pricing, and Absence of Manipulation

Under perfect competition, optimal liquidity provision, and fair pricing constraint, mechanical properties hold:

• Martingale Condition: Enforcement of $E[p_{t+1}|\mathcal{F}_t]=p_t$ for all $t$ rules out profitable manipulation, even with concave bare impact. Over large $N$ independent trends, total impact becomes additive, eliminating round-trip arbitrage (Donier, 2012).
• Impact Partition: Price moves decompose into mechanical (transient, impact kernel-shaped) and informational (permanent, linear) components (Donier et al., 2014). Mechanical impact arises from liquidity consumption; only persistent directional information leaves lasting price shifts.
• Closed-Form Prediction: Across models (Minority Game (Barato et al., 2011), stochastic differential games (Singh, 2021), coarse-grained order-driven equilibrium (Said, 2022)), aggregation of mechanical effects reproduces empirical impact curves including temporary build-up, power-law decay, and permanent plateau.

6. Empirical Regularities and Calibration

Mechanical market impact models reproduce and explain key stylized facts:

• Temporary and Transient Impact: Intraday metaorders yield impact $I_{\mathrm{temp}} \propto R^{1/2}T^{-\gamma}$ with duration reinforcement ($\gamma\simeq0.25$). Power-law transient curves $$I_{\mathrm{trans}}(s)\propto s^{\alpha^{\mathrm{(tr)}}}$$ with $0.5\leq\alpha^{\mathrm{(tr)}}\leq0.8$ describe the impact life cycle (Bacry et al., 2014).
• Decay Regimes: Post-execution impact decays in two regimes: slow power-law ($\sim t^{-0.5}$) immediately after, then faster at longer lags. Daily-scale impact splits into systematic and idiosyncratic components, the latter mean-reverting after de-biasing for subsequent metaorder flow.
• Equation Table: Impact Laws Across Mechanical Models | Model Framework | Impact Law | Principal Exponent | |:--------------------------------|:------------------------------|:---------------------------:| | LLOB/diffusion-reaction | $I(Q)\propto\sqrt{Q}$ | Square-root ($\delta=1/2$) | | Dimensional analysis | $G(Q)\propto\sigma\sqrt{Q/V}$ | Square-root | | Minority game | $I^*=(hT)/(P(1+\chi))$ | Linear for predictable MG | | Bayesian updating | $I(V)\propto\sqrt{V}$ | Square-root |

7. Synthesis and Implications for Practice

Mechanically derived market impact laws—particularly the universal square-root law—reflect a fundamental microstructural response of the matching engine and supply/demand ecology, rather than informational detection, adversarial response, or specific strategic behavior. Properties such as concave impact, optimal execution savings due to decay, absence of price manipulation, universality across markets and regimes, and direct links to volatility are direct consequences of liquidity mechanics, order-flow autocorrelation, fair pricing, and equilibrium risk transfer.

A consequence is that risk models, execution algorithms, and regulatory frameworks can be formulated on purely mechanical grounds, obviating the need for trader-specific information assumptions. The observed excess volatility, impact decay patterns, and dynamic market resilience are thus best interpreted as emergent from supply-side competition, autocorrelated order flow, and microstructural rules rather than informational asymmetry.

The body of work (Donier, 2012, Maitrier et al., 22 Feb 2025, Donier et al., 2014, Pohl et al., 2017, Singh, 2021, Barato et al., 2011, Zhou, 2012, Bacry et al., 2014, Durin et al., 2023, Jusselin et al., 2018) establishes a rigorous foundation for the mechanical origin of market impact, providing comprehensive explanatory, predictive, and prescriptive models for the design and analysis of modern financial markets.