Proxy Metaorders: Definition and Impact

• Proxy metaorders are algorithmically reconstructed sequences that approximate large trading orders split into smaller child orders.
• They enable empirical estimation of market impact using the square-root law, capturing concave price dynamics and decay profiles.
• These methods support optimal execution and liquidity modeling by extracting key trade statistics from anonymized public data.

A proxy metaorder is a statistical or algorithmic reconstruction of a large trading order that has been fragmented into sequential executions. Such proxies are employed when direct identification of true metaorders is impossible—the precise parent order, trader, or algorithm is not observable in the market data. Proxy metaorders play a central role in empirical finance by enabling the estimation of market impact functions, cost models, and liquidity profiles from anonymized or aggregated public trade data. The definition, theoretical modeling, and practical implications of proxy metaorders are deeply informed by a large body of research (Farmer et al., 2011, Maitrier et al., 23 Mar 2025, Maitrier et al., 9 Jun 2025, Maitrier et al., 5 Sep 2025).

1. Concept and Identification of Proxy Metaorders

Metaorders are large orders, typically from institutional traders, that are incrementally executed by splitting into many smaller ("child") market or limit orders. In proprietary datasets, metaorder boundaries may be provided via trader IDs, brokerage codes, or order management system (OMS) tracking. In public trade data, true metaorders are hidden; proxies must be reconstructed algorithmically.

Proxy metaorder identification is performed by algorithmic grouping of temporally and directionally contiguous orders—typically all buy (or sell) orders executed within a certain time window or by a "synthetic" agent. Robust proxy generation methods include sequential grouping based on random mapping functions (assigning synthetic trader IDs in (Maitrier et al., 23 Mar 2025)) or time-thresholding algorithms in synthetic order flow (Maitrier et al., 5 Sep 2025).

A stylized pseudo-code for generating proxy metaorders from public trade data (see (Maitrier et al., 23 Mar 2025, Maitrier et al., 5 Sep 2025)) is:

def generate_proxy_metaorders(trades, num_traders, freq_dist): # Assign synthetic trader IDs based on freq_dist # Group consecutive orders of same sign by the same trader into metaorders # Output: metaorder start/end times, volume, execution trajectory

The reconstructed proxies retain statistically accurate features such as the true metaorder size distribution, average duration, and participation rates, provided the mapping parameters are chosen suitably.

2. Market Impact Laws for Proxy Metaorders

Empirical studies and theoretical models converge on the "square-root law" as the fundamental stylized fact for metaorder impact. Both real and proxy metaorders exhibit a concave market impact profile: the expected peak price change $I(Q)$ satisfies

$I(Q) = Y \sigma \sqrt{Q/V_D}$

where $Q$ is metaorder volume, $V_D$ is daily traded volume, $\sigma$ is daily volatility, and $Y$ is a numerical prefactor (typically $Y\in [0.5,1]$) (Farmer et al., 2011, Bacry et al., 2014, Donier et al., 2014, Maitrier et al., 23 Mar 2025).

The concave profile is captured during execution, with instantaneous impact scaling as

$$ℐ(\phi Q) = \sqrt{\phi} \cdot I(Q), \qquad \phi \in [0,1]$$

This functional form is robustly recovered by proxy metaorder algorithms from public trade data (Maitrier et al., 23 Mar 2025, Maitrier et al., 5 Sep 2025), independent of the grouping method.

Permanent impact—the lasting price change after execution—is observed to plateau at a level $\sim 2/3$ of the peak temporary impact for a wide range of assets and metaorder sizes, consistent with theoretical predictions for Pareto-distributed metaorder sizes (exponent $\beta \simeq 1.5$) (Farmer et al., 2011, Said et al., 2018, Bucci et al., 2019).

3. Dynamic Models and Microstructure Foundations

Theoretical frameworks modeling proxy metaorder impact include:

• Stylized equilibrium models: These posit Nash equilibria among informed traders, fair pricing conditions, and martingale properties; impact scaling emerges from metaorder size distributions and stopping probabilities (Farmer et al., 2011).
• Hawkes process models: Self-exciting processes govern market order arrivals; nearly unstable kernels produce long-memory order flow and power-law impact exponents (Jaisson, 2014, Bacry et al., 2014). Proxy metaorders need not be detected; their impact emerges directly from aggregate observable order flow.
• Propagator and decay models: Impact decays post-execution according to a power-law kernel,

$$\mathcal{I}(Q, z) = I(Q) [z^{1-\beta} - (z-1)^{1-\beta}], \qquad z = t/T \geq 1$$

with $\beta \simeq 0.2$ (Bucci et al., 2019, Maitrier et al., 23 Mar 2025). Empirically, impact relaxes to 2/3 by end-of-day and to $\sim 1/2$ after $\sim$50 days.

• Zero Intelligence and Bayesian models: Microstructural explanations demonstrate that local price impact (per child order) can be linear, but aggregate (proxy) impact is concave due to liquidity rebalancing and order-book mean reversion (Nadtochiy, 2020, Saddier et al., 2023, Ravagnani et al., 7 Mar 2025). Bayesian market-makers set prices by inferring persistent order flow, resulting in statistical square-root scaling without requiring metaorder identification.

4. Proxy Construction from Public Data: Algorithms and Empirical Validation

The main challenge addressed by recent work is realistic proxy metaorder generation from anonymous trade data, overcoming limitations of proprietary datasets (Maitrier et al., 23 Mar 2025, Maitrier et al., 5 Sep 2025). The method involves:

• Random mapping function: Orders are tagged by synthetic trader IDs drawn from a distribution (e.g., power law or uniform).
• Order grouping: Consecutive orders of the same sign by a synthetic trader are grouped as a metaorder.
• Extraction of metaorder statistics: Start/end times, volume, and execution profiles are computed.

Empirical validation across futures and equities demonstrates recovery of (i) square-root market impact, (ii) concave intra-execution profile, and (iii) realistic impact decay. The prefactor $Y$ and exponent $\beta$ agree with proprietary-data studies.

Critical to success is preserving the chronological order of trades and selecting suitable grouping/threshold parameters. Sufficient dataset size and market liquidity are required for robust inference.

5. Crowding, Co-impact, and Correlation Effects

Market impact for proxy metaorders is sensitive to overlapping execution by multiple investors ("co-impact") (Bucci et al., 2018). The total observed impact is a function of the net aggregate order flow:

$$I_N = Y \left( \sum_{i=1}^N \varphi_i \right)^{\cdot 1/2}$$

where $\varphi_i$ are signed volume fractions and $N$ is the number of contemporaneous metaorders. Empirically, three regimes are observed:

Regime	Impact Scaling	Typical $N$ / $\varphi$
Square-root	$I \propto \sqrt{\varphi}$	Large $N$, large $\varphi$
Linear	$I \propto \varphi$	Small $N$, intermediate $\varphi$
Finite intercept $I_0$	$I \to I_0$ as $\varphi\to 0$	High sign correlation

Co-impact is dominated by the sign-correlation coefficient $C_e$ among metaorders. The effect is a non-zero baseline cost for small proxy metaorders in crowded markets.

6. Methodological Challenges and Model Calibration

Proxy metaorder estimation from public data faces several challenges:

• Inability to disentangle metaorder overlap: Long memory in order flow autocorrelation, observed in market data, is driven by overlapping metaorders; statistical models calibrated on public data may incorrectly attribute autocorrelation to child order impact alone (Naviglio et al., 28 Jan 2025).
• Linear vs. concave estimation: Classical models may predict linear price impact and limited post-execution reversion, contrary to the concave and decaying profiles observed empirically.
• Modified Transient Impact Models: To recover realistic market impact, new models introduce a tunable fraction $\alpha$ of child orders triggering additional order flow. Under critical kernel conditions and $\alpha\approx1$, impact becomes permanent; with less triggering ($\alpha<1$), concavity and decay are restored.
• Parameter calibration: Accurate recovery of the true scaling exponents, decay parameters, and prefactors requires careful calibration against empirical datasets (either synthetic or real).

7. Practical Implications and Applications

Proxy metaorder analysis enables:

• Transaction cost modeling: Square-root law and decay profiles can be incorporated into risk and cost estimates for large orders.
• Optimal execution strategy design: Understanding concavity encourages order splitting, timing, and liquidity-seeking behavior.
• Liquidity provision calibration: For market-makers, modeling the nonlinear risk profile is essential for strategy design, especially in the presence of proxy-detected metaorders.
• Research reproducibility: Public-data proxy reconstruction facilitates broader, higher-quality empirical studies, overcoming limitations of proprietary datasets and enabling standardized benchmarks (Maitrier et al., 23 Mar 2025, Maitrier et al., 5 Sep 2025).
• Volatility decomposition: Recent work supports the "order-driven" theory or excess volatility, attributing most observed variance to superposed proxy metaorders rather than exogenous fundamental news (Maitrier et al., 9 Jun 2025).

Summary

Research demonstrates that proxy metaorders—algorithmically reconstructed sequences of trades representing the execution of large, hidden parent orders—exhibit the core statistical laws of market impact: concave (square-root) scaling, realistic post-execution decay, and persistence governed by overlapping and correlated order flow. Theoretical models and empirical algorithms enable recovery of metaorder statistics from public data, providing actionable insights for trading, risk management, and market microstructure research. The universality of the square-root law and the success of proxy construction highlight the mechanical origin of short-term price impact and support the feasibility of metaorder impact estimation even in the absence of direct trader identification (Farmer et al., 2011, Maitrier et al., 23 Mar 2025, Maitrier et al., 9 Jun 2025, Maitrier et al., 5 Sep 2025).