LOB-Impact Benchmark: Emergent Market Impact in Generative Limit Order Book Models

1 Introduction

• Distributional benchmarks leave counterfactual response to an agent’s own orders largely unmeasured, although mispricing that response can produce policies that are wrong for real markets.Existing benchmarks measure response only for a single event.Source: §1 · p. 2
• The square-root law relates expected price change to metaorder size, daily volume, volatility, and a dimensionless amplitude; its concavity reflects liquidity replenishment and resilience.Persistent trade signs coexist with diffusive prices because the market absorbs and relaxes from each push.Source: §1 · p. 2
• Earlier injected-metaorder studies report square-root-shaped or near-square-root impact, but do not separate mechanical price impact from other participants’ reactions.A historical trace cannot provide the counterfactual; in a generator, the model itself supplies the response whose resilience is being measured.Source: §1 · p. 2
• The benchmark injects metaorders into five neural state-space generators and identifies an impact law in all 25 model–stock pairs, pairing each build-up injection with a blind run to isolate model response from book mechanics.It adds automatic per-stock calibration and reports a number only when a law is identified.Source: §1 · p. 2

2.1 Benchmark design

• Free-generation rollouts compare trade-sign persistence with price diffusivity, measure uninjected drift and single-trade response, and serve as a first filter before intervention.A replay can pass this phase while showing no metaorder impact, so an intervention is required to establish impact.Source: §2.1 · p. 2
• Build-up injections alternate aggressive children with generated messages at 10% participation; visible and blind runs on matched windows and seeds isolate the model’s indirect response.The blind orders fill in the simulator but are hidden from model inputs; buy and sell responses are antisymmetrised, while free drift is measured separately.Source: §2.1 · p. 2
• Impact is fitted on cumulative participation prefixes, with an identifiability gate applied before reporting the exponent and amplitude.The gate rejects amplitudes below the floor or optima at grid bounds; planted laws and null trials validate detection, while kernel baselines test recovery of known relaxation.Source: §2.1 · pp. 3–4
• Post-execution relaxation is measured by the settled impact level and its slope, with days pooled on a common executed-fraction clock.The analysis compares settling levels with real metaorders and kernel baselines, and marks cells where fast and slow days still differ materially.Source: §2.1 · pp. 3–4

2.2 Models under test

• The model panel accepts any message-level generator and includes replay, heuristic, kernel, and five neural generators with continuous hidden state across generated and injected messages.The replay and kernel baselines span no causal response, permanent impact, and built-in resilience; neural models share a corpus that includes AMD.Source: §2.2 · p. 4, Table 1 · p. 5

2.3 Data preparation

• Per-stock calibration uses median market-order size for child volume, daily volume for participation normalization, and market-order share to set child spacing.The 100-child metaorder is fixed across models; selected stocks span distinct generation burdens, with AMD in the high-OTR cohort at ΔN = 401 ± 69.Source: §2.3 · p. 4, Table 2 · p. 4, Figure 3 · p. 5

3.1 Phase 1: Free generation

• On AMD, every neural generator fails C1: none combines the real stream’s persistent signs and diffusive price behavior without free drift.The real stream has γflow = 1.66 and H = 0.54; neural generators drift from +9 to −251 ticks per rollout.Source: §3.1 · p. 5

3.2 Phase 2: Metaorder response

• On AMD, the five neural generators have identified exponents of 0.66–0.93, with 95% day-clustered intervals of ±0.05–0.06, all far from 0.5.The S5 family is closest to 0.5, but no interval comes near it.Source: §3.2 · p. 5
• On AMD, neural amplitudes are 2.2–8.6 and a 100-child metaorder moves simulated prices 17–70 bps, against at most 11 bps for any baseline.The neural amplitudes are reported as four to seventeen times what a real metaorder pays per unit of participation.Source: §3.2 · pp. 5, 8

3.4 Extending to a larger universe

• Across all 25 neural model–stock pairs, impact laws are identified with exponents above 0.5; C1, C4, and C5 fail, while C2 and C3 pass.The exponents range from 0.63 to 1.21, and all 25 day-clustered intervals exclude 0.5.Source: Table 3 · p. 7, §3.4 · p. 8

3.2 Phase 2: Metaorder response

• On AMD, neural exponents overlap replay-kernel exponents, but neural amplitudes exceed every baseline; visible peaks are at least 3× the blind floor in all 25 cells.This contrast is the benchmark’s evidence of impact beyond book mechanics.Source: §3.2 · p. 8

3.3 Phase 3: Relaxation

• On AMD, four neural models end above peak after execution; GDN ends below peak, but its −127-tick free drift prevents attributing that decline to resilience.No neural cell passes the relaxation criterion.Source: §3.3 · p. 8

3.4 Extending to a larger universe

• On EA, outside the training corpus, neural amplitudes are Y = 12–50; the observed out-of-distribution result preserves the law but not its calibration.For the four training-corpus stocks, reported amplitude ranges differ by stock.Source: §3.4 · p. 8
• Distributional ranking and impact-law ranking are unstable or negatively correlated in the reported comparisons, so a better distributional mimic can sit farther from the impact law.The reported correlations range from +0.54 on GOOG to −0.71 on MSFT; within MSFT neural rows, the correlation is −0.90.Source: §3.4 · p. 8

• Across the neural panel, impact exists and is the generator’s own, but its dynamics are not market-realistic.Source: §4 · p. 8
• Persistent flow with insufficient replenishment makes impact build too steeply and prevents credible relaxation, consistent with the joint C4/C5 failure.Source: §4 · p. 8
• Execution cost models depend on impact shape, amplitude, and relaxation, so wrong endogenous impact can reward schedules a real book would punish.Source: §4 · p. 8
• At checkpoints, Phase 1 should target stock-specific empirical flow persistence and Hurst values, repairing replenishment rather than eliminating flow memory; Phases 2–3 test build-up and relaxation.Source: §4 · p. 8
• Only one stock is shown band by band; all results use one participation rate, child size, aggressive children, metaorder shape, month, and exchange.Source: §4 · p. 8
• A month atypical by about one within-month standard deviation would move the exponent past its half-width cap.Source: §4 · p. 9

• Five next-token-trained neural message generators produced identifiable impact laws in all 25 model–stock pairs, while a replay unable to react did not.Source: §5 · p. 9
• The models’ impact builds more steeply than the square-root reference and shows no credible post-execution relaxation.Source: §5 · p. 9
• The authors tentatively attribute the failures to insufficient replenishment linked to limited long-horizon memory from still-short context windows.Source: §5 · p. 9