LOB-Impact Benchmark: Emergent Market Impact in Generative Limit Order Book Models
Authors: George Nigmatulin, Valentin Mohl, Sascha Frey, Kang Li, Aramis Marti-Shahandeh, Mihai Cucuringu, Anisoara Calinescu, Jakob Foerster, Stefan Zohren
Organizations: British Open-ended Learning and Discovery Lab (BOLD), Department of Engineering Science, University of Oxford, Oxford, United Kingdom · British Open-ended Learning and Discovery Lab (BOLD), Department of Computer Science, University of Oxford, Oxford, United Kingdom · British Open-ended Learning and Discovery Lab (BOLD), Department of Statistics, University of Oxford, Oxford, United Kingdom · Department of Mathematics, University of California Los Angeles, Los Angeles, United States · Department of Statistics & Oxford Man Institute of Quantitative Finance, University of Oxford, Oxford, United Kingdom
Realistic market impact is essential for generative limit order book (LOB) models to serve as world models for trading agents. If a simulator misprices this mechanism, agents learn policies that are optimal in the simulator but wrong in the real market. Existing metrics assess generated streams purely through distributional comparison with historical data, leaving counterfactual response to an agent's own orders largely unexplored. Here, we show that market impact emerges naturally in neural LOB generators trained solely for next-token prediction. We inject execution-style metaorders into the generated message flow of five autoregressive models across five NASDAQ stocks and measure the response using estimators from the empirical impact literature, validated on baselines whose physics is known in advance. Our benchmark evaluates any message-level generator in three phases: free generation (reading order-flow persistence against price diffusivity); metaorder injection (yielding the impact law's exponent and amplitude); and a post-execution window (measuring relaxation). In addition, comparing each visible injection to a blind, simulator-only control isolates the model's learned response from standard book mechanics. For every model on every stock, we measure how the impact responds as a function of cumulative metaorder participation. Interestingly, we observe emergent impact in generative models of market data, with exponents in the range of 0.63 to 1.21, slightly above the traditional 0.5 of the square-root law of impact. In the family of models tested, we are so far unable to observe impact decay during relaxation. We tentatively attribute this to the still-short context windows of the models. We release the benchmark as open source to validate future generative models for market data.
Figures & tables
Figure 1. The benchmark and its counterfactual probe. Left: four inputs, three phases, and the claim ladder they decide, each row reading claim ⋅ variable ⋅ pass criterion . Phase 1 generates freely and decides the balance (C1); Phase 2 injects the build-up and decides existence (C2), attribution (C3) and the square-root law (C4); Phase 3 injects, then falls silent, and decides relaxation (C5). Schedules print as (what is injected; children i ; silent blocks c ). Every build-up injection runs twice, visible and blind : same windows and seeds, the orders fill in the simulator while neither model input (message stream, book state) carries them, and the visible-over-blind ratio attributes the response to the model itself (C3). Table 3 prints the resulting ladder for every model × stock. Right: the injection protocol. After a conditioning window ( Context ), aggressive children are injected ΔN model-generated messages apart; the post-execution phase injects none. The traced paths are the real-market expectation the benchmark tests against: a transient response decaying towards a permanent level. The sell experiment mirrors the buy; the two combine by antisymmetrisation (Eq. 2 ), and the experiment repeats over many windows (stacked frames). Two-part schematic. Left: a flow diagram of the benchmark, from four inputs through three phases (free generation, metaorder injection, post-execution) to five claims, C1 to C5, each with its variable and pass criterion. Right: timelines of the injection protocol, showing a context window, child orders injected at regular message intervals during execution, and a silent post-execution window, with the expected price path rising during execution and decaying afterwards.
Figure 2. The estimator’s input on AMD. Rows: the Historic replay (negative control, top) and S5-120M (bottom). Columns: raw signed points (lnq,I) (left) and equal-count bins with signed conditional means (right). Half the replay’s points lie at or below zero; signed means cancel where there is no signal, so the replay’s means straddle zero while S5-120M’s rise. The estimator built on these means is defined in the text. Four panels for AMD in a two-by-two grid; the top row is the historic replay and the bottom row is the S5-120M generator. The left panels show raw signed impact points against log participation. The right panels show equal-count bins with signed means, which straddle zero for the replay and rise for S5-120M.
Model
Class
How it works
Trained / calibrated on
Gen.
Historic
replay
Replays the real stream unchanged; cannot react.
real held-out data
—
Heuristic
replay + shift
Replay; a consumed best level shifts all later messages.
rule fixed a priori
—
Propagator
replay + kernel
Replay + one-tick kicks decaying as 32+31ℓ−1/2 .
Table 1. Models under test. Top block: non-neural baselines, calibrated (where anything is fitted) on the evaluation stock itself. Bottom block: neural generators sharing one corpus (merged cell) that includes AMD, so every neural measurement is in-distribution. “Gen.” is the wall-clock for one build-up rollout ( ∼40 k messages): replay baselines transform a stored stream (—), parametric baselines run on CPU, neural models on one H100. † Fine-tuned with a 2000-message context and run with a 4000-message window.
Figure 3. The universe and the calibration it fixes (day-mean over 487 names, January 2026). Left: the median market-order size, which sets the child volume. Middle: the market-order fraction of the message stream, the denominator of Eq. 3 . Right: the resulting ΔN , the messages the generator must produce between children at η=10% ; shaded bands are the OTR cohorts, vertical lines the universe median and the five selected stocks. Since ΔN is the generation burden of the same experiment, the five span the universe minimum (EA), the modal band (NVDA, MSFT), the high-OTR cohort (AMD) and its tail (GOOG). Three histograms over 487 stocks: the median market-order size, the share of messages that are market orders, and the number of generated messages between child orders. Shaded bands mark OTR cohorts, and vertical lines mark the universe median and the five selected stocks.
Stock
Cohort
Child ( ± std)
ΔN ( ± std)
ϕ
gen
EA
Low OTR
17(±4)
122(±15)
7.49%
12 k
NVDA
Mid (modal)
31(±7)
250(±35)
3.65%
25 k
MSFT
Mid (modal)
19(±3)
261(±59)
3.68%
26 k
AMD
High OTR
20(±3)
401(±69)
2.30%
40 k
GOOG
High OTR (tail)
23(±2)
729(±183)
1.30%
73 k
Table 2. Selected stocks, one per cohort (day-mean, whole shares, January 2026). “Child” is the median market-order size, used as the child size; ± = std across days; “gen” is the messages a 100 -child build-up metaorder must generate. Figures 2 and 4 show AMD; the others are in Table 3 only.
Figure 4. The AMD measurements, all three phases in one figure ( 5×2 ). Top, Phase 1 : pseudo-metaorders on the no-injection arm (the pipeline returns drift, not impact); the trade-sign autocorrelation; the mid-displacement variance (slope =2H ); the drift-corrected own-trade response R(ℓ) . Middle, Phase 2 : antisymmetrised I(k) against the ⋅ -law reference (full range inset), the master curve, and the binned estimator along the metaorder ( δ(k) and Y(k) , both against 0.5 ). Bottom, Phase 3 : I(k) past the end of execution, and the master curve against the theoretical reference (dashed: v build-up, then power-law decay to 32 ). Apart from GDN’s drift, only the curves with resilience typed in come down. Ten panels for AMD in five rows of two. The top rows show Phase 1: pseudo-metaorder impact without injection, trade-sign autocorrelation, mid-price variance scaling and the own-trade response. The middle rows show Phase 2: impact against a square-root reference, a master curve, and binned estimates of exponent and amplitude along the metaorder. The bottom row shows Phase 3: impact after execution ends and the master curve against a theoretical decay reference.
Model
Par.
Ctx
KS
L1
WS
γfl
H
drift
δ
Y
v/b
e/p
slp
C1–C5
AMD
target
—
—
low
low
low
1.66
0.54
≈0
0.5 (exp)
0.5 (exp)
—
32 (exp)
∼−1.5 (exp)
—
Historic
—
—
.060
.162
.096
1.66
0.54
+1
( 0.11 ) u
( 0.03 ) u
—
( −0.00 ) u
( −0.5 ) u
✓ ✗ ⋅⋅⋅
Heuristic
2 f
∞
.060
.162
.096
1.66
0.54
+1
0.81
1.46
—
0.95
−0.3c
✓ ✓ ⋅ ✗ ✗
Propagator
2 f
∞
.060
.162
.096
1.66
0.54
+1
0.76
0.98
—
0.89
−0.5
✓ ✓ ⋅ ✗ ✗
OW
2 f
exp.
.041
.106
.070
1.66
0.54
+1
0.73
0.96
—
0.81
−0.3
✓ ✓ ⋅ ✗ ⋅
Table 3. All five stocks, one row per model, the same measurements everywhere (the AMD block is the one read band by band in Section 3 ). Columns: Par. = parameters; Ctx = conditioning window; LOB-Bench KS/L1/WS on free-generation rollouts against the real stream, 500-message windows, lower is better; Phase-1 flow balance ( γfl = trade-sign memory, H = diffusivity) and free drift (ticks); Phase-2 build-up ( δ = binned exponent, where the direct- L2 profile identifies one; ± = half-width of the 95% day-clustered bootstrap interval, the trading day being the resampling unit; Y = amplitude at the data edge; v/b = visible peak over blind floor, the C3 ratio); Phase-3 relaxation (e/p = end-over-peak; slp = post-execution slope in % of peak per execution duration). target rows: the stock’s real stream in identical windows; (exp) = the empirical value for real markets, from the literature. C1–C5 = the claim ladder of Figure 1 , one verdict per claim (✓ pass, ✗ fail, ⋅ not decidable) on the criteria of Section 2 : C1 the balance holds ; C2 impact exists ; C3 not book mechanics ; C4 the square-root law ; C5 the impact relaxes . The blind arm behind C3 runs for the neural panel only. Shaded bold : the two numbers behind the neural panel’s failed C4 and C5 – δ against the 0.5 of the square-root law, e/p against the real-market 32 relaxation (the kernel rows bound this schedule) – the one signature all 25 neural cells repeat. Markers: “—” does not exist; n.i. = not identified; ( ⋅ ) u = below the identifiability or sample-size requirement, shown for reference only; c by construction; m duration-mixed; f fixed a priori; † 2k fine-tune at 4000; ‡ EA out-of-distribution.
We propose ReLOBGen, a method for generating limit order book (LOB) messages that are replayable by construction. Replayability is required for closed-loop market simulation, yet existing LOB message generators may produce non-replayable raw messages, i.e., messages inconsistent with the current market state. These generators therefore rely on post-hoc correction or rejection followed by resampling, which may alter the replayed message distribution or increase inference cost. ReLOBGen instead ensures replayability during generation: it selects the referenced order from the resting orders in the current LOB and then generates the remaining message fields to be consistent with that order and the market state. For realistic reference selection, ReLOBGen samples from a learned distribution over eligible resting orders, efficiently computed from cached order representations and a context-dependent query. It then enforces the consistency of the remaining fields by masking out invalid tokens. Together, these components enable efficient generation of realistic messages without post-hoc correction or resampling. In 500-message rollouts, ReLOBGen achieves 100% replayability, improves market realism, particularly for top-of-book statistics and the relative prices of LOB messages, and provides a 2.7-3.6× speedup per replayed message over the LOBS5 baseline.
Junoh Kang, Kiseop Lee, Bohyung Han
ECE & 3IPAI, Seoul National University · Department of Statistics, Purdue University · 3IPAI, Seoul National University
Deep Limit Order Book forecasting models capture nonlinear market dynamics, but their ability to quantify the effects of counterfactual order book messages has not been systematically validated. We introduce a model-agnostic framework that compares a trained forecaster's predictive distributions before and after injecting mechanically valid counterfactual messages, defining short-horizon model-implied market impact. A Transformer-based forecaster recovered scenario rankings with a Spearman correlation of 0.99 and 97.2% directional agreement with realized historical outcomes among non-neutral scenarios. Observation-level analysis further showed that estimated impacts captured incremental sequence-dependent variation beyond scenario identity and the pre-event forecast. These results provide evidence that pretrained Limit Order Book forecasters can be repurposed for scenario-conditioned response modeling without retraining.
Eljas Linna, Kestutis Baltakys, Derrick Manoharan +2
Data Science Research Center, Tampere University, Finland
Generative models of limit orderbook (LOB) data have advanced rapidly, but their evaluation often focuses on stylised facts and selected market statistics. These measures provide useful diagnostics but may not capture the joint temporal and cross-level structure of order-book trajectories. We introduce LOB-ID, an embedding-based framework that adapts the Fréchet Inception Distance (FID) and Monge Inception Distance (MIND) to LOB data. To obtain domain-specific embeddings, we train the DeepLOB architecture on four months of Level-2 order-book data for five equities. We show that LOB-ID is stable across time, instruments, and embedding checkpoints, and rises monotonically under controlled distortions. We then construct a moment-matching attack against FID and a deep-book perturbation that evades statistic-based evaluation. MIND remains substantially more sensitive to both distortions. Finally, we score five generative LOB models, spanning stochastic baselines and deep learning approaches, and find that LOB-ID ranks them in line with the joint temporal and cross-level structure each captures by construction.