Order Book Generators Reproduced Market Impact With an Exponent Just Above 0.5

A group from Oxford's BOLD lab (Department of Engineering Science and Department of Computer Science; the authors include Stefan Zohren) has released LOB-Impact, a benchmark. It tests something that standard quality metrics for generative limit order book models do not capture: how a simulator reacts to orders from the trading agent itself. The result: market impact emerges on its own in neural generators, even though they were trained only to predict the next message. The exponent of the impact law came out slightly above 0.5, that is, above the classic square-root-law value. The work will be presented at the 7th ACM International Conference on AI in Finance (ICAIF ’26) in Milan on 14–17 November 2026.

Why test the response to your own orders

Generative order book models (GANs, recurrent and state-space generators, transformers, diffusion models) are currently evaluated by how closely their message flow resembles the historical one. Benchmarks such as LOB-Bench compare the distributions of spread, volumes, inter-event times and imbalance. But people want to use these simulators as "world models" for reinforcement learning. If a simulator misjudges the price of the agent's own trades, the agent will learn a strategy that is optimal inside the simulator and wrong in the real market. Distributional metrics do not detect this skew.

The reference for the test is the square-root law: the expected price change caused by a large order is proportional to daily volatility and to the square root of the ratio of order size to daily volume. It has been confirmed on equities, futures and the foreign exchange market. According to the paper, it rests on two forces: order flow is persistent (the sign of the next trade is predictable from the previous one), yet the price is still diffusive, because the market restores liquidity and "absorbs" the shock.

Three phases and a blind control

The benchmark takes a model, a stock, a calibration on that stock's data and a budget of runs. It then proceeds in three phases. In the first, the model generates flow freely, and the benchmark measures the decay of trade-sign autocorrelation, the Hurst exponent of the mid-price and the free drift.

In the second, a metaorder is inserted into the generated flow: after a context window of real flow, aggressive "child" orders of median size alternate with the model's messages. A single run lasts from 12 to 73 thousand messages, depending on the stock, far longer than the 4000-message window on which the neural models were trained. In the third phase, the injections stop and relaxation is measured.

Buys and sells are mirrored and pooled, so direction-independent drift is reduced.

The main technique is a "blind" run on the same windows and seeds. The orders are executed in the simulator, but neither the messages nor the order book state that the model sees know about them. The ratio of the "visible" response to the "blind" one shows how much of the impact the model itself created and how much came from order book mechanics. This matters: simple replay baselines with no model inside them sometimes also have an identifiable law, so the exponent alone is not enough for attribution.

Before interpreting results, the authors validated the estimator. Laws with a known exponent, injected into the real point cloud, are recovered with a slight downward bias, so the reported exponents are conservative. In null trials, the test found no spurious laws. Baseline models such as the Obizhaeva–Wang kernel (a standard order book model with exponential liquidity recovery) correctly recover the persistence built into them on all five stocks.

What worked and what did not

The models were trained on data from January 2022 to December 2025, with January 2026 held out for evaluation. The models tested include S5-120M and a Mamba-3-based model, both with a 4000-message window; one run on an H100. In total there are five neural generators and five NASDAQ stocks (AMD and others). Historical replay passes the first phase perfectly but shows no impact at all, which is exactly why an intervention is needed to detect it.

In every cell where the law was identified, the neural models produced impact with an exponent slightly above 0.5. The weak spot is relaxation: in reality, impact falls to roughly two-thirds of its peak within one to two execution durations, and to half by the tenth. The models' neural decay kernel came out constant under two independent methods. The authors cautiously link this to the short context windows.

The limitations are clear: the paper shows that the effect is present within the protocol, but it does not prove that an agent trained in such a simulator would be profitable on a live exchange. The practical takeaway for developers of trading simulators is that counterfactual impact checks are worth adding to flow-plausibility metrics. The authors are releasing the benchmark code as open source.