Limit Order Book Forecasting

Modeling limit order book (LOB) dynamics is central to market forecasting and execution-strategy evaluation. While diffusion models achieve strong empirical performance, existing LOB snapshot models typically lack a learned recurrent encoder and formal statistical guarantees. We introduce LOBGrad, a conditional DDPM framework for probabilistic LOB forecasting that combines an autoregressive RNN encoder with a diffusion-based generator. In particular, we establish an excess-risk oracle inequality for the one-step DDPM noise-prediction objective under exponential mixing and regularity assumptions. We further show that, under contractive dynamics, the dependence on the conditioning-window length can be reduced to a dependence on the latent-state dimension. Experiments on BTC/USDT, GOOGL, and META show that LOBGrad improves on its encoder-free diffusion ablation across all reported queue metrics and obtains competitive results against other baselines. It also reproduces the key stylized facts of LOB microstructure, and yields reasonably calibrated queue-based predictive intervals. A stylized downstream task illustrates the potential value of the generated forecasts.

We study the conditional distribution of limit-order volume submitted over the next five minutes, given the current order book and recent order flow. The task requires capturing price placement, total volume, and predictive uncertainty jointly. Coordinate-wise diffusion losses do not explicitly account for price displacement, while normalizing fields to unit mass removes information about total volume. Agreement on selected summary statistics also does not establish fidelity to the joint field distribution. We introduce OrderFlowDiT, a conditional Diffusion Transformer for unnormalized bid and ask volume-at-price fields, trained with an unbalanced optimal-transport (UOT) auxiliary loss. We also develop a full-field evaluation framework that combines a transport cost sensitive to price placement and total volume with proper scoring rules and calibration diagnostics. In the primary comparison on 1950 validation conditions across 30 U.S. equities, adding UOT reduces full-field transport cost by 45.8% and the transport energy score by 26.1% relative to training with mean squared error alone. Both measures improve for every stock. These findings support incorporating price geometry and volume into generative modeling, with field accuracy and predictive uncertainty evaluated together.

Score-based diffusion methods usually perturb data with Brownian Gaussian noise, corresponding to a light-tailed forward stochastic differential equation (SDE). For market time series this SDE can be a poor abstraction: limit order book (LOB) order flow is heavy-tailed, bursty, and occasionally jump-like. We study STOCHLOB, a training-only score-matching framework that changes the forward diffusion corruption law while leaving the deployed predictor deterministic. Instead of Brownian-only Gaussian corruption, the noising process uses two non-Gaussian SDE-inspired alternatives: jump-Lévy noise through a compound-Poisson variance mixture, and clipped $α$-stable noise through a positive-stable Gaussian scale mixture. These corruptions inject finite jumps and heavy-tailed residuals into training without requiring reverse-time diffusion sampling at inference. The resulting score target has a useful interpretation: its radial factor $h_t(r)$ is the posterior expected corruption precision, so large residuals are weighted according to the latent corruption scale that most plausibly generated them. On chronologically held-out Coinbase LOB data, the proposed variants remain competitive on uncorrupted BTC and altcoin test horizons and improve several realized tail-window stress diagnostics. An exploratory Feishu A-share transfer study further tests whether the probability outputs remain usable under transaction costs and trading-rule constraints. Code and configuration files are available at https://github.com/ArshiAbolghasemi/Penny.

Autoregressive transformers trained on limit order book (LOB) message streams are increasingly proposed as foundation models for market microstructure. Because the generative process of real markets is unknown, these models can be judged only by the realism of their output, leaving open what they actually learn. We make the question answerable by inverting the setup: we train GPT-style models on order flow from simulators whose latent parameters are known, so that the Bayes-optimal predictor, the generating state, and the exact data entropy serve as references. Measured against them, next-token pretraining on order flow implements amortized Bayesian inference over market regimes. The learned predictive matches the exact Bayes optimum to within a millinat per token and approaches the oracle floor even on a nonstationary family whose latent state drifts within an episode, a regime that requires filtering. The latent regime is linearly decodable from frozen activations, well above baselines, so a frozen-model read-out calibrates simulator parameters more accurately than windowed maximum likelihood at a fraction of the cost. In our experiments these capabilities are not produced by scale: capacity and token volume saturate early, and it is the diversity of training regimes, not the token count, that determines generalization to unseen markets. On real NASDAQ data, zero-shot transfer is set by how well the simulator prior covers real flow: an enriched mixture nearly closes the gap to an in-domain baseline, and pretraining beats from-scratch training at every budget across held-out tickers. Known ground truth turns average next-token loss, which is uninformative near the entropy floor, into an exact comparison against the Bayes-optimal and oracle predictors.

Forecasting short-horizon mid-price movements from limit order book (LOB) data is central to algorithmic trading, yet most deep LOB forecasters are point predictors: they output a direction or a displacement, but never indicate which of their forecasts can be trusted. We introduce UQ-LOB, a lightweight, encoder-agnostic uncertainty quantification module that attaches to any pretrained LOB encoder and, in the spirit of attentive neural processes, conditions each forecast on a context set of recently completed windows whose outcomes are already realised. The UQ-regression variant outputs a calibrated Gaussian over the future tick displacement, while the UQ-classification variant outputs a categorical distribution over down/up/stationary. Both expose a scalar confidence (predicted signal-to-noise ratio or class probability) that supports selective prediction. On 5.2 billion LOB events across seven cryptocurrency assets and horizons of 5, 10 and 15 seconds, UQ-regression attains near-nominal 68% interval coverage, and restricting to the most confident 10% of predictions raises directional macro F1 by 0.11-0.15 for UQ-regression and 0.05-0.11 for UQ-classification, at every horizon. On large, economically meaningful moves, the tightest confidence tier reaches a directional F1 of 0.88 (down) and 0.83 (up) at the 5-second horizon.

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.

A large language model (LLM) trained on synthetic limit order book (LOB) data achieves near perfect scores in generating valid sequences of LOB events. However, the LLM's implicit world model fails to learn the state of the LOB. This deficiency leads to biased estimates and spurious predictability in using the LLM to forecast future LOB events. Our analysis uses novel tests of an LLM's world model, extending prior work from deterministic settings to the stochastic dynamics needed for the LOB.

We study whether a scaling-law-style inference-compute frontier appears in limit order book prediction. Using FI-2010 and a suite of models ranging from small decision trees to neural LOB architectures, we find that the realized empirical frontier of predictive loss versus structural forward work is well summarized by a power law. In particular, with MLPLOB held out as an architecture family, a power-law fit to the low- and mid-compute non-MLPLOB frontier extrapolates across multiple orders of magnitude and attains $R^2=0.941$ on the excluded high-compute MLPLOB target frontier. A similar exercise in latency space gives substantially weaker results, showing that latency is not merely noisy compute. We use this gap to motivate FastBiNLOB, a dense axis-separable LOB mixer built from hardware-friendly temporal and feature mixing operations. In a five-seed experiment, FastBiNLOB exceeds the published $y_{10}$ and $y_{100}$ macro-F1 targets at notably lower latency than existing published SOTA architectures.

Modeling the dynamics of financial Limit Order Books (LOB) at the message level is challenging due to irregular event timing, rapid regime shifts, and the reactions of high-frequency traders to visible order flow. Previous LOB models require cumbersome data representations and lack adaptability outside their original tasks, leading us to introduce LOBERT, a general-purpose encoder-only foundation model for LOB data suitable for downstream fine-tuning. LOBERT adapts the original BERT architecture for LOB data by using a novel tokenization scheme that treats complete multi-dimensional messages as single tokens while retaining continuous representations of price, volume, and time. With these methods, LOBERT achieves leading performance in tasks such as predicting mid-price movements and next messages, while reducing the required context length compared to previous methods.

Real-time market prediction services need correct predictions before a decision deadline; a correct prediction delivered late is not usable. TIP-Search studies time-predictable inference scheduling over fixed market predictors under uncertain load. It filters conformal latency-quantile feasible models, dispatches over finite workers, and uses shielded constrained online experts to trade accuracy, queue pressure, and deadline risk. On the optimized deployable pool, TIP-Search reaches 0.994 raw accuracy and 0.991 timely accuracy. On official TLOB FI-2010 h=10, TIP-Search++ raises timely accuracy from 0.156 to 0.239 and deadline satisfaction from 0.391 to 0.962. In matched h10 profiled systems replay, OCO-ACPO reaches 0.303 timely accuracy and 0.951 deadline satisfaction, with paired gains over RAMSIS/SneakPeek/utility-style comparators of $+0.00285$ timely accuracy ($p=0.0118$) and $+0.0146$ deadline satisfaction ($p=1.5{\times}10^{-5}$). SA-OCO-ACPO improves timely/deadline service by 0.188--0.417 over CPO under nonstationary stress. The claim is a systems scheduling result, not a broad LOB classifier leaderboard.