Daily AI Trading Paper

Today’s paper is about a very specific but very important forecasting problem in limit-order-book markets: given the current book and recent order flow, can we predict not just whether liquidity will arrive over the next few minutes, but how much will arrive and where it will be posted in price space? That matters because in electronic markets, liquidity is not static. Traders care about how quickly the book replenishes after trades and cancellations, where new orders are likely to cluster, and how aggressive those orders will be relative to the mid-price. For execution, market making, and short-horizon trading, the shape of incoming limit-order supply affects expected costs and the available opportunity set. A forecast of the future book state alone is not enough, because the book at the end of a horizon reflects not only new submissions, but also cancellations and executions. This paper instead focuses on one component of the future trading environment: the new limit-order volume submitted over the next five minutes, split into bid and ask fields across price levels. The key modeling issue is that this is not just a scalar prediction problem. We want a joint distribution over two price-by-side fields. That means the model needs to capture total submitted volume, the allocation of that volume across prices, and uncertainty about both. This is where many standard approaches fall short. If you normalize each field to unit mass, you lose information about how much volume was actually submitted. If you only forecast total volume, you lose the price geometry. And if you only match a few summary statistics, you can still miss the true joint distribution. Two forecasts can have the same mean field but very different uncertainty and very different probabilities of low-volume or shifted-volume outcomes. The paper’s broader context sits at the intersection of market microstructure and generative modeling. There is a long line of work on queue-reactive models, Hawkes-style order-flow models, agent-based simulators, and more recent data-driven generators for messages, book states, and market dynamics. Diffusion models are attractive here because they can generate complex conditional distributions, not just point estimates. But the usual diffusion training loss is coordinate-wise mean squared error on denoising, which does not explicitly care about price displacement. If volume moves from one price level to a nearby one, MSE treats that as just a set of coordinate errors, not as a geometrically small shift. In market terms, though, moving volume one tick away is much less severe than moving it many ticks away. That is the central motivation for introducing an optimal-transport-based auxiliary loss. The model, called OrderFlowDiT, is a conditional Diffusion Transformer. It generates unnormalized bid and ask volume-at-price fields over a fixed price grid relative to the mid-price at the forecast origin. The fields are unnormalized on purpose: the model keeps absolute shares, so total submitted volume is part of the prediction target, not something normalized away. Each coordinate in the field records cumulative new limit-order volume submitted on one side within a price bin during the five-minute forecast window. The input condition is the current order book snapshot plus recent order flow, which is consistent with the idea that both state and flow history matter for liquidity dynamics. The main methodological addition is an unbalanced optimal transport, or UOT, auxiliary loss. Standard optimal transport compares two measures by matching mass across locations according to a ground cost, which is exactly the kind of geometry-aware comparison we want for price levels. But standard balanced transport assumes equal total mass, which is not appropriate here because predicted and realized submission fields can have different total volumes. Unbalanced optimal transport relaxes that requirement by allowing mass mismatch through divergence penalties. In this paper, the UOT term is implemented as a debiased unbalanced Sinkhorn divergence on the decoded denoised fields, and it is added to the diffusion MSE objective. So the model is still trained as a diffusion model, but the auxiliary signal explicitly tells it to care about both spatial displacement and volume mismatch. That distinction is important. MSE alone can learn to place mass roughly in the right region, but it is blind to the geometry of the price grid. The UOT term gives the model a more market-relevant notion of error: if the forecast puts volume at the wrong price, the penalty depends on how far off it is, and if the total volume is wrong, that also matters. In other words, the training objective is aligned more closely with the structure of order-book supply. The evaluation framework is also a contribution. The authors argue that standard market-simulation-style checks, such as matching summary statistics, are not enough to establish fidelity of the full conditional distribution. So they combine several complementary metrics. One is a mass-aware full-field transport cost that penalizes both price displacement and total volume differences. They also use transport and joint-field energy scores, which are proper scoring rules for predictive distributions, along with CRPS for side volumes and calibration diagnostics. That is a good fit for a generative forecasting problem, because the goal is not only to generate realistic samples, but to represent uncertainty faithfully. Experimentally, the paper compares MSE-only training against MSE plus UOT on 1,950 validation conditions across 30 U.S. equities. The primary reported result is that adding the UOT auxiliary loss substantially improves full-field accuracy. Relative to the MSE-only baseline, Base-UOT reduces full-field transport cost by 45.8% and transport energy score by 26.1%. Importantly, both of those measures improve for every stock in the test set, which suggests the gain is not driven by a small subset of names. The paper also reports improvements in joint-field energy and log-side-volume CRPS, indicating that the model is not only better at placing mass on the price grid, but also better calibrated on side volumes. There is also a fine-tuned variant using Group Relative Policy Optimization, or GRPO, applied to diffusion generation. That variant improves transport energy and volume CRPS further, but it comes with trade-offs: full-field cost, joint-field energy, and pooled volume-rank calibration worsen, and ensemble dispersion is reduced. So the paper does not present GRPO as a universal improvement; rather, it shows that reward-based fine-tuning can shift the balance between accuracy and diversity, and those trade-offs need to be judged metric by metric. What should we take away from the results? First, the paper supports the idea that geometry matters in order-flow generation. If you are forecasting volume-at-price fields, it is not enough to ask whether the model gets the right total volume or the right marginal statistics. You need a loss and an evaluation metric that understand where the volume sits on the price ladder. Second, the results show that adding a transport-based auxiliary objective can materially improve a conditional diffusion model without abandoning the probabilistic framing. Third, the evaluation itself is a useful contribution because it separates field fidelity from uncertainty quality. A model can look good under one score and still be overconfident or too concentrated under another. There are also limitations to keep in mind. The target here is submission flow, not the entire order-book evolution. Cancellations and executions are outside the modeled quantity, so a generated submission field does not by itself produce a full market simulator or determine the next book state. The forecast horizon is fixed at five minutes, so the setup is tailored to short-horizon liquidity supply rather than longer-term dynamics. And while the paper uses a richer evaluation suite than many prior studies, any forecast of a high-dimensional field remains sensitive to how the price grid is chosen, how calibration is measured, and how much sample diversity the generator preserves. Still, the main message is clear: for limit-order supply forecasting, the combination of diffusion modeling and unbalanced optimal transport gives a more market-aware way to generate probabilistic price-volume fields. The paper shows that if you want to model liquidity provision realistically, you should treat price placement, total volume, and uncertainty as a single joint problem, not as separate afterthoughts.