Predicting Order Book Dynamics with KANFormer

Author: Denis Avetisyan

A new deep learning model leverages survival analysis to forecast the likelihood of limit orders being filled, offering a powerful tool for algorithmic trading.

The KANFormer architecture processes local observation blocks-snapshots $𝐗_{LOB} \in \mathbb{R}^{L\times(4n+4)}$-through dilated causal convolutions and KAN-Transformers to generate embeddings, concurrently encoding agent actions $𝐀_{actions} \in \mathbb{R}^{L\times(d+5)}$ via a KAN-Transformer, then concatenating these representations with queue position to predict density and survival functions.

KANFormer integrates Kolmogorov-Arnold Networks with limit order book features and agent actions to predict fill probabilities and time-to-fill.

Accurately predicting order execution remains a challenge in dynamic limit order books despite advances in high-frequency trading. This paper introduces KANFormer, a novel deep learning model for predicting fill probabilities via survival analysis that integrates market dynamics with agent-level actions. By combining dilated convolutional networks, Transformer encoders, and Kolmogorov-Arnold Networks, KANFormer demonstrably outperforms existing methods in both calibration and discrimination using CAC 40 futures data. Can this approach, leveraging rich market signals and expressive neural architectures, unlock more nuanced and interpretable predictions of transactional likelihood in complex financial environments?

The Imperative of Accurate Time-to-Fill Prediction

Predicting the duration until an order is filled – the $Time-to-Fill$ – represents a critical component of successful automated trading strategies, directly impacting execution costs and profitability. However, traditional statistical methods often falter when applied to the intricacies of the $Limit Order Book$ (LOB). The LOB, a complex system of competing buy and sell orders, introduces non-linear dynamics and interdependencies that challenge the assumptions of linear models. Factors such as order size, price level, market volatility, and the actions of other traders all contribute to a constantly shifting landscape, making accurate $Time-to-Fill$ prediction a significant hurdle. Consequently, strategies relying on simplistic models may experience suboptimal execution, increased slippage, and ultimately, reduced returns, highlighting the need for more sophisticated predictive approaches capable of navigating the LOB’s inherent complexity.

A fundamental difficulty in predicting order execution times stems from the pervasive issue of censoring within market data. This phenomenon occurs when an order remains unfulfilled within the observation period, meaning its true time-to-fill is unknown – it’s only known that it exceeds the length of the data collection window. Standard statistical models, designed to analyze complete datasets, struggle with this incomplete information, potentially leading to biased estimates and inaccurate predictions. Unlike typical regression problems, simply excluding censored observations introduces a selection bias, while treating the observation window as the fill time underestimates actual execution delays. Consequently, specialized techniques capable of handling this right-censored data – such as survival analysis or modified regression approaches – are essential to derive reliable insights from limit order book dynamics and ultimately, to improve trade execution strategies.

Predicting order execution times demands sophisticated modeling capable of navigating the intricacies of censored data – instances where orders remain unfilled within the observed timeframe. Standard statistical techniques often falter when confronted with this incompleteness, necessitating approaches that account for the probability of non-fulfillment. Beyond simply addressing censoring, effective prediction hinges on understanding the dynamic interplay of agent actions within the Limit Order Book (LOB). Each order submission, cancellation, or modification by various participants influences the likelihood of any given order being filled, creating a complex web of dependencies. Models must therefore move beyond static snapshots of the LOB and instead incorporate the sequential nature of agent behavior, recognizing that current conditions are shaped by past actions and will influence future outcomes. Successfully capturing this dynamic – the constant negotiation and competition among agents – is paramount to achieving accurate time-to-fill predictions and optimizing trade execution strategies.

The limit order book (LOB) displays concurrent buy (green) and sell (purple) orders, with each square representing a single unit.

Survival Analysis: A Logically Consistent Approach to Order Execution

Survival analysis is a statistical method originally developed for modeling time-to-event outcomes, such as patient survival in medical studies, and is directly applicable to order execution modeling. Unlike traditional regression techniques, survival analysis explicitly addresses censoring, a condition where the fill event is not observed within the observation window – for example, an order remaining unfilled at the end of the trading day. This is crucial in order book data where many orders do not immediately fill. The core output of a survival model is the estimated survival function, $S(t)$, which represents the probability that an order remains unfilled at time $t$. By modeling the time-to-fill distribution, and appropriately handling censored observations, survival analysis provides a robust framework for estimating the probability of order execution and understanding factors influencing fill times.

The performance of survival models in order execution is quantitatively assessed using metrics such as the C-index and the Rank Correlation Loss for Likelihood (RCLL). The C-index, ranging from 0.5 to 1.0, represents the probability that a randomly selected filled order will have a shorter time-to-fill than a randomly selected censored order; values above 0.5 indicate predictive power beyond random chance. RCLL, calculated as $RCLL = – \frac{1}{N} \sum_{i=1}^{N} log(P(T_i = t_i | T_i \ge t_i))$, measures the calibration of predicted probabilities against observed outcomes, with lower values indicating better calibration. These metrics facilitate model comparison and iterative refinement, ensuring the reliability of time-to-fill predictions and the optimization of trading strategies.

Framing order fill as a time-to-event allows the application of established survival analysis techniques, such as the Kaplan-Meier estimator and Cox proportional hazards models, to estimate the probability of an order being filled within a specified timeframe. This approach moves beyond simple fill rates by quantifying the time dependency of execution and accommodating censored orders – those that haven’t filled by the observation end date. By modeling the hazard rate – the instantaneous probability of fill at a given time – traders can predict fill probabilities under varying market conditions and optimize order placement strategies, including price and quantity, to minimize time-to-fill and maximize execution efficiency. Furthermore, the resulting hazard function can be incorporated into algorithmic trading systems to dynamically adjust order parameters based on predicted execution times.

Metrics were tracked across a 20-point prediction timeline to evaluate performance.

KANFormer: A Model Architected for Precise Time-to-Fill Estimation

KANFormer is a deep learning model designed for predicting fill probabilities, and it uniquely integrates the strengths of two distinct neural network architectures. It utilizes the `Transformer Architecture`, known for its ability to model long-range dependencies in sequential data through self-attention mechanisms. Complementing this, KANFormer incorporates `Kolmogorov-Arnold Networks` (KANs), which are provably universal function approximators and excel at representing complex, non-linear relationships with fewer parameters than traditional deep neural networks. This combination aims to leverage the Transformer’s sequential modeling capabilities alongside the KAN’s efficient function approximation to achieve accurate and robust fill probability predictions.

KANFormer utilizes a Deep Convolutional Cascade (DCC) to process Limit Order Book (LOB) snapshots, enabling the model to efficiently capture local temporal patterns within the order book data. The DCC architecture processes sequential LOB snapshots, extracting features that represent changes in bid and ask prices, volumes, and order book depth. Crucially, the DCC integrates information regarding agent actions – specifically, the orders placed by each agent – and the agent’s current queue position within the order book. This combined input allows KANFormer to model the interplay between order book dynamics, agent behavior, and the agent’s standing, providing a more comprehensive representation of the factors influencing time-to-fill estimates.

KANFormer demonstrates a substantial improvement in Time-to-Fill prediction accuracy as measured by several key metrics. The model achieves an average Area Under the Curve (AUC) of 0.76, surpassing the performance of the ConvTrans model described in arroyo2024deep. Further evaluation indicates KANFormer attains the highest C-index among compared models, and exhibits superior calibration with a Negative Right-Censored Log-Likelihood (RCLL) of 0.53. These results collectively indicate KANFormer’s ability to not only predict fill times with greater accuracy, but also to provide well-calibrated probability estimates.

Analysis of the ConvTrans model (arroyo2024deep) reveals that feature importance evolves consistently across different prediction horizons, as indicated by averaged SHAP values calculated from the test set.

Interpreting KANFormer: Unveiling the Determinants of Predictive Power

The predictive power of KANFormer hinges on its ability to discern subtle patterns within market data, a process now illuminated through the application of $SHAP$ values. This analytical technique reveals that KANFormer prioritizes order imbalances – the discrepancy between buy and sell orders – as a primary driver of its predictions. Specifically, significant shifts in order flow strongly influence the model’s assessment of potential price movements. Complementing this, the model demonstrably focuses on key price levels, identifying areas of support and resistance as critical determinants. By quantifying the contribution of each feature, $SHAP$ analysis confirms that KANFormer doesn’t simply memorize data, but rather learns to interpret the interplay between order dynamics and price action, offering a nuanced understanding of market behavior.

The value of KANFormer extends beyond predictive accuracy; its interpretability empowers traders with crucial insight into the reasoning behind each forecast. Rather than simply receiving a signal, users gain an understanding of why the model anticipates a particular market movement, pinpointing the specific order book imbalances and price levels that drive its conclusions. This transparency facilitates more informed decision-making, allowing traders to validate the model’s logic against their own market expertise and refine strategies accordingly. By revealing the causal factors behind predictions, KANFormer moves beyond a “black box” approach, fostering confidence and enabling a synergistic relationship between human intuition and artificial intelligence in the pursuit of profitable trading.

The successful deployment of any predictive model, particularly within the high-stakes environment of financial trading, hinges on establishing user confidence and seamless integration. KANFormer’s design prioritizes a transparent view into its decision-making process, moving beyond a ‘black box’ approach. This interpretability isn’t merely an academic exercise; it allows traders to understand why a specific prediction was made, examining the factors that contributed to the outcome. Consequently, this clarity fosters trust in the model’s recommendations, encouraging informed decision-making and facilitating a more effective integration of KANFormer into existing, real-world trading systems. Without this level of insight, models risk being underutilized or outright rejected, regardless of their underlying predictive power.

Analysis of SHAP values reveals how the importance of different features evolves across varying prediction horizons using our model.

Future Directions: Expanding the Horizon of Order Execution Modeling

The confluence of KANFormer, a novel attention mechanism adept at capturing long-range dependencies in time series data, and advanced reinforcement learning (RL) presents a compelling avenue for the future of automated trade execution. Researchers are poised to investigate architectures where KANFormer serves as a state representation module within an RL agent, allowing the agent to more effectively discern intricate patterns and anticipate market movements. This integration could enable the agent to learn optimal trading policies – deciding when, how much, and what to trade – directly from market data, adapting to changing conditions without explicit programming. The resulting system promises to move beyond traditional order execution strategies, potentially minimizing transaction costs, reducing market impact, and maximizing profitability by leveraging the predictive power of deep learning with the decision-making capabilities of reinforcement learning. Such an approach could unlock more nuanced and responsive trading algorithms capable of navigating complex market dynamics.

Predictive accuracy in order execution modeling stands to gain significantly from the application of deep learning survival models, such as DeepHit. Traditionally, survival analysis has been employed to estimate the time until an event-in this context, order fulfillment or cancellation-but these models often rely on parametric assumptions that may limit their adaptability to complex financial data. DeepHit, and similar architectures, bypass these limitations by leveraging the power of deep neural networks to directly learn hazard functions from data, enabling a more nuanced understanding of order lifespan. This approach allows for the incorporation of high-dimensional features and non-linear relationships, potentially leading to more precise predictions of order execution probabilities and improved algorithmic trading strategies.

The convergence of survival analysis and deep learning presents a compelling pathway towards significantly improved order execution strategies and overall market efficiency. Traditional survival models, adept at predicting time-to-event occurrences like order fulfillment, often rely on assumptions that limit their adaptability to complex financial data. Integrating these methods with the feature extraction capabilities of deep learning algorithms, however, allows for the modeling of non-linear relationships and high-dimensional data, leading to more accurate predictions of order execution probabilities. This enhanced predictive power enables the development of dynamic trading strategies that intelligently adapt to changing market conditions, minimize execution costs, and ultimately contribute to a more fluid and efficient marketplace. Such an approach transcends simple price prediction, focusing instead on the timing of trade completion – a critical, yet often overlooked, component of successful execution.

The pursuit of accurate fill probability prediction, as demonstrated by KANFormer, aligns with a fundamentally mathematical endeavor. The model’s architecture, integrating Kolmogorov-Arnold Networks and survival analysis, isn’t merely about achieving empirical success; it’s about constructing a provably robust system for understanding market dynamics. As Marvin Minsky observed, “Common sense is the collection of things everyone knows but no one can explain.” KANFormer strives to explain fill probabilities, moving beyond simple observation through a rigorous framework. The careful feature engineering and the application of survival analysis techniques represent a logical decomposition of the problem, aiming for a solution grounded in demonstrable principles rather than opaque correlations. The efficacy of SHAP values further reinforces this commitment to interpretability and mathematical clarity, showcasing the model’s internal logic.

Where Do We Go From Here?

The pursuit of predicting order book dynamics, as exemplified by KANFormer, inevitably confronts the fundamental limitations of inductive reasoning. A model, however elegantly constructed, remains a statistical approximation of a profoundly complex system. While the integration of survival analysis offers a potentially more robust framework for estimating time-to-fill probabilities, the crucial question of reproducibility persists. If a model’s predictions are sensitive to minor variations in training data or hyperparameter selection – a common affliction – then its practical value diminishes rapidly. Determinism is not merely a desirable trait; it is a prerequisite for any system intended to inform actual trading decisions.

Future work must move beyond simply demonstrating improved accuracy on historical data. Rigorous sensitivity analysis, coupled with formal verification techniques, is essential. The use of SHAP values, while valuable for feature importance assessment, does not address the core issue of model stability. A truly elegant solution would not only predict what will happen, but also quantify the certainty of that prediction.

The inherent noise within limit order books presents an ongoing challenge. Models can approximate, but they cannot eliminate, the unpredictability that arises from the confluence of rational and irrational agent behaviors. Perhaps the ultimate goal is not to predict the market, but to create a model that accurately reflects its inherent uncertainty – a paradoxical ambition, to be sure, but one that aligns with the fundamental principles of mathematical rigor.

Original article: https://arxiv.org/pdf/2512.05734.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/