Fill Probability

The foundational input to fill probability is queue position—the ordinal rank of a resting limit order at a given price level. Since limit order books operate on strict price-time priority, an order's position in the queue determines how many shares must trade before it reaches the front.

• Snapshot inference: Uses order book snapshots and trade prints to estimate position by tracking cumulative volume additions and cancellations at the price level
• Message-level reconstruction: Parses every add, cancel, and execution message from the exchange feed to maintain an exact queue counter
• Partial fills as signals: A partial execution reveals that the order has reached the front of the queue, resetting the position estimate

Accurate queue estimation is the difference between a fill probability of 5% and 95% for a passive order sitting deep in the book.

Fill probability is heavily conditioned on the bid-ask imbalance—the ratio of resting liquidity on the buy side versus the sell side at the best prices. A strong imbalance predicts the direction and aggressiveness of the next trade.

• Volume imbalance ratio: (Bid Volume - Ask Volume) / (Bid Volume + Ask Volume) at the top N levels of the book
• Depth-weighted imbalance: Extends the ratio deeper into the book, weighting each level by its distance from the mid-price to capture latent liquidity pressure
• Trade arrival correlation: High buy-side imbalance correlates with increased probability of aggressive market buy orders, improving fill odds for resting sell limit orders

These signals are particularly potent in the seconds immediately following a quote change, before the imbalance mean-reverts.

The stochastic intensity of trade arrivals—how frequently market orders execute against the book—directly governs the hazard rate of a limit order being filled. This is typically modeled as a Hawkes process or non-homogeneous Poisson process.

• Baseline intensity: Captures the average trade frequency for the instrument, which varies dramatically by time of day (U-shaped intraday pattern)
• Self-excitation: A trade begets more trades; the Hawkes kernel captures clustering behavior where volatility events cascade
• Size distribution: Not all trades are equal—a single 10,000-share market order can consume multiple queue positions instantly, while 100-share trades barely move the queue

The fill probability over a horizon T is the complement of the survival probability: P(fill) = 1 - exp(-∫₀ᵀ λ(t) dt), where λ(t) is the conditional trade intensity.

A limit order faces two competing risks: execution and cancellation. Fill probability must account for the fact that other traders can cancel their orders (improving your queue position) or undercut your price (stranding your order behind a new best quote).

• Cancellation rate estimation: Models the probability that orders ahead of you in the queue are canceled before execution, effectively advancing your position without a trade
• Quote competition: A new limit order placed at a better price resets the queue entirely—your fill probability drops to near zero unless you reprice
• Flickering quotes: In high-frequency environments, quotes that appear and disappear within milliseconds create false signals of queue advancement

Sophisticated fill probability models treat these as competing risks in a multi-state survival framework, where the order can transition to filled, canceled, or stranded states.

Fill probability is not a single number—it is a term structure that varies dramatically with the specified time window. A limit order has a very different probability of filling in 100 milliseconds versus 10 minutes.

• Ultra-short horizon (< 1 sec): Dominated by queue position and immediate trade arrival; useful for latency-sensitive market-making strategies
• Medium horizon (1-60 sec): Incorporates order book replenishment, cancellation dynamics, and short-term volume predictions
• Long horizon (> 1 min): Heavily influenced by price drift, volatility forecasts, and the probability that the mid-price moves away from the limit price entirely

The term structure of fill probability is the key input for Almgren-Chriss-style optimal execution models, where the trader balances the certainty of immediate market orders against the cost savings of patient limit orders.

Modern fill probability estimation has moved beyond parametric models to gradient-boosted trees and deep learning architectures that ingest raw order book state and output calibrated probabilities.

• Feature engineering: Inputs include queue position, order book imbalance at multiple depths, recent trade velocity, volatility regime indicators, and time-of-day fixed effects
• Calibration: Raw model scores are transformed via Platt scaling or isotonic regression to ensure the predicted probabilities match empirical fill frequencies
• Online adaptation: Models retrain on streaming data to adapt to regime changes—a fill probability model calibrated during low-volatility periods will be miscalibrated during a volatility event

These models are deployed with microsecond inference latency, often on FPGA or ASIC hardware, to inform real-time order routing decisions across fragmented liquidity venues.