2.4 Queue Position & Fill Probability

Basic Prerequisites: Probability Theory and Market Microstructure & Order Types. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

A passive limit order does not fill on arrival. It joins a queue and waits for enough order outflow from the front to reach it. Under price-time priority (FIFO) that outflow is trades (market orders) plus cancellations that happen ahead of you. Your fill odds are therefore not a property of the price - they are a property of your queue position.

This folder is the queue-and-fill topic-folder for Pillar 2. It is a hub: it gives you (a) the fast formula lookup below (job #1) and (b) routes you to six sub-pages that go from zero-knowledge intuition, through the mechanics of the book and the matching engine, to analytic fill-probability models, queue-reactive dynamics, the failure modes that bleed a desk dry, and the modern extensions (multi-venue overbooking, OFI price impact, Hawkes arrivals).

The one-sentence essence. “Your fill probability is governed by the outflow process in front of you, not by how attractive your price looks: under FIFO you fill when $\xi \ge x$ (cumulative outflow $\xi$ ahead exceeds your queue position $x$), so the queue position $x$ is the single control variable a passive trader actually manages - and it is exactly the variable a naive backtest ignores.”

Scope note (vs the siblings). This folder is the micro-event view - will this order fill, and when. For the aggregate view of price movement see Optimal Execution & Almgren–Chriss (scheduling a block) and Market Impact & Depth (how fills move the price). The three meet at the order-flow-imbalance variable.

Primary verified sources: Cont, Stoikov & Talreja (2010); Cont & Kukanov (2017); Cont, Kukanov & Stoikov (2014); Gould et al. (2013); Lo, MacKinlay & Zhang (2002); Rosu (2009); Foucault, Pagano & Roell, Market Liquidity Ch 4–6; Hasbrouck, Empirical Market Microstructure Ch 6–10. All numbers below were re-executed and reproduced (see §3).

2. Mathematical Ground Truth & Derivations

Notation. $x$ = orders (shares) ahead of your order in the FIFO queue at your price; $Q$ = total depth at that price; $L$ = your order size; $\mu$ = market-order (trade) arrival rate at your price; $\lambda$ = limit-order arrival rate at your price; $\theta$ = per-order cancellation rate; $\xi$ = cumulative outflow from the front of the queue over $[0,T]$ ($\xi =$ trades $+$ cancels ahead); $p$ = per-tick trade-arrival probability (discrete models).

The fill condition (price-time priority). Under strict FIFO, an order of size $L$ at queue position $x$ receives

You start filling once outflow passes $x$, and you finish once it passes $x+L$. This is the fill function of Cont & Kukanov (2017) and the object every queue model is trying to predict.

Quick-Reference Lookup - the fast facts of this folder (all reproduced in §3):

Critical framing caveat. $x$ (queue position) and $Q$ (depth) are different variables and both matter. Deep total depth is good for spreads but bad for you if $x$ is large: your fill odds fall with $x$ but the adverse-selection penalty falls with $Q$. Calibrating on $Q$ alone is the single most common modelling error.

3. Computational Implementation - the fill-probability engine

Runs on the standard library only here (pages 04 and 06 additionally use numpy). It reproduces every verified number above: the negative-binomial closed form matches the Monte Carlo fill rate, the mean-field ODE brackets the Gillespie simulation, and the exact Markov-chain mid-move probability matches simulation to ~2 decimals (within $0.002$).

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the full failure analysis lives in 05 · Failure Modes & Practice. In one line each:

1. The standing-queue delusion - a backtest that treats “a trade printed at my price” as “I was filled” overstates the fill rate ($1.0000$ vs true $0.7939$ here) and books P&L on trades that never happened.
2. Latency & stale-quote pick-off - the fraction of fills that are adverse grows with your round-trip latency ($P(\text{picked off})$ goes $0.025\to 0.993$ as latency goes $0.05\to 10$ ms in the toy model).
3. Adverse selection at the fill - $\mathbb{E}[\Delta M_T\mid \text{filled}]<0$: passive fills are systematically followed by unfavorable moves (fill $-$ $0.0119 vs no-fill + $0.0166).
4. Cancel-and-reinsert is not free - re-joining the queue after a cancel puts you at the back; every refresh trades queue position for a marginally better price.

5. References

• Cont, Rama; Stoikov, Sasha; Talreja, Rishi - “A stochastic model for order book dynamics,” Operations Research 58(3), 549–563 (2010). *The foundational tractable queue model: independent Poisson queues, birth–death first-passage, Laplace-transform conditional probabilities.
• Cont, Rama; Kukanov, Arseniy - “Optimal order placement in limit order markets,” Quantitative Finance 17(4), 553–571 (2017). *The fill function $(\xi-x)^+
• Cont, Rama; Kukanov, Arseniy; Stoikov, Sasha - “The price impact of order book events,” Journal of Financial Markets 17, 47–88 (2014). *The linear OFI price-impact law $\Delta P=\beta \text{OFI}/\text{depth}$
• Lo, Andrew W.; MacKinlay, A. C.; Zhang, June - “Econometric models of limit-order executions,” Journal of Financial Economics 65(1), 31–71 (2002). The classic empirical survival/hazard model of limit-order execution times - “how long until I fill or cancel.” ADV
• Gould, Martin D.; Porter, M. A.; Williams, S.; McDonald, M.; Fenn, D. J.; Howison, S. D. - “Limit order books,” Quantitative Finance 13(11), 1709–1742 (2013). Structured empirical survey: price-time vs pro-rata, cancel-to-trade ratios, latency effects on conditional event studies. arXiv:1012.0349. ADV
• Rosu, Ioanid - “A dynamic model of the limit order book,” Review of Financial Studies 22(11), 4601–4641 (2009). Equilibrium LOB; the “hump” shape in depth vs distance and the trade-off between matching optimism and fill pessimism. ADV
• Foucault, Thierry; Pagano, Marco; Roell, Ailsa - Market Liquidity: Theory, Evidence, and Policy (OUP, 2013)
• Hasbrouck, Joel - Empirical Market Microstructure (OUP, 2007)

6. Connected Graph Bridges

• Foundational base: Probability Theory (Poisson processes, first passage, Laplace transforms) · Stochastic Calculus (CTMC generators)
• Sibling in-pillar: Market Microstructure & Order Types (order-type menu, maker–taker fees) · Low-Latency Systems Architecture (how latency enters the queue) · Optimal Execution & Almgren–Chriss · Hardware Acceleration & FPGA
• Market-making view: Adverse Selection & Glosten–Milgrom · Avellaneda–Stoikov & Optimal Quoting · Market Impact & Depth

Beginner: start at 01 · Practitioner: start at 05