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Fill Probabilities in a Limit Order Book with State-Dependent Stochastic Order Flows

Felix Lokin, Fenghui Yu

TL;DR

This paper develops a tractable framework for estimating fill probabilities of limit orders across depth levels in a limit order book by modeling the book as interacting state-dependent queues. It derives semi-analytical expressions for key execution quantities using first-passage times and Laplace-transform techniques, enabling analysis of mid-price moves and fill probabilities at the best and one level deeper. The framework accommodates various intensity specifications (Models I–III) and is validated with FX spot data, showing good empirical alignment and indicating that deeper-level fills are typically negligible. The results provide practical tools for execution optimization and offer a unified lens to recover existing models as special cases.

Abstract

This paper studies the fill probabilities of limit orders placed at different price levels in a limit order book. These probabilities play a central role in execution optimization, as limit orders are not guaranteed to be executed and inherently involve a trade-off between execution cost and execution risk. We model the limit order book within a general state-dependent stochastic framework, representing its dynamics as a collection of interacting queuing systems while incorporating key stylized market features. Within this framework, we derive semi-analytical expressions for several quantities of interest under state-dependent order flows, including the probability of a mid-price change, the fill probabilities of orders placed at the best quotes, and those of orders placed deeper in the book before the opposite best quote moves. While the framework can be extended to even deeper price levels, the corresponding fill probabilities are typically negligible. We validate the proposed model through extensive numerical experiments using real foreign exchange spot market data. The results demonstrate that the model remains tractable while capturing essential order book dynamics, and that the derived expressions achieve good accuracy in estimating fill probabilities.

Fill Probabilities in a Limit Order Book with State-Dependent Stochastic Order Flows

TL;DR

This paper develops a tractable framework for estimating fill probabilities of limit orders across depth levels in a limit order book by modeling the book as interacting state-dependent queues. It derives semi-analytical expressions for key execution quantities using first-passage times and Laplace-transform techniques, enabling analysis of mid-price moves and fill probabilities at the best and one level deeper. The framework accommodates various intensity specifications (Models I–III) and is validated with FX spot data, showing good empirical alignment and indicating that deeper-level fills are typically negligible. The results provide practical tools for execution optimization and offer a unified lens to recover existing models as special cases.

Abstract

This paper studies the fill probabilities of limit orders placed at different price levels in a limit order book. These probabilities play a central role in execution optimization, as limit orders are not guaranteed to be executed and inherently involve a trade-off between execution cost and execution risk. We model the limit order book within a general state-dependent stochastic framework, representing its dynamics as a collection of interacting queuing systems while incorporating key stylized market features. Within this framework, we derive semi-analytical expressions for several quantities of interest under state-dependent order flows, including the probability of a mid-price change, the fill probabilities of orders placed at the best quotes, and those of orders placed deeper in the book before the opposite best quote moves. While the framework can be extended to even deeper price levels, the corresponding fill probabilities are typically negligible. We validate the proposed model through extensive numerical experiments using real foreign exchange spot market data. The results demonstrate that the model remains tractable while capturing essential order book dynamics, and that the derived expressions achieve good accuracy in estimating fill probabilities.
Paper Structure (37 sections, 7 theorems, 116 equations, 9 figures, 8 tables)

This paper contains 37 sections, 7 theorems, 116 equations, 9 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Limit Order Book Model
  3. Limit Order Book as a Stochastic Model
  4. Dynamics of State-Dependent Order Flows
  5. Examples of Order Flow Models
  6. Model I
  7. Model II
  8. Model III
  9. Preliminaries on State-Dependent Queuing Systems
  10. First-Passage Times of State-Dependent Birth--Death Processes
  11. First-Passage Times of State-Dependent Pure-Death Processes
  12. Fill Probabilities in a Limit Order Book
  13. Probability of a Change in Mid-Price
  14. Fill Probability at the Best Quotes
  15. Fill Probability at a Price Level Deeper than the Best Quotes
  16. ...and 22 more sections

Key Result

Proposition 1

Let $\sigma_A$ and $\sigma_B$ denote the first-passage times to $0$ of the best ask and best bid queues, respectively. For $i\in\{A,B\}$, let $\hat{f}^{s_0}_{\sigma_i}(s)$ denote the Laplace transform of the density of $\sigma_i$, conditional on the initial spread $s_0\geq 1$. Then Define and let $i,j\in\{A,B\}$ with $i\neq j$. Then the conditional probabilities in pricemoveprob and pricemovepro

Figures (9)

  • Figure 1: Schematic representation of the order book dynamics at the best bid $Q= Q_{p_B}$.
  • Figure 2: Schematic representation of the order book dynamics at the best ask $Q= Q_{p_A}$.
  • Figure 3: Market order arrival rates (buy and sell) per second, by week.
  • Figure 4: Distribution of executed limit orders by their distance (in ticks) from the best quote at execution.
  • Figure 5: Distribution of executed limit orders by their distance (in ticks) from the best quote at submission.
  • ...and 4 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Lemma 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Proposition 3
  • Remark 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • ...and 1 more