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

Paper: arXiv 2403.02572

Abstract

This paper focuses on computing the fill probabilities for limit orders positioned at various price levels within the limit order book, which play a crucial role in optimizing executions. We adopt a generic stochastic model to capture the dynamics of the order book as a series of queueing systems. This generic model is state-dependent and also incorporates stylized factors. We subsequently derive semi-analytical expressions to compute the relevant probabilities within the context of state-dependent stochastic order flows. These probabilities cover various scenarios, including the probability of a change in the mid-price, the fill probabilities of orders posted at the best quotes, and those posted at a price level deeper than the best quotes in the book, before the opposite best quote moves. These expressions can be further generalized to accommodate orders posted even deeper in the order book, although the associated probabilities are typically very small in such cases. Lastly, we conduct extensive numerical experiments using real order book data from the foreign exchange spot market. Our findings suggest that the model is tractable and possesses the capability to effectively capture the dynamics of the limit order book. Moreover, the derived formulas and numerical methods demonstrate reasonably good accuracy in estimating the fill probabilities.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematics such as state-dependent birth-death processes, Laplace transforms, and continued fractions to derive semi-analytical fill probabilities. It also demonstrates empirical rigor by validating the model with extensive numerical experiments on real FX spot market data, assessing accuracy in capturing limit order book dynamics.

Research Flowchart

  flowchart TD
  A["Research Goal: Compute fill probabilities<br>for limit orders in a Limit Order Book"] --> B
  subgraph B ["Methodology: Queueing System Model"]
      B1["Stochastic Order Flows"]
      B2["State-Dependent Dynamics"]
      B3["Stylized Factors"]
  end
  B --> C["Input: FX Spot Market Data"]
  C --> D["Computations: Semi-Analytical Expressions"]
  D --> E{"Outcomes & Findings"}
  E --> F1["Probability of Mid-Price Change"]
  E --> F2["Fill Probabilities: Best Quotes"]
  E --> F3["Fill Probabilities: Deeper Levels"]
  F1 & F2 & F3 --> G["Model Validation: High Accuracy<br>Tractability & Effective Dynamics Capture"]

Cited References (30)

1. A deep learning approach to estimating fill probabilities in a limit order book (2022) — link
2. A generalized birth–death stochastic model for high-frequency order book dynamics (2012) — link
3. A new metric of absolute percentage error for intermittent demand forecasts (2016) — link
4. A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions (2009) — link
5. A Stochastic Model for Order Book Dynamics (2010) — link
6. AFM Market Watch Edition 8 - Algorithmic Trading (2023) — link
7. Algorithmic and High-Frequency Trading (2015)
8. Computing Laplace Transforms for Numerical Inversion Via Continued Fractions. (1999) — link

0 of 30 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.