Diffusion Approximations for High-Frequency Limit Order Book Dynamics

Summary

The document models the bid and ask queues in a liquid market where orders arrive at high frequency. It derives a functional central limit theorem for their joint dynamics and shows that, as order arrival frequency increases, the book can be approximated by a Markovian jump-diffusion in the positive orthant. The process characteristics are expressed in terms of statistical properties of the underlying order flow, allowing the framework to accommodate varied distributions and temporal dependence.

These approximations make it possible to analyze quantities such as the probability of a price increase and the time until the next price move, conditional on the current book state. The stated framework applies to a broad class of stochastic order-flow models, including Poisson point processes, self-exciting processes, and ACD-GARCH models. The document describes theoretical results and potential analytical uses rather than reporting a specific empirical trading test. Its approximation depends on high order-arrival frequency, and the brief description does not state how accuracy varies across particular markets or parameter choices.

Key ideas

• A functional central limit theorem describes the joint dynamics of bid and ask queues.
• At high order-arrival frequency, the limit order book is approximated by a Markovian jump-diffusion.
• The approximation supports conditional analysis of price direction and time to the next price move.
• The framework covers order-flow models including Poisson, self-exciting, and ACD-GARCH processes.

Tags

# Order book dynamics in liquid markets: limit theorems and diffusion approximations


# Order book dynamics in liquid markets: limit theorems and diffusion approximations









We propose a model for the dynamics of a limit order book in a liquid market where buy and sell orders are submitted at high frequency. We derive a functional central limit theorem for the joint dynamics of the bid and ask queues and show that, when the frequency of order arrivals is large, the intraday dynamics of the limit order book may be approximated by a Markovian jump-diffusion process in the positive orthant, whose characteristics are explicitly described in terms of the statistical properties of the underlying order flow. This result allows to obtain tractable analytical approximations for various quantities of interest, such as the probability of a price increase or the distribution of the duration until the next price move, conditional on the state of the order book. Our results allow for a wide range of distributional assumptions and temporal dependence in the order flow and apply to a wide class of stochastic models proposed for order book dynamics, including models based on Poisson point processes, self-exciting point processes and models of the ACD-GARCH family.

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