[Paper Review] A level-1 Limit Order book with time dependent arrival rates

Jonathan A. Chávez-Casillas, Robert J. Elliott+2 more|arXiv (Cornell University)|Apr 21, 2017

Complex Systems and Time Series Analysis3 citations

TL;DR

This paper extends Cont and de Larrard's (2013) level-1 limit order book model by introducing time-dependent arrival rates for limit and market orders, enabling more realistic modeling of high-frequency trading dynamics. The key contribution is proving that the conditional diffusion limit of the price process converges to a Brownian meander under these time-inhomogeneous conditions, with explicit parameter estimation methods provided for empirical implementation using high-frequency data.

ABSTRACT

We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.

Motivation & Objective

• To develop a more realistic stochastic model for limit order book dynamics by replacing constant arrival rates with time-dependent rates.
• To extend the Cont and de Larrard (2013) level-1 model to incorporate periodic, time-inhomogeneous arrival processes for limit and market orders.
• To derive the asymptotic behavior of the price process under time-dependent intensities, showing convergence to a Brownian meander.
• To provide a practical framework for estimating model parameters using empirical high-frequency data.
• To validate the model with real Facebook trading data, demonstrating its ability to capture price trends and volatility patterns.

Proposed method

• Models bid and ask queue sizes as independent continuous-time birth-death processes with time-dependent intensities λb(t), λa(t), µb(t), µa(t) for limit and market orders.
• Introduces random extinction times σ(b,n) and σ(a,n) for each queue, with price changes occurring at the minimum of these times.
• Defines the price process S_t as a step process that increases or decreases by δ when the ask or bid queue is depleted, respectively.
• Uses a renewal-type structure with renewal times Vn = Σ_{k=1}^{n-1} τk to model the sequence of price changes.
• Derives the distribution of the first price change time τ1 under time-inhomogeneous Poisson processes, using survival probabilities and transition matrices.
• Employs a scaling normalization (v=1) to ensure parameter identifiability and enables estimation via empirical averages of order counts over time intervals.

Experimental results

Research questions

• RQ1 How does the introduction of time-dependent arrival rates affect the asymptotic behavior of the price process in a level-1 limit order book model?
• RQ2 Can the conditional diffusion limit of the price process still be characterized as a Brownian meander when arrival intensities are time-inhomogeneous?
• RQ3 What is the impact of time-varying order arrival rates on the volatility and drift of the price process in high-frequency trading environments?
• RQ4 How can the model parameters (λb, λa, µb, µa) be consistently estimated from high-frequency order book data?
• RQ5 To what extent does the model reproduce observed price trends and volatility patterns in real market data?

Key findings

• The conditional diffusion limit of the price process under time-dependent arrival rates converges to a Brownian meander, extending the result of Cont and de Larrard (2013) to non-homogeneous intensities.
• The model's asymptotic behavior depends on the relative magnitudes of the arrival rates; when ˆλa < ˆµa and ˆλb < ˆµb, the unconditioned limiting process is a Brownian motion with non-zero volatility.
• Empirical estimation using Facebook data (Nov 3–7, 2014) shows that the bid queue is typically depleted before the ask queue, explaining the observed downward price trend on November 3rd.
• The estimated volatility ˆσ = 0.0066 from the transition matrix method is close to the pooled high-frequency estimate of 0.0066, validating the analytical estimation approach.
• The ratio Λa/Ma > 1 on November 3rd indicates higher ask-side order flow, consistent with the observed price decline, confirming the model's predictive power.
• The model achieves parameter identifiability through normalization (v=1), ensuring that key parameters like c0 and c1 remain invariant under scaling of the time-dependent rate function.