A time-dependent Markovian model of a limit order book

TL;DR

This paper introduces a time-dependent Markovian model of a limit order book to analyze how microscopic order dynamics can lead to complex diffusion behaviors in asset prices, supported by empirical validation.

Contribution

It proposes a simple, microscopic, time-dependent model of order arrivals and cancellations, bridging the gap between order-level dynamics and diffusion limits of prices.

Findings

Model captures time-dependent order flow dynamics.

Empirical studies validate model assumptions.

Price processes may converge to complex diffusions.

Abstract

This paper considers a Markovian model of a limit order book where time-dependent rates are allowed. With the objective of understanding the mechanisms through which a microscopic model of an orderbook can converge to more general diffusion than a Brownian motion with constant coefficient, a simple time-dependent model is proposed. The model considered here starts by describing the processes that govern the arrival of the different orders such as limit orders, market orders and cancellations. In this sense, this is a microscopic model rather than a ``mesoscopic'' model where the starting point is usually the point processes describing the times at which the price changes occur and aggregate in these all the information pertaining to the arrival of individual orders. Furthermore, several empirical studies are performed to shed some light into the validity of the modeling assumptions and to verify whether certain stocks satisfy the conditions for their price process to converge to a more complex diffusion.