A Mathematical Approach to Order Book Modeling

TL;DR

This paper models the order book as a multidimensional Markov chain, analyzing its stationary distribution and showing the price process converges to Brownian motion, bridging microscopic and macroscopic descriptions of price formation.

Contribution

It introduces a mathematical framework for order book modeling using Markov chains and proves convergence results, connecting agent-based models with stochastic differential equations.

Findings

Stationary distribution exists under certain cancellation structures.

Rescaled price process converges to Brownian motion.

Numerical simulations align with market data.

Abstract

Motivated by the desire to bridge the gap between the microscopic description of price formation (agent-based modeling) and the stochastic differential equations approach used classically to describe price evolution at macroscopic time scales, we present a mathematical study of the order book as a multidimensional continuous-time Markov chain and derive several mathematical results in the case of independent Poissonian arrival times. In particular, we show that the cancellation structure is an important factor ensuring the existence of a stationary distribution and the exponential convergence towards it. We also prove, by means of the functional central limit theorem (FCLT), that the rescaled-centered price process converges to a Brownian motion. We illustrate the analysis with numerical simulation and comparison against market data.