A Semi-Markovian Modeling of Limit Order Markets

TL;DR

This paper extends a stochastic limit order book model to incorporate arbitrary inter-arrival time distributions and dependencies between events, using Markov renewal processes to better match empirical data while maintaining analytical tractability.

Contribution

It introduces a generalized framework for limit order book modeling with non-exponential inter-arrival times and event dependencies, supported by explicit Laplace transform expressions.

Findings

Model calibrated to major tech stocks data from 2012.

Maintains analytical tractability with explicit Laplace transforms.

Bid-ask spread remains constant at one tick in the model.

Abstract

R. Cont and A. de Larrard (SIAM J. Finan. Math, 2013) introduced a tractable stochastic model for the dynamics of a limit order book, computing various quantities of interest such as the probability of a price increase or the diffusion limit of the price process. As suggested by empirical observations, we extend their framework to 1) arbitrary distributions for book events inter-arrival times (possibly non-exponential) and 2) both the nature of a new book event and its corresponding inter-arrival time depend on the nature of the previous book event. We do so by resorting to Markov renewal processes to model the dynamics of the bid and ask queues. We keep analytical tractability via explicit expressions for the Laplace transforms of various quantities of interest. We justify and illustrate our approach by calibrating our model to the five stocks Amazon, Apple, Google, Intel and Microsoft on June 21^{st} 2012. As in R. Cont and A. de Larrard, the bid-ask spread remains constant equal to one tick, only the bid and ask queues are modeled (they are independent from each other and get reinitialized after a price change), and all orders have the same size.