Group for Research in Decision Analysis

A semi-Markovian modeling of limit order books: A survey of recent findings and results

Anatoliy Swishchuk University of Calgary, Canada

R. Cont and A. de Larrard (SIAM J. on Finan. Math., 2013) introduced a Markovian stochastic model for the dynamics of a limit order book (LOB) and computed various quantities of interest such as the probability of a price increase or the diffusion limit of the price process.

In this talk, we consider several extensions of their model suggested by empirical observations. One of them is to 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 (not independent). The dynamics of the bid and ask queues are modeled by Markov renewal process and the mid-prices - by a semi-Markov process. We justify and illustrate the approach by calibrating our model to the five stocks, Amazon, Apple, Google, Intel, Microsoft, on June 21st, 2012 (Lobster data), to the 15 stocks from Deutsche Boerse Group (September 23d, 2013), and to Cisco asset (November 3d, 2014). As in Cont & de Larrard, the bid-ask spread remains constant equal to one tick and all orders have the same size. Different quantities of interest and diffusion limit are obtained.

The second extension is associated with the case when the price changes are not fixed at one tick. And the third one is related to the case with arbitrary number of states for the price changes. For both cases the justification, diffusion limits, implementations and numerical results are presented for different LOB data: Lobster data, and Cisco, Facebook, Intel, Liberty Global, Liberty Interactive, Microsoft, Vodafone from 2014/11/03 to 2014/11/07. (The talk is based on two research papers written with my ex- and current students: Nelson Vadori, Julia Schmidt, Katharina Cera and Tyler Hofmeister).


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