One-level limit order books with sparsity and memory

Motivated by Cont and Larrard (2013)'s seminal Limit Order Book (LOB) model, we propose two continuous-time models for the level I of a LOB in which the arrivals of limit orders, market orders, and cancellations are assumed to be mutually independent, memoryless, and stationary, but, unlike the aforementioned paper, the proposed models also account for some of the sparsity and memory exhibited by real LOB dynamics. Specifically, the first proposed model allows for variable price shifts after each price change in order to account for some of the larger-than-usual "gaps" between levels (sparsity property) that has been observed in some empirical studies. A more realistic approach is pursued in a second model by keeping the information about the standing orders at the opposite side of the book after each price change (memory property), and also incorporating arrivals of new orders within the spread, which in turn leads to a variable spread. To illustrate the applicability of the latter model, analytical expressions for some important quantities of interest, such as the distribution of the time span between price changes and the probability of consecutive price increments conditioned on the current state of the book, are derived. In spite of the inherent model complexity, the long-run asymptotic behavior of the resultant mid-price process is fully characterized for both models and, hence, our analysis shed further light on the relation between the macro price dynamics and some more detailed LOB features than those considered in earlier works. The asymptotic results are illustrated with a numerical Monte Carlo study for which an efficient simulation scheme is also developed.