A one-level limit order book model with memory and variable spread

Motivated by Cont and de Larrard (2013)'s seminal Limit Order Book (LOB) model, we propose a new model for the level I of a LOB in which the arrivals of orders and cancellations are still assumed to be mutually independent, memoryless, and stationary, but, unlike the aforementioned paper, the information about the standing orders at the opposite side of the book after each price change and the arrivals of new orders within the spread are incorporated. Our main result gives a diffusion approximation for the mid-price process, which sheds further light on the relation between the mid-price behavior at low frequencies and some LOB features not considered in earlier works. To illustrate the applicability of the proposed framework, we also develop a feasible method to compute several 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. These LOB model features are relevant in many applications such as high frequency trading and intraday risk management. The proposed method is also used to develop an efficient simulation scheme for the price dynamics, which is then applied to assess numerically the accuracy of the diffusion approximation.