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

Abstract

We propose a new model for the level I of a Limit Order Book (LOB), which incorporates 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. Our main result gives a diffusion approximation for the mid-price process. To illustrate the applicability of the considered framework, we also propose 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. The proposed method is used to develop an efficient simulation scheme for the price dynamics, which is then applied to assess numerically the accuracy of the diffusion approximation.

A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics

“…Example 1.7 (Reduced form order book models). Reduced form order book models (Cont and De Larrard, 2012;Cont and de Larrard, 2013;Chavez-Casillas and Figueroa-Lopez, 2017;Huang et al, 2017) focus on the dynamics of best bid/ask queues. We can recover this class of models as a special case of our framework by choosing L := θ to be the tick size, H a := L 2 (0, L),…”

“…Although joint modeling of order flow at all price levels in the limit order book is more appealling, (S)PDE models have lacked the analytical and computational tractability needed for applications; as a result, most analytical results have been derived using reduced-form models of the best bid-ask queues (Cont and De Larrard, 2012;Cont and de Larrard, 2013;Chavez-Casillas and Figueroa-Lopez, 2017;Huang et al, 2017).…”

A high‐frequency trade execution model for supervised learning

“…Many quantities, such as the probability of price movements given the state of the limit order, are relevant for trading and intraday risk management. The complex relation between order book dynamics and price movements has been the focus of econometric and stochastic modeling (see Chavez‐Casillas & Figueroa‐Lopez, ; Cont & de Larrard, ; Cont, Stoikov, & Talreja, ; Cont et al., ; Engle & Russell, and references therein). For analytic tractability, these models assume a data generation process and typically estimate quantities based on asymptotic limits of diffusion processes.…”

Optimal placement of a small order in a diffusive limit order book

“…In this study, we discuss the optimal strategy when the price dynamics follow a diffusive model such as a Brownian motion (BM) or a Geometric Brownian motion (GBM). A BM model, often called the Bachelier model, can be seen as a reasonable approximation of asset price dynamics at intermediate intraday time horizons (see, e.g., Ch avez-Casillas & Figueroa-L opez, 2017;Cont & De Larrard, 2013). Also, bridging with the work of Guo et al (2017), a BM with 0 drift (respectively, nonzero drift) appears as the limit of a symmetric (respectively, asymmetric) correlated random walk when the time step between price changes and the tick size goes to 0 in a certain way (cf.…”