Scaling Limit of a Limit Order Book Model via the Regenerative Characterization of Lévy Trees

Abstract

Abstract. We consider the following Markovian dynamic on point processes: at constant rate and with equal probability, either the rightmost atom of the current configuration is removed, or a new atom is added at a random distance from the rightmost atom. Interpreting atoms as limit buy orders, this process was introduced by Lakner et al. [Lakner et al. (2016) by showing that, in the case where the mean displacement at which a new order is added is positive, the measure-valued process describing the whole limit order book converges to a simple functional of this reflected Brownian motion. Our results make it possible to derive useful and explicit approximations on various quantities of interest such as the depth or the total value of the book.Our approach leverages an unexpected connection with Lévy trees. More precisely, the cornerstone of our approach is the regenerative characterization of Lévy trees due to Weill [Weill M (2007) Regenerative real trees. Ann. Probab. 35:2091-2121], which provides an elegant proof strategy which we unfold.

A diffusion approximation for limit order book models

“…[10,14]), there is little work on the derivation of a measure valued diffusion limit starting from a microscopic ("event-by-event") description of the limit order book. Two exceptions are the particular models considered in [1] and [17]. The work [1] extends the models in [9] and [8] by introducing additional noise terms in the pre-limit in which case the dynamics can then be approximated by an SPDE in the scaling limit.…”

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

“…Lakner et al [30] derived a high frequency limit for a one-sided order book model under the assumption that on average investors place their limit orders above the current best ask price. The opposite case when orders are placed in the spread with higher probability is analyzed in [29], where the authors use a coupling between a simple one-sided limit order book model and a branching random walk to characterize the diffusion limit. Bayer et al [5] extends the models in [21,22] by introducing additional noise terms in the pre-limit in which case the dynamics can then be approximated by an SPDE in the scaling limit.…”

Second order approximations for limit order books

“…Furthermore, a diffusive behaviour for the volumes at the top of the book together with a non-diffusive price process was derived in [6]. In [20] the authors study a one-sided measure-valued order book model, for which the scaling limit is given by a diffusive price process together with a block-shaped order book for the volumes. A more macroscopic perspective has been adopted in [19] and [24] where order book dynamics are described as the solution to SPDEs.…”