# Diffusion Scaling of a Limit-Order Book Model: The Symmetric Case

Trading of ﬁnancial instruments has moved away from the trading ﬂoor and onto electronic exchanges. These exchanges typically operate as continuous double auctions and this move has been accompanied by a huge increase in the number of orders submitted and in the amount of information available to participants. Modelling the ﬂow of orders and organizing this information is of vital importance to practitioners who are concerned with volatility, prediction, market microstructure, and optimal execution of large orders, and to those who design or regulate order book protocols. It is difficult to build models of the limit-order book that capture all of the important features, especially strategic play among agents. Smith et al. [21] have shown in a simulation study of a model based on Poisson arrivals that such simpliﬁed models can mimic the dynamics of real limit-order books. Even these simpliﬁed models are difﬁcult to analyze. Cont et al. [3] provide a Laplace transform analysis of such a model, and this analysis depends critically on the Poisson assumption. In this thesis we begin to adapt ideas from heavy trafﬁc queueing theory, which typically require only that arrival processes are renewal processes. This thesis is designed to establish that such an approach is viable, not build a realistic limit-order book model. For this reason, we consider the simplest model in which a nontrivial diffusion limit can be obtained, and in this simplest model, the assumption of Poisson arrivals is retained. It is to be expected, but not established here, that the results obtained can be generalized to permit renewal arrivals of orders and cancellation waiting times that are not exponential. The focus of this thesis is a sequence of limit-order book models that follow the dynamics proposed in [3], in which orders of equal size arrive according to Poisson processes whose rates are keyed off of the opposite best price. The heavy-trafﬁc scaled sequence is shown to converge to a simple model in which the scaled number of orders at each price level follow diffusion or jump-diffusion processes. In establishing this limit, new techniques are developed for proving convergence of sequences of “locally unbalanced” Markov chains when the method of inﬁnitesimal generators does not apply and for establishing state-space collapse by extending the “crushing” argument of Peterson [20]. A weaker than usual form of convergence in distribution on D[0,∞), the space of càdlàg functions on[0,∞), is used in proving convergence to the jump-diffusion processes. Cont and de Larrard [2] have also computed a diffusion-scaled limit of a limit-order book in which the only orders in the book are at the bid and ask prices, and when the queue at one of these prices vanishes, both the bid and ask prices move one tick and the system reinitializes. In the model of this thesis, the system continues without reinitializing when one of these queues vanishes.

## History

## Date

2019-01-30## Degree Type

- Dissertation

## Department

- Mathematical Sciences

## Degree Name

- Doctor of Philosophy (PhD)