High Frequency Asymptotics for the Limit Order Book

Abstract

We study the one-sided limit order book corresponding to limit sell orders and model it as a measure-valued process. Limit orders arrive to the book according to a Poisson process and are placed on the book according to a distribution which varies depending on the current best price. Market orders to buy periodically arrive to the book according to a second, independent Poisson process and remove from the book the order corresponding to the current best price. We consider the above described limit order book in a high frequency regime in which the rate of incoming limit and market orders is large and traders place their limit sell orders close to the current best price. Our first set of results provide weak limits for the unscaled price process and the properly scaled measure-valued limit order book process in the high frequency regime. In particular, we characterize the limiting measure-valued limit order book process as the solution to a measure-valued stochastic differential equation. We then provide an analysis of both the transient and long-run behavior of the limiting limit order book process.

A functional limit theorem for limit order books with state dependent price dynamics

“…Despite the considerable empirical evidence that the state of the order book, especially order imbalance at the top of the book, has a noticeable impact on order dynamics (see [2,4,12] and references therein) the order flow in most limit order book models either follows independent Poisson dynamics or depends on the price process only as in [11,14,15,16]. Notable exceptions are the papers by Abergel and Jeddi [1], where Hawkes-type dynamics are used, Huang et al [17] and Huang and Rosenbaum [18] where the ergodicity of a general Markovian order book model is studied and the diffusivity of the rescaled price process in this general framework is derived, and Horst and Kreher [13] who obtained a deterministic scaling limit for LOBs with fully state dependent event dynamics.…”

Ergodicity and Diffusivity of Markovian Order Book Models: A General Framework

“…In the recent years, many works have been devoted to the description of order book dynamics. Order book models can be essentially divided into two types: economic models, where one focuses on the behaviors of individual agents and their optimal decisions, see for example Parlour (1998), Foucault (1999) and Roşu (2009); statistical models, where the order flows are seen as random processes, see Smith, Farmer, Gillemot, and Krishnamurthy (2003), Cont, Stoikov, and Talreja (2010), Abergel and Jedidi (2011), Cont and De Larrard (2013), Lakner, Reed, and Stoikov (2013), Lachapelle, Lasry, Lehalle, and Lions (2014), Bayer, Horst, and Qiu (2015) and Abergel and Jedidi (2015). With the notable exception of Abergel and Jedidi (2015), where the authors consider the case of Hawkes-type dynamics, these models usually assume Poisson flows 1 for the order arrival processes.…”

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

“…As mentioned earlier, we consider in the present paper the case E(log M ) > 0. The case E(log M ) < 0 is studied in Lakner, Reed and Stoikov (2016) and exhibits a fundamentally different behavior. Namely, when E(log M ) < 0 the entire limit order book appropriately scaled is asymptotically concentrated at the price and the latter converges to a monotonically decreasing process.…”

“…When E(log M ) > 0 this is no longer the case, and the main challenge of the present paper is to show that the price process converges to a geometric reflected Brownian motion. Because of this fundamentally different asymptotic behavior, the techniques developed in Lakner, Reed and Stoikov (2016) cannot be directly applied to study the case E(log M ) > 0.…”