Second order approximations for limit order books

Abstract

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g. the volume standing at the top of the book). We study the fluctuations of the price and volume process relative to their first order approximation given in ODE-PDE form under two different scaling regimes. In the first case we suppose that price changes are really rare, yielding a constant first order approximation for the price. This leads to a measure-valued SDE driven by an infinite dimensional Brownian motion in the second order approximation of the volume process. In the second case we use a slower rescaling rate, which leads to a non-degenerate first order approximation and gives a PDE with random coefficients in the second order approximation for the volume process. Our results can be used to derive confidence intervals for models of optimal portfolio liquidation under market impact.2010 Mathematics Subject Classification. 60F17, 91G80. Key words and phrases. Functional central limit theorem, second order approximation, high frequency limit, limit order book.Financial support through the CRC TRR 190 is gratefully acknowledged.

Diffusion Limit of Poisson Limit-Order Book Models

“…We do not scale the price ticks. By scaling price ticks as well as time and order volume, one can obtain limiting models that are governed by partial differential equations in the case of fluid scaling (see Gao & Deng [23], Horst & Kreher [31], and Horst & Paulsen [33]) and by measure-valued stochastic differential equations or stochastic partial differential equations in the case of diffusion scaling or multiple time scales (see Bayer, Horst & Qiu [6], Horst & Kreher [32], Lakner, Reed & Stoikov [46], and Lakner, Reed & Simatos [45]). Kirilenko, Sowers & Meng [42] develop a model with three time scales and both fluid and diffusion scalings.…”

Jump diffusion approximation for the price dynamics of a fully state dependent limit order book model

“…Figure 3. The idea of the shadow book is taken from [17] and is subsequently used in [14,15]. It has to be understood as a technical tool to model the (conditional) distribution of the size of limit order placements inside the spread.…”

“…Deriving a deterministic high frequency limit for limit order book models guarantees that the scaling limit approximation stays tractable in view of practical applications. Such an approach is pursued by Horst and Paulsen [17], Horst and Kreher [14,15], and Gao and Deng [11]. In [14] a weak law of large numbers is established for a limit order book model with Markovian dynamics depending on prices and standing volumes simultaneously.…”

A Scaling Limit for Limit Order Books Driven by Hawkes Processes

“…Their scaling limits required two time scales: a fast time scale for cancelations and limit order placements outside the spread, and a comparably slow time scale for market order arrivals and limit order placements in the spread. The different times scales had at least two drawbacks: first, they imply that the proportion of market orders and spread placements is negligible in the limit; second, as shown in the recent paper [20], they make it impossible to obtain a non-degenerate second-order approximation for the full LOB dynamics. Our scaling limit does not require different time scales.…”