Second-Order Approximation of Limit Order Books in a Single-Scale Regime

Ulrich Horst; Dörte Kreher; Konstantins Starovoitovs

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arXiv · 2308.00805

Second-Order Approximation of Limit Order Books in a Single-Scale Regime

Ulrich Horst·Dörte Kreher·Konstantins Starovoitovs

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Abstract

We establish a first- and second-order approximation for an infinite dimensional limit order book model in a single (critical) scaling regime where market and limit orders arrive at a common time scale. With our choice of scaling we obtain non-degenerate first- and second-order approximations for the price and volume dynamics. While the first-order approximation is given by a coupled ODE-PDE system, the second-order approximation is described in terms of an infinite-dimensional stochastic evolution equation driven by a cylindrical Brownian motion. The driving noise processes exhibit a non-trivial correlation in terms of the model parameters. We prove that the evolution equation has a unique solution and that the sequence of standardized limit order book models converges weakly to the solution of the evolution equation. The proof uses a non-standard martingale problem. We calibrate a linearized model to market data and explain how our model can be used for deriving confidence intervals of portfolio liquidation values.

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Ulrich Horst, Dörte Kreher, Konstantins Starovoitovs. 2023-08-01. Second-Order Approximation of Limit Order Books in a Single-Scale Regime. https://arxiv.org/abs/2308.00805

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