A Law of Large Numbers for Limit Order Books

Ulrich Horst () and Michael Paulsen ()
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Ulrich Horst: Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany
Michael Paulsen: Institut für Mathematik, Humboldt-Universität zu Berlin, 10099 Berlin, Germany

Mathematics of Operations Research, 2017, vol. 42, issue 4, 1280-1312

Abstract: We define a stochastic model of a two-sided limit order book in terms of its key quantities best bid [ ask ] price and the standing buy [ sell ] volume density . For a simple scaling of the discreteness parameters, that keeps the expected volume rate over the considered price interval invariant, we prove a limit theorem. The limit theorem states that, given regularity conditions on the random order flow, the key quantities converge in probability to a tractable continuous limiting model. In the limit model the buy and sell volume densities are given as the unique solution to first-order linear hyperbolic PDEs, specified by the expected order flow parameters. We calibrate order flow dynamics to market data for selected stocks and show how our model can be used to derive endogenous shape functions for models of optimal portfolio liquidation under market impact.

Keywords: limit order book; scaling limit; averaging principle; queueing theory (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (14)

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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:42:y:2017:i:4:p:1280-1312

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