Stochastic dynamics of limit order books: a journey across time scales

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• Rama Cont (University of Oxford)
• Tuesday 06 August 2024, 16:00-16:30
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SSDW02 - Stochastic reflection

The intraday dynamics of electronic financial markets is the outcome  of a complex interplay between buy and sell orders submitted across a wide range of frequencies by various market participants. The state of such a market may be represented by the limit order book, a  multiclass queueing system indexed by positive integers (price levels), whose stochastic evolution summarizes the dynamics of supply and demand. Given the very high frequency of order arrivals, heavy traffic limits and diffusion approximations provide relevant descriptions for price and order flow dynamics at various time resolutions. The multiplicity of time scales at play and the availability of detailed data down to microsecond  level make limit order book dynamics an interesting laboratory for asymptotic methods in queueing theory. One approach is to describe the dynamics of the full limit order book by a stochastic PDE , for which different formulations  have been proposed. We argue that the ‘correct’ limit is a spatially inhomogeneous limit which leads to a ‘degenerate’ stochastic moving -boundary problems with white noise perturbations on the boundary. A more tractable approach is to zoom in on the ‘boundary layer’ corresponding to the queues with the best buy and sell prices (‘top’ of the order book) and describe their stochastic dynamics as a tandem queue model whose state space is the positive quadrant. Depletion of the buy and sell queues then lead to discontinuous stochastic reflection at the boundary, and in the heavy traffic limit one retrieves a regulated Brownian motion inside the quadrant with stochastic discontinuous reflection on the boundary. Price dynamics is then described by a the difference of two boundary occupation time processes. If time permits we will also discuss a host of related open problems relevant for applications.

This talk is part of the Isaac Newton Institute Seminar Series series.

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