Correlation emergence and the Epps effect in two coupled limit order books

Abstract

We give a unified analytic account of correlation emergence and the Epps effect in two coupled limit order books. The Epps effect is the empirical reduction in measured cross-asset log-return correlation at short aggregation scales, or equivalently the recovery of measured correlation as the aggregation interval increases. The model starts from a fixed-grid discrete random walk for order flow in operational time, with creation, cancellation and diffusion. A pair-trader coupling between the books is introduced at the level of order creation, and calendar time is imposed through separate observation clocks. We clarify how the operational-time model reduces to coupled reaction--diffusion equations with a moving reaction boundary defining the model log-mid-price. Using a regularised local-response representation of the coupling, we derive approximate closed-form expressions for realised correlations as a function of aggregation time. Here the Epps effect is shown to arise from two distinct mechanisms: asynchronous observation clocks (subordination), finite coupling response times, and their combination.

16 pages, 1 figure, 7 appendices. Revised numerical implementation to conform to the fixed operational-time lattice and explicit calendar-time subordination, corrected numerical source and coupling treatment; the analytical results and main conclusions are unchanged. Reproducibility code: https://github.com/timgebbie/correlation-emergence-reproducibility