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 model starts from a discrete random-walk description of order flow with creation, cancellation and diffusion. A pair-trader coupling between the books is introduced at the level of order creation. We clarify how the discrete model reduces to coupled reaction--diffusion equations with a moving reaction boundary defining the transaction 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 three distinct mechanisms: asynchronous event clocks (subordination), finite coupling response times, and their combination.

• The paper develops an analytic coupled reaction–diffusion model showing that cross-asset correlation emerges endogenously from order-flow interactions rather than being imposed as a fixed parameter.
• The paper decomposes short-horizon correlation loss into asynchronous event clocks, finite coupling response times, and their interaction, with each mechanism producing explicit scale-dependent correlation curves.
• The paper shows that heavy-tailed waiting times further slow correlation emergence, while regularised coupling and weak first-moment matching are necessary for a valid discrete-to-continuum limit.

This paper develops an analytic account of how cross-asset correlation emerges endogenously from order-flow interactions between two limit order books, and how this emergence produces the Epps effect — the well-documented attenuation of realised correlation at short aggregation horizons (2606.14182). The work extends a numerical simulation framework for coupled reaction–diffusion order books (Bauer et al., 2024) to an approximate closed-form theory, decomposing the Epps effect into three distinct mechanisms: asynchronous event clocks (subordination), finite coupling response times, and their combination.

Model construction

The starting point is a discrete random-walk description of signed order density $\varphi^{(j)}_i$ on a price lattice, with creation, cancellation, and nearest-neighbour diffusion governed by transition probabilities. The master equation includes a memory kernel $K_{n-m}$ associated with Sibuya waiting times, which produces fractional (anomalous) diffusion in the continuum limit, together with a random driving force $F^{(j)}$ and an incremental source term $c^{(j,k)}$. Under the scaling $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$, with $0<\alpha\le 1$, the discrete dynamics converge to a reaction–diffusion equation

$$\partial_t \varphi^{(j)} = D_\alpha\, \partial_x^2\bigl[D_t^{1-\alpha}\varphi^{(j)}\bigr] - \nu_j \varphi^{(j)} + c^{(j,k)}(x,t).$$

A central modelling choice is that the transaction price is not imposed exogenously but defined as a moving reaction boundary $p_j(t) = \{x : \varphi^{(j)}(x,t)=0\}$ separating buy- and sell-dominated regions of the book. This makes price dynamics implicit in the evolution of the order density and enables the local linearisation of the zero-crossing condition that underpins the analytic results.

Pair-trader coupling

The inter-book interaction represents a pairs trader who observes transient mid-price deviations $\Delta p_{jk} = p^{(j)} - p^{(k)}$ and submits side-selective orders near the reaction boundary of the richer book to push prices back together. In the numerical model of (Bauer et al., 2024), this coupling is piecewise, with branches involving inverse powers of the spread. For the analytic derivation, the authors introduce a regularised local-response representation:

$$\ell^{(j,k)}(x,t) = \gamma_{jk}\, z_{jk}(t)\, q_j(y_j)\, W(y_j, z_{jk};\varepsilon),$$

where $K_{n-m}$0, $K_{n-m}$1 is an odd source-shape kernel, and $K_{n-m}$2 is a smooth side-selection function with hard-threshold limit $K_{n-m}$3. The authors are explicit that this regularisation is not merely notational: pointwise equivalence with the legacy coupling fails near $K_{n-m}$4 because branches containing $K_{n-m}$5 or $K_{n-m}$6 have no finite limit there, and unregularised branches can scale as $K_{n-m}$7 in the joint spread–grid limit, so that the diffusion limit and symmetric-spread limit do not commute. Equivalence is instead established weakly: both couplings must project onto the reaction-front displacement mode with the same first moment, $K_{n-m}$8. For the Gaussian-shaped kernel, the side-selected first moment evaluates explicitly to $K_{n-m}$9. A practical matching condition $F^{(j)}$0 is given for resolving the smoothed selector on the lattice.

Analytic decomposition of the Epps effect

The core result is a set of approximate closed-form expressions for realised correlation $F^{(j)}$1 as a function of aggregation scale $F^{(j)}$2, each isolating one attenuation mechanism.

Subordination-only mechanism. With prices evolving as correlated Brownian motions in operational time but observed through independent clocks $F^{(j)}$3, the conditional covariance equals $F^{(j)}$4 times the Lebesgue measure of the overlap of the two sampled clock intervals. This yields the exact factorisation $F^{(j)}$5 under assumptions (A1)–(A4), where all approximation enters through evaluation of the overlap factor. For independent Poisson refresh clocks with pooled rate $F^{(j)}$6,

$F^{(j)}$7

which behaves as $F^{(j)}$8 at small $F^{(j)}$9 and recovers $c^{(j,k)}$0 as $c^{(j,k)}$1. For inverse-stable fractional clocks, exponential survival factors are replaced by Mittag–Leffler functions, giving $c^{(j,k)}$2.

Coupling-only mechanism. Linearising the zero-crossing condition via the frozen-slope (sharp-interface) approximation, and projecting the coupling onto the translational mode of the reaction front, yields a mean-reverting price-level system

$c^{(j,k)}$3

so the spread follows an Ornstein–Uhlenbeck process with relaxation rate $c^{(j,k)}$4. Convolving the resulting exponential cross-covariance kernel with the triangular window-overlap weight gives

$c^{(j,k)}$5

The structural identity of this curve with the subordination case is notable: finite response time and asynchronous sampling generate the same functional form of correlation build-up, through different microstructural channels. An important consequence stated by the authors is that even perfectly synchronous observation does not eliminate short-horizon correlation decay if the coupling response itself is slow — a mechanism absent from earlier reduced-form derivations attributing attenuation solely to sampling or lead–lag effects (Mastromatteo et al., 2010, Chang et al., 2020).

Combined mechanism. When both effects operate, the leading-order separable approximation is

$c^{(j,k)}$6

with Mittag–Leffler replacements $c^{(j,k)}$7 under fractional relaxation. The small-scale asymptotics show that fractional time changes the initial build-up from order $c^{(j,k)}$8 to order $c^{(j,k)}$9, so heavy-tailed waiting times slow short-horizon correlation emergence further.

Relation to prior literature

The paper positions itself at the intersection of three strands: reaction–diffusion latent order book models (Donier et al., 2014, Benzaquen et al., 2017), reduced-form Epps-effect explanations based on non-synchronous trading and estimator bias [epps1979; lo1990; hayashi2005; (Chang et al., 2020)], and discrete-event representations of high-frequency markets. Its distinguishing claim is that correlation here is an output of the dynamics rather than a fixed primitive as in Hawkes-process or multivariate diffusion models (Bacry et al., 2015). The work also complements arguments that the Epps effect reflects the discrete-event nature of markets (Chang et al., 2020) by showing precisely how discreteness survives the diffusion limit and what additional regularity assumptions are required to pass through it.

Limitations and open questions

Several limitations are conceded directly. The combined formula is a first-order separable approximation that is exact only when the clock process and coupling response are independent; it fails when pair-trader activity is itself triggered by the same events that generate observations, or when clocks modify local coupling intensity. The equality $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$0 in the single-clock scaling is a modelling assumption rather than a mathematical necessity. Variance growth is assumed approximately linear over the aggregation scales considered, and drifts are dropped on the grounds that their covariance contribution is lower order. The frozen-slope approximation treats the reaction-front shape as fixed during the response calculation, and the derivation requires only that $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$1 be odd and locally linear — but the equivalence between the legacy discrete coupling and the regularised analytic coupling holds only in a weak, first-moment sense under a specified joint limiting order, not pointwise. Finally, the results are population-level approximations derived analytically; the paper does not present new calibration against empirical data, leaving open the question of whether the fitted parameters $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$2, $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$3, and $D_\alpha = \frac{r}{2}\Delta x^2/\Delta t^\alpha$4 recover realistic Epps-effect curves in actual markets.

Conclusion

The paper provides a unified analytic treatment showing that the Epps effect arises from three distinguishable mechanisms within a single coupled reaction–diffusion framework: clock asynchrony, finite coupling relaxation, and their interaction, each producing explicit scale-dependent correlation curves with common functional form. By making the regularisation required for the discrete-to-continuum passage explicit, it clarifies which features of the effect are discretisation artefacts and which are genuine continuum behaviour.