Modeling microstructure noise with mutually exciting point processes

E. Bacry; S. Delattre; M. Hoffmann; J. F. Muzy

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Modeling microstructure noise with mutually exciting point processes

E. Bacry, S. Delattre, M. Hoffmann, J. F. Muzy

arXiv:1101.3422v1Jan 18, 2011q-fin.TRmath.PR

TL;DR

The paper asks how tick-level price models can represent microstructure noise and cross-asset decorrelation while recovering coarse-scale diffusion. It develops marked Hawkes point-process models for positive and negative jumps, derives closed-form multiscale second-order properties, and compares them with futures data. The model reproduces major stylized facts, although its bivariate fits do not capture exact empirical signature plots and correlation functions.

Problem
Tick-level discreteness and microstructure noise complicate volatility and correlation estimation, while existing latent-price models do not faithfully reproduce microscopic price behavior.
Method
The paper constructs univariate and multivariate marked Hawkes point-process models whose coupled upward and downward jump intensities control price behavior across scales.
Results
Closed-form second-order expressions recover signature-plot behavior and the Epps effect across scales, with empirical comparisons on Euro-Bund and Euro-Bobl futures.
Takeaways & Limitations
The framework provides a tractable way to study how microscopic mean reversion aggregates into cross-correlated Brownian diffusion behavior.
Takeaways & Limitations
The exponential kernels and imposed symmetries are arbitrary, and the bivariate model has difficulty accounting for exact empirical signature plots and correlation functions.

Abstract

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We introduce a new stochastic model for the variations of asset prices at the tick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair of assets). The construction is based on marked point processes and relies on linear self and mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By coupling suitably the stochastic intensities of upward and downward changes of prices for several assets simultaneously, we can reproduce microstructure noise (i.e. strong microscopic mean reversion at the level of seconds to a few minutes) and the Epps effect (i.e. the decorrelation of the increments in microscopic scales) while preserving a standard Brownian diffusion behaviour on large scales. More effectively, we obtain analytical closed-form formulae for the mean signature plot and the correlation of two price increments that enable to track across scales the effect of the mean-reversion up to the diffusive limit of the model. We show that the theoretical results are consistent with empirical fits on futures Euro-Bund and Euro-Bobl in several situations.

1 Introduction

The paper addresses how to model tick-level price dynamics while connecting microscopic microstructure effects to coarse-scale diffusion and cross-asset behavior. It proposes Hawkes-process models whose aggregation captures microstructure noise and the Epps effect across scales.

Motivation: High-frequency price modeling is difficult because discrete trade arrivals and price changes introduce strong small-scale mean reversion, complicating volatility estimation.At very fine scales, realized volatility increases as the sampling interval decreases, unlike the flat behavior expected under Brownian diffusion.
Existing approaches: Additive latent-price noise models separate an unobserved Brownian efficient price from observed noisy prices, but do not faithfully reproduce tick-level discreteness.The paper specifically notes that this approach can force the correlation function to diverge as τ → 0.
The Epps effect: Cross-asset correlations typically increase with coarser sampling and nearly vanish at very high frequency, producing the Epps effect.This creates a high-frequency correlation-estimation problem despite clearly correlated intraday movements in Bund and Bobl prices.
Research gap: Existing point-process approaches model high-frequency arrivals or order-book behavior but generally do not address how statistical price properties evolve across resolution scales.The paper positions its contribution as the next step: linking fine-scale price dynamics to intermediate and asymptotic behavior.
Proposed approach: The proposed fine-to-coarse model uses marked multivariate Hawkes processes for positive and negative price jumps, coupling intensities across assets.This construction is designed to reproduce microscopic mean reversion and decorrelation while retaining large-scale Brownian diffusion.
Paper scope: The paper derives closed-form second-order quantities across time scales and lags, supporting signature plots, the Epps effect, lead-lag analysis, and empirical comparisons.It covers univariate and bivariate models, simulation, parameter estimation, diffusive limits, and comparisons with real data.

2 The model in the univariate case

The univariate model represents upward and downward price changes with mutually exciting point processes, using Hawkes intensities to generate microscopic mean reversion while retaining diffusive large-scale behavior. Closed-form signature-plot results support estimation and simulation of the model.

Construction: The price process is constructed as the difference between counting processes for positive and negative jumps.The two point processes are defined over a time horizon and represent the positive and negative variations of one asset.
Construction: Mutually exciting Hawkes intensities make downward activity increase after upward movement, and upward activity increase after downward movement.This selective excitation implements the model’s mean-reverting mechanism.
Construction: Stationarity and stability require the excitation kernel to satisfy ||ϕ||1 < 1.Under this condition, the mutually exciting point processes are well defined and admit stationary increments.
Signature plot: A causal right-sided exponential kernel yields a closed-form expression for the mean signature plot under the stability condition.The resulting proposition describes the scale-dependent signature plot analytically.
Signature plot: The signature plot crosses over from microstructural variance at small scales to diffusive variance at large scales.This crossover is the model’s scale-dependent representation of microscopic mean reversion and macroscopic diffusion.
Estimation and simulation: Both maximum-likelihood and regression estimators recover the model parameters accurately in simulations, while the regression estimator also fits the theoretical signature plot well.The regression estimator can use uniformly sampled data and is described as faster than maximum likelihood estimation.

3 The model in the bivariate case

The bivariate extension couples four Hawkes point processes associated with upward and downward movements of two assets. Its covariance formulas describe signature plots, the Epps effect, and lead-lag behavior across scales, with simulations and estimators supporting the construction.

3.1 Definition: The bivariate model uses four mutually exciting point processes for the positive and negative variations of two asset prices.The two price processes are each built from upward and downward counting processes, then coupled through their intensities.
3.1 Definition: Cross-asset coupling retains upward-X1-upward-X2 and downward-X1-downward-X2 interactions while excluding opposite-direction cross-couplings.This choice is intended to account for mean reversion and cross coupling between the assets.
3.2 Computation of the signature plot and the Epps effect: The covariance matrix Ckl(τ) captures multiscale self- and cross-correlations, including diagonal signature plots and off-diagonal Epps effects.Its lag dependence can also be used to assess lead-lag asymmetry.
3.2 Computation of the signature plot and the Epps effect: Proposition 3.1 provides an explicit expression for the covariance Ckl(τ) as a function of the time scale, with constants given in an appendix.The fully symmetric case has a corresponding covariance expression in Corollary 3.2.
3.2 Computation of the signature plot and the Epps effect: The model reproduces the Epps effect by evaluating ρ(τ) = C12(τ)/C11(τ) across time scales.Simulations compare estimated correlations with analytical curves for asymptotic correlations including 0.15, 0.40, and 0.65.
3.3 Numerical simulations and parameter estimation: Maximum-likelihood and regression estimators both produce quite accurate parameter estimates with errors of the same magnitude.The bivariate model targets both individual signature plots and the cross-asset Epps effect.
3.3 Numerical simulations and parameter estimation: Increasing α13 from 0.01 to 0.05 changes the asymptotic large-scale correlation from 0.15 to 0.65 and makes the simulated processes appear more correlated.The figures use α12 = 0.23, β = 0.11, and µ = 0.015 in both cases.

4 Diffusive (large scale) limit of the model

The paper examines whether the multivariate point-process construction has a macroscopic limit as the observation scale grows. Its second-order results yield a diffusive limit, and semimartingale limit theorems identify a multivariate Brownian motion with the appropriate covariance matrix.

4 Diffusive (large scale) limit of the model: The large-scale question concerns the existence and properties of the N-variate process obtained as T →∞.The analysis focuses on the macroscopic limit of normalized processes and their second-order correlation properties.
4 Diffusive (large scale) limit of the model: In both the univariate and bivariate models, the second-order results establish the existence of a diffusive limit.The bivariate limit is characterized by a covariance matrix.
4 Diffusive (large scale) limit of the model: The simulated Epps effect compares estimated correlation coefficients with expected analytical curves across asymptotic correlations from 0.15 to 0.65.This connects the microscopic scale-dependent correlation behavior with the model’s large-scale correlation setting.
4 Diffusive (large scale) limit of the model: Semimartingale limit theorems can rigorously yield a limiting multivariate Brownian motion with the appropriate covariance matrix.The paper states that detailed descriptions and proofs are presented in a forthcoming paper.

5 Comparison to empirical data

The empirical comparison uses Euro-Bund and Euro-Bobl tick data across two datasets. Despite the model’s simplifying assumptions, its fitted signature plots and cross-asset correlations broadly capture observed behavior across scales.

Interpretation: The empirical comparison is qualitative because the model uses arbitrary exponential kernels and symmetries that may not be suitable in practice.The authors describe the construction as a first-brick model for understanding aggregation from discrete price changes to coarser-scale diffusion.
Data: The study analyzes tick-by-tick last traded prices for Eurex Euro-Bund and Euro-Bobl futures contracts.The contracts represent long-term and medium-term German debt instruments, respectively.
Data: Dataset I covers 21 days from 11/01/2009 to 12/15/2009, sampled from 9am to 11pm on the 12/2009 contract.
Data: Dataset II covers 41 days from 06/01/2009 to 08/01/2009, selecting the most liquid June or September 2009 maturity each day.
Signature plots and Epps effect: For Euro-Bund and Euro-Bobl jointly, the large-scale correlation is close to ρ = 0.77, and the simple bivariate model captures variance and covariance features from small to large scales.The comparison includes the estimated Epps effect and fitted signature plots for both assets.

6 Conclusion and prospects

The paper presents Hawkes-process price models that yield closed-form scale-dependent second-order properties and reproduce signature-plot and Epps-effect patterns, while remaining a simple approximation of market dynamics.

Conclusion and prospects: The bivariate tick-by-tick model uses Hawkes self- and mutual excitation to connect microscopic price changes with large-scale diffusion behavior.The framework is designed to address fine-to-coarse market dynamics using interpretable parameters.
Conclusion and prospects: Closed-form expressions describe the model’s second-order properties across time scales, including signature plots and cross-asset correlations.The model can be simulated and estimated by maximum likelihood or moment methods.
Conclusion and prospects: The 2D model does not fully reproduce the exact empirical behavior of signature plots and correlation functions.The authors characterize it as a simple framework rather than a complete account of observed market dynamics.
Conclusion and prospects: Mean reversion is softened at large scales by diffusion, but the influence of the excitation kernel remains.Its residual large-scale effect can be quantified from microscopic and macroscopic variance expressions.
Conclusion and prospects: The estimated mean-reversion parameter distributions peak near x = 1/3 for Bund and Bobl contracts, with means x = 0.29 and x = 0.36.The paper interprets this pattern as suggesting a strength of microstructural mean reversion associated with minimizing long-term volatility.
Prospects: Future work includes lead-lag effects, nonparametric kernel estimation, volatility nonstationarity and long-range correlations, and exogenous-news effects.These topics are identified as extensions of the parametric Hawkes framework.

Appendix 1: Signature plot in the univariate case

The univariate appendix derives the signature plot by exploiting symmetry between positive and negative jump-counting processes and solving the associated covariance relations for an exponential kernel.

Appendix 1: For an exponential kernel, the univariate signature-plot shape can be computed directly from the model’s covariance equations.The derivation begins by defining an auxiliary quantity and applying the model relations.
Appendix 1: Symmetry between the two jump-counting processes provides the signature-plot expression for τ > 0.The processes correspond to the positive and negative price movements in the univariate construction.
Appendix 1: The covariance calculation separates the continuous covariance of jump densities into Mij(t) and the difference M(t) = M11(t) − M12(t).This difference is used to characterize the relevant covariance structure for the signature plot.
Appendix 1: Conditional means with respect to the future filtration are averaged unconditionally to derive equations for Mij(τ) and M(τ).The resulting relation for M(τ) is then solved using an exponential ansatz.

Appendix 2: Correlation function in the multivariate case

The multivariate appendix derives cross-asset correlation functions by transforming covariance equations into coupled linear systems, solving them, and inverting the resulting Laplace transforms.

Appendix 2: The system solution uses the mean intensity vector Λ and an auxiliary vector v, with components of J evaluated at kernel-related transform arguments.The appendix then determines four constants from the transformed system.
Appendix 2: The covariance matrix C is expressed through the continuous covariance matrix M, a kernel matrix K, and an auxiliary quantity J.These objects organize the calculation of correlations across assets and lags.
Appendix 2: The matrix M satisfies an integral equation whose negative-lag values follow from Mij(−v) = Mji(v).This symmetry relates the covariance at opposite lags while exchanging asset indices.
Appendix 2: Applying a unilateral Laplace transform converts the covariance integral equation into a coupled linear system for the transformed matrix M.A corresponding linear system is also obtained for the transformed auxiliary quantity J.
Appendix 2: After solving for the constants and inverse-transforming, the appendix obtains the functions Kρν(t) and the correlation matrix Cαβ(t, τ).A final double integral yields the stated correlation expressions.