Estimation of the lead-lag parameter from non-synchronous data
Résumé
We propose a simple continuous time model for modeling the lead-lag effect between two financial assets. A two-dimensional process (X-t, Y-t) reproduces a lead-lag effect if, for some time shift theta is an element of R, the process (X-t, Yt+theta) is a semi-martingale with respect to a certain filtration. The value of the time shift theta is the lead-lag parameter. Depending on the underlying filtration, the standard no-arbitrage case is obtained for theta = 0. We study the problem of estimating the unknown parameter theta is an element of R, given randomly sampled non-synchronous data from (X-t) and (Y-t). By applying a certain contrast optimization based on a modified version of the Hayashi-Yoshida covariation estimator, we obtain a consistent estimator of the lead-lag parameter, together with an explicit rate of convergence governed by the sparsity of the sampling design.