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We introduce a new stochastic model for the variations of asset prices at thetick-by-tick level in dimension 1 (for a single asset) and 2 (for a pair ofassets). The construction is based on marked point processes and relies onlinear self and mutually exciting stochastic intensities as introduced byHawkes. We associate a counting process with the positive and negative jumps ofan asset price. By coupling suitably the stochastic intensities of upward anddownward changes of prices for several assets simultaneously, we can reproducemicrostructure noise (i.e. strong microscopic mean reversion at the level ofseconds to a few minutes) and the Epps effect (i.e. the decorrelation of theincrements in microscopic scales) while preserving a standard Browniandiffusion behaviour on large scales. More effectively, we obtain analyticalclosed-form formulae for the mean signature plot and the correlation of twoprice increments that enable to track across scales the effect of themean-reversion up to the diffusive limit of the model. We show that thetheoretical results are consistent with empirical fits on futures Euro-Bund andEuro-Bobl in several situations.

Modeling microstructure noise with mutually exciting point processes