  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.
