Modeling microstructure noise with mutually exciting point processes
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 withthe 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 increment sin 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.
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