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Modelling microstructure noise with mutually exciting point processes. (English) Zbl 1280.91073
Summary: 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 mutually exciting stochastic intensities as introduced by Hawkes. We associate a counting process with the positive and negative jumps of an asset price. By suitably coupling 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 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 us 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.

MSC:
91B25 Asset pricing models (MSC2010)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G50 Sums of independent random variables; random walks
62P05 Applications of statistics to actuarial sciences and financial mathematics
91B84 Economic time series analysis
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