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We introduce a new model for describing the fluctuations of a tick-by-ticksingle asset price. Our model is based on Markov renewal processes. We considera point process associated to the timestamps of the price jumps, and marksassociated to price increments. By modeling the marks with a suitable Markovchain, we can reproduce the strong mean-reversion of price returns known asmicrostructure noise. Moreover, by using Markov renewal processes, we can modelthe presence of spikes in intensity of market activity, i.e. the volatilityclustering, and consider dependence between price increments and jump times. Wealso provide simple parametric and nonparametric statistical procedures for theestimation of our model. We obtain closed-form formula for the mean signatureplot, and show the diffusive behavior of our model at large scale limit. Weillustrate our results by numerical simulations, and that our model isconsistent with empirical data on the Euribor future.

Semi Markov model for market microstructure