Clustering and Mean Reversion in a Hawkes Microstructure Model

This paper provides explicit formulas for the first and second moments and the autocorrelation function of the number of jumps over a given interval for the multivariate Hawkes process. These computations are possible thanks to the affine property of this process. We unify the stock price models of Bacry et al. (2013a) and Da Fonseca and Zaatour (2013) both of them based on the Hawkes process, the first one having a mean reverting behaviour whilst the second one has a clustering behaviour, and build a model having these two properties. We compute various statistics as well as the diffusive limit for the stock price that determines the connection between the parameters driving the high frequency activity to the daily volatility. Lastly, the impulse function giving the impact on the stock price of a buy/sell trade is explicitly computed