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, Quantitative Finance, 13, 65–77) and Da Fonseca and Zaatour (forthcoming, Journal of Futures Markets) both of them based on the Hawkes process, the first one having a mean reverting behavior while the second one a clustering behavior, 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. © 2014 Wiley Periodicals, Inc. Jrl Fut Mark