This study provides explicit formulas for the moments and the autocorrelation function of the number of jumps over a given interval for a self‐excited Hawkes process. These computations are possible thanks to the affine property of this process. Using these quantities an implementation of the method of moments for parameter estimation that leads to an fast optimization algorithm is developed. The estimation strategy is applied to trade arrival times for major stocks that show a clustering behavior, a feature the Hawkes process can effectively handle. As the calibration is fast, the estimation is rolled to determine the stability of the estimated parameters. Lastly, the analytical results enable the computation of the diffusive limit in a simple model for the price evolution based on the Hawkes process. It determines the connection between the parameters driving the high‐frequency activity to the daily volatility. © 2013 Wiley Periodicals, Inc. Jrl Fut Mark 34:548–579, 2014