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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 (2014, 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 35:813–838, 2015

Consistently fitting vanilla option surface is an important issue in derivative modelling. In this paper, we consider three different models: local and stochastic volatility, local correlation, hybrid local volatility with stochastic rates, and address their exact, nonparametric calibration. This calibration process requires solving a nonlinear partial integro-differential equation. A modified alternating direction implicit algorithm is used, and its theoretical and numerical analysis is performed.

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The aim of this paper is to develop a multi‐asset model based on the Hawkes process describing the evolution of assets at high frequency and to study the lead–lag relationship as well as the correlation between the assets within this framework. We compute several statistical quantities and the covariance matrix associated with the diffusive limit of the model so that the relation between the parameters driving the assets at high and low frequencies is explicit. We illustrate the results using several financial assets quoted in the Eurex market and show how the model captures the lead–lag relationship between them. © 2016 Wiley Periodicals, Inc. Jrl Fut Mark 37:260–285, 2017