  We introduce and establish the main properties of QHawkes ("Quadratic"
Hawkes) models. QHawkes models generalize the Hawkes price models introduced in
E. Bacry et al. (2014), by allowing all feedback effects in the jump intensity
that are linear and quadratic in past returns. A non-parametric fit on NYSE
stock data shows that the off-diagonal component of the quadratic kernel indeed
has a structure that standard Hawkes models fail to reproduce. Our model
exhibits two main properties, that we believe are crucial in the modelling and
the understanding of the volatility process: first, the model is time-reversal
asymmetric, similar to financial markets whose time evolution has a preferred
direction. Second, it generates a multiplicative, fat-tailed volatility
process, that we characterize in detail in the case of exponentially decaying
kernels, and which is linked to Pearson diffusions in the continuous limit.
Several other interesting properties of QHawkes processes are discussed, in
particular the fact that they can generate long memory without necessarily be
at the critical point. Finally, we provide numerical simulations of our
calibrated QHawkes model, which is indeed seen to reproduce, with only a small
amount of quadratic non-linearity, the correct magnitude of fat-tails and time
reversal asymmetry seen in empirical time series.
