Quadratic Hawkes processes for financial prices

Abstract

We introduce and establish the main properties of QHawkes (‘Quadratic’ Hawkes) models. QHawkes models generalize the Hawkes price models introduced in Bacry and Muzy [Quant. Finance, 2014, 14(7), 1147–1166], by allowing feedback effects in the jump intensity that are linear and quadratic in past returns. 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 being at the critical point. A non-parametric fit of the QHawkes model 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. 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.

Keywords:

• Market microstructure
• Hawkes processes
• Financial prices
• Volatility modelling
• Time-reversal asymmetry
• Pearsons diffusion
• High-frequency trading

Acknowledgements

We want to thank R. Chicheportiche, J. Gatheral, S. Hardiman, Th. Jaisson, I. Mastromatteo and M. Rosenbaum for many insightful discussions on these issues. We also thank the referees whose remarks helped improve the quality of our manuscript.

Notes

No potential conflict of interest was reported by the authors.

1 For various generalizations of the Multifractal Random Walk, see the in-depth review proposed in Bacry et al. (2008). Note that some empirical ‘fatness’ can be reproduced by the log-normal MRW. Indeed, although the theoretical tail exponent is much too large, the long memory of the MRW does generate ‘non ergodic’ empirical exponents that match the ones observed in financial time series (Bacry et al. 2010).

2 Although time-asymmetric MRWs can be constructed explicitly, see Bacry et al. (2012b).

3 In fact, if the kernels K, , etc. to arbitrary order are all diagonal, the model boils down to a Hawkes process with leverage, i.e. , with adequately redefined kernels and L such that to ensure positivity of the intensity.

4 Note that iff has unit jumps .

5 In the case of linear Hawkes processes, this condition is also sufficient to obtain stationarity in the case (whereas the case is more subtle, see Brémaud and Massoulié (2001)).

6 Although Zumbach describes this effect at the daily time scale, whereas we will here study intra-day time scales.