Quadratic Hawkes Processes for Financial Prices

24 Pages Posted: 26 Sep 2015

See all articles by Pierre Blanc

Pierre Blanc

Ecole des Ponts ParisTech

Jonathan Donier

Université Paris VI Pierre et Marie Curie

Jean-Philippe Bouchaud

Capital Fund Management

Date Written: September 25, 2015

Abstract

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.

Keywords: Hawkes processes, financial prices, volatility modelling, time-reversal asymmetry, Pearsons diffusion, high frequency trading, order splitting, fat tails

Suggested Citation

Blanc, Pierre and Donier, Jonathan and Bouchaud, Jean-Philippe, Quadratic Hawkes Processes for Financial Prices (September 25, 2015). Available at SSRN: https://ssrn.com/abstract=2665669 or http://dx.doi.org/10.2139/ssrn.2665669

Pierre Blanc

Ecole des Ponts ParisTech ( email )

6-8 avenue Blaise-Pascal, Cité Descartes
Champs-sur-Marne
Marne-la-Vallée Cedex 2, 77455
France

Jonathan Donier (Contact Author)

Université Paris VI Pierre et Marie Curie ( email )

175 Rue du Chevaleret
Paris, 75013
France

Jean-Philippe Bouchaud

Capital Fund Management ( email )

23 rue de l'Université
Paris, 75007
France
+33 1 49 49 59 20 (Phone)

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