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 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.

Heterogeneous Criticality in High Frequency Finance: A Phase Transition in Flash Crashes

“…To achieve a distribution of log-returns close to normal and stationary in time, we must normalize returns locally to account for two known regularities: daily seasonalities and long memory effects [ 22 , 23 , 24 , 25 ]. Mid-price returns have been shown to have approximate zero mean but a non-stationary variance due to the above [ 26 ].…”

Radical Complexity

“…More precisely, past price trends, whether up or down, lead to higher future volatility but not the other way round. In 2019, following some work by P. Blanc, J. Donier and myself [ 6 ], A. Dandapani, P. Jusselin and M. Rosenbaum proposed to describe financial time series with what they called a “Quadratic Rough Heston Model” [ 7 ], which is a synthesis of all the ideas reviewed above. It is probably the most realistic model of financial price series to date.…”

Exact asymptotic solutions to nonlinear Hawkes processes: a systematic classification of the steady-state solutions

“…However, beyond the derivation of general conditions for existence, obtaining analytical solutions of these models is very difficult due to the complex interplay between their nonlinear and non-Markovian structures. Only a few studies exist, such as the analysis of the stability of these processes (conditions for non-explosiveness) [19], a special solution for the ZHawkes (Zumbach Hawkes) processes with an exponential memory in the diffusive limit [28], and an asymptotic analysis for high-baseline intensity using the functional central limit theorem [29].…”

“…As an improved model, a nonlinear version of the Hawkes process was introduced by Blanc, Donier, and Bouchaud in Ref. [28], where the intensity dynamics is given by a quadratic extension to the standard Hawkes process,…”

“…This model is called the quadratic Hawkes (QHawkes) processes and has been theoretically analyzed in Ref. [28].…”