  Using microscopic price models based on Hawkes processes, it has been shown
that under some no-arbitrage condition, the high degree of endogeneity of
markets together with the phenomenon of metaorders splitting generate rough
Heston-type volatility at the macroscopic scale. One additional important
feature of financial dynamics, at the heart of several influential works in
econophysics, is the so-called feedback or Zumbach effect. This essentially
means that past trends in returns convey significant information on future
volatility. A natural way to reproduce this property in microstructure modeling
is to use quadratic versions of Hawkes processes. We show that after suitable
rescaling, the long term limits of these processes are refined versions of
rough Heston models where the volatility coefficient is enhanced compared to
the square root characterizing Heston-type dynamics. Furthermore the Zumbach
effect remains explicit in these limiting rough volatility models.
