Asymmetric super-Heston-rough volatility model with Zumbach effect as scaling limit of quadratic Hawkes processes

Hawkes processes were first introduced to obtain microscopic models for the rough volatility observed in asset prices. Scaling limits of such processes leads to the rough-Heston model that describes the macroscopic behavior. Blanc et al. (2017) show that Time-reversal asymmetry (TRA) or the Zumbach effect can be modeled using Quadratic Hawkes (QHawkes) processes. Dandapani et al. (2021) obtain a super-rough-Heston model as scaling limit of QHawkes processes in the case where the impact of buying and selling actions are symmetric. To model asymmetry in buying and selling actions, we propose a bivariate QHawkes process and derive a super-rough-Heston model as scaling limits for the price process in the stable and near-unstable regimes that preserves TRA. A new feature of the limiting process in the near-unstable regime is that the two driving Brownian motions exhibit a stochastic covariation that depends on the spot volatility.