Reconciling Rough Volatility with Jumps

We reconcile rough volatility models and jump models using a class of reversionary Heston models with fast mean reversions and large vol-of-vols. Starting from hyper-rough Heston models with a Hurst index [Formula: see text]-, we derive a Markovian approximating class of one-dimensional reversionary Heston-type models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale [Formula: see text] and an unconstrained parameter [Formula: see text]. Sending [Formula: see text] to 0 yields convergence of the reversionary Heston model toward different explicit asymptotic regimes based on the value of the parameter [Formula: see text]. In particular, for [Formula: see text], the reversionary Heston model converges to a class of Lévy jump processes of normal inverse Gaussian type. Numerical illustrations show that the reversionary Heston model is capable of generating at-the-money skews similar to the ones generated by rough, hyper-rough, and jump models.