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 H ∈ (−1/2, 1/2), we derive a Markovian approximating class of one dimensional reversionary Hestontype models. Such proxies encode a trade-off between an exploding vol-of-vol and a fast mean-reversion speed controlled by a reversionary timescale ϵ > 0 and an unconstrained parameter H ∈ R. Sending ϵ to 0 yields convergence of the reversionary Heston model towards different explicit asymptotic regimes based on the value of the parameter H. In particular, for H ≤ −1/2, 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.