Reconciling rough volatility with jumps

Eduardo Abi Jaber; Nathan De Carvalho

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arXiv · 2303.07222

Reconciling rough volatility with jumps

Eduardo Abi Jaber·Nathan De Carvalho

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Abstract

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 $H \in (-1/2,1/2)$, 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 time-scale $\epsilon>0$ and an unconstrained parameter $H \in \mathbb R$. Sending $\epsilon$ 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 $H \leq -1/2$, the reversionary Heston model converges to a class of L\'evy 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.

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Eduardo Abi Jaber, Nathan De Carvalho. 2023-03-13. Reconciling rough volatility with jumps. https://arxiv.org/abs/2303.07222

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