Connecting Rough Volatility Models with Lévy Jump Processes

Summary

This work connects rough volatility and jump models through reversionary Heston-type models with rapid mean reversion and large volatility-of-volatility. It begins with hyper-rough Heston specifications and constructs a one-dimensional Markovian approximating family. A time-scale parameter controls the reversion speed and the associated volatility behavior, while the model also allows the Hurst parameter to range beyond its original interval.

As the time scale tends to zero, the authors derive distinct explicit limiting regimes depending on the Hurst parameter. For parameter values at or below the stated threshold, the limit is a family of Normal Inverse Gaussian-type Lévy jump processes. Numerical illustrations show that the reversionary models can produce at-the-money volatility skews resembling those from rough, hyper-rough, and jump models. The evidence described is numerical; no calibration results, market sample, or empirical forecasting comparison is provided.

Key ideas

• Fast mean reversion and large volatility-of-volatility provide a link between rough volatility and jump models.
• A one-dimensional reversionary Heston family approximates hyper-rough Heston dynamics.
• Different limiting regimes arise as the reversionary time scale approaches zero.
• For sufficiently low Hurst parameter values, the limit is a Normal Inverse Gaussian-type Lévy process.
• Numerical examples show similar at-the-money skews across the reversionary, rough, hyper-rough, and jump models.

Tags

# Reconciling rough volatility with jumps


# 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 $H \in (-1/2,1/2)$, 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 $ε>0$ and an unconstrained parameter $H \in \mathbb R$. 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 $H \leq -1/2$, 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.

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