Local volatility under rough volatility

Florian Bourgey, Stefano De Marco, Peter K. Friz and Paolo Pigato

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Abstract: Several asymptotic results for the implied volatility generated by a rough volatility model have been obtained in recent years (notably in the small-maturity regime), providing a better understanding of the shapes of the volatility surface induced by rough volatility models, and supporting their calibration power to S&P500 option data. Rough volatility models also generate a local volatility surface, via the so-called Markovian projection of the stochastic volatility. We complement the existing results on the implied volatility by studying the asymptotic behavior of the local volatility surface generated by a class of rough stochastic volatility models, encompassing the rough Bergomi model. Notably, we observe that the celebrated "1/2 skew rule" linking the short-term at-the-money skew of the implied volatility to the short-term at-the-money skew of the local volatility, a consequence of the celebrated "harmonic mean formula" of [Berestycki, Busca, and Florent, QF 2002], is replaced by a new rule: the ratio of the at-the-money implied and local volatility skews tends to the constant 1/(H + 3/2) (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.

Date: 2022-04, Revised 2022-11
New Economics Papers: this item is included in nep-ets, nep-ore and nep-rmg
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Citations: View citations in EconPapers (3)

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Journal Article: Local volatility under rough volatility (2023)
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:2204.02376

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