Local volatility under rough volatility

TL;DR

This paper investigates the asymptotic behavior of the local volatility surface generated by rough stochastic volatility models, revealing a new relationship between implied and local volatility skews that depends on the roughness parameter H.

Contribution

It extends existing asymptotic results to local volatility surfaces in rough volatility models, introducing a new skew ratio rule dependent on the regularity index H.

Findings

The local volatility skew ratio tends to 1/(H + 3/2) as maturity approaches zero.

The classical 1/2 skew rule is replaced by a new rule involving the roughness index H.

The results encompass the rough Bergomi model and generalize the harmonic mean formula.

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.