Local volatility under rough volatility

Abstract

AbstractSeveral 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, supporting their calibration power to SP500 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 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 et al. (2002). Quantitative Finance, 2, 61–69], is replaced by a new rule: the ratio of the at‐the‐money implied and local volatility skews tends to the constant (as opposed to the constant 1/2), where H is the regularity index of the underlying instantaneous volatility process.