On the skew and curvature of implied and local volatilities

TL;DR

This paper explores the relationship between local and implied volatility surfaces at short maturities, revealing how rough volatility characteristics influence skew and curvature through advanced mathematical techniques.

Contribution

It introduces a Malliavin calculus-based framework that recovers the $rac{1}{H+3/2}$ rule linking local and implied volatility skews in rough volatility models.

Findings

Reveals the $rac{1}{H+3/2}$ rule for short-end skew slopes.

Expresses implied volatility curvature in terms of local volatility features.

Shows the dependence of these relationships on the Hurst parameter H.

Abstract

In this paper, we study the relationship between the short-end of the local and the implied volatility surfaces. Our results, based on Malliavin calculus techniques, recover the recent $\frac{1}{H+3/2}$ rule (where $H$ denotes the Hurst parameter of the volatility process) for rough volatilitites (see Bourgey, De Marco, Friz, and Pigato (2022)), that states that the short-time skew slope of the at-the-money implied volatility is $\frac{1}{H+3/2}$ the corresponding slope for local volatilities. Moreover, we see that the at-the-money short-end curvature of the implied volatility can be written in terms of the short-end skew and curvature of the local volatility and viceversa, and that this relationship depends on $H$.