On the Skew and Curvature of the Implied and Local Volatilities

ABSTRACTIn 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 1H+3/2 rule (where H denotes the Hurst parameter of the volatility process) for rough volatilities (see F. Bourgey, S. De Marco, P. Friz, and P. Pigato. 2022. “Local Volatility under Rough Volatility.” arXiv:2204.02376v1 [q-fin.MF] https://doi.org/10.48550/arXiv.2204.02376.), that states that the short-time skew slope of the at-the-money implied volatility is 1H+3/2 of 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 vice versa. Additionally, this relationship depends on H.KEYWORDS: Stochastic volatilitylocal volatilityrough volatilityMalliavin calculus Disclosure statementNo potential conflict of interest was reported by the author(s).Notes1 For example we can suppose a local volatility flat between 0 to first Monte Carlo, or even to use a discretization of the following asymptotic expression limT→0I0(T,K)=log⁡(KST)∫STk1σ(T,u)du 2 Another way to get the skew is to take the derivative with respect to K in E((ST−K)+)=BS(T,K,I(T,K)). Then we get the following expression for the skew ∂kI(T,K)=−E(I(ST>K))−∂KBS(T,K,I(T,K)∂σBS(T,K,I(T,K)). Notice that the term E(I(ST>K)) can be estimated in the same simulation where we get the price of the option. We have checked both approaches and we have confirmed that they lead to identical resultsAdditional informationFundingThis work was supported by Agencia Estatal de Investigación [grant number PID2020-118339GB-I00].