On the skew and curvature of implied and local volatilities

2022/05/23 by Alòs, Elisa, García-Lorite, David, Pravosud, Makar
#60G22 #91-10 #FOS: Economics and business #G.3 #J.4 #Mathematical Finance (q-fin.MF)

paper · doi:10.48550/arxiv.2205.11185

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 (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 (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.

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