The smile of stochastic volatility : Revisiting the Bergomi-Guyon expansion

We revisit the so-called Bergomi-Guyon expansion (Bergomi and Guyon, Stochastic volatility's orderly smiles, Risk, May 2012). The expansion provides the smile of implied volatility at second order in the volatility of volatility for general stochastic volatility models, including variance curve models. First, we present a new derivation of the price expansion which relies on the probabilistic interpretation of the solutions to the partial differential equations satisfied by the correction terms, and on the remarkable fact that the greeks that naturally show up in the expansion are martingales in the unperturbed state. Second, we introduce the concept of implied spot-variance covariance by showing that a unique time-homogeneous instantaneous spot-variance covariance can be inferred from the market term-structure of at-the-money (ATM) skew, based on the Bergomi-Guyon expansion at first order; in particular we show that, at first order, the only time-homogeneous models consistent with a power-law decay of the ATM skew have a spot-volatility covariance of the rough volatility type. We also suggest alternatives to rough volatility that account for the empirical observation that the term-structure of ATM skew, even though it globally decays like a power law, does not seem to blow up for vanishing maturity. Third, we use the Bergomi-Guyon expansion to provide a new, economical derivation of the structural link between the ATM skew and the skewness of log-returns at first order in general stochastic volatility models. Finally we complement Bergomi and Guyon (2012) (1) by proving how the price expansion translates into the implied volatility expansion, and (2) by deriving the explicit expansion and providing some extra numerical experiments in the case of the two-factor Bergomi model