We've walked a million miles for one of these smiles
We derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and to congenial numerical treatments.
We show that the skew and the curvature of the smile can be computed as exotic options, for which the Hedged Monte Carlo method is particularly well suited.
When applied to options on the S&P index, we find that the skew and the curvature of the smile are very poorly reproduced by the standard Edgeworth (cumulant) expansion. Most notably, the relation between the skew and the skewness is inverted at small and large vols, a feature that none of the models studied so far is able to reproduce. Furthermore, the around-the-money curvature of the smile is found to be very small, in stark contrast with the highly kurtic nature of the returns.