Deconstructing the Volatility Smile

Abstract

This paper investigates the relationship between the implied volatility smile and the underlying joint density of two quantities characterizing the stochastic volatility process - namely the mean integrated variance, $\frac{1}{T}\int_0^T\sigma_s^2ds$, and the stochastic integral $\int_0^T\sigma_s dW_{s}^{\sigma}$. A simple form of this joint density is proposed which, when fit to the zero correlation smile and a single non-zero correlation smile, will then generate to good agreement the smile for an arbitrarily chosen correlation. Further, the method complements and extends the work of \cite{carr_lee_robust} and \cite{friz_gatheral} to non-zero correlation. In doing so, it allows for the study of volatility derivatives in the quanto case which is particularly relevant in the foreign exchange markets.

Keywords: volatility Smile, SABR model, Heston model, volatility swap, quanto

JEL Classification: G12