# Deconstructing the Volatility Smile

32 Pages Posted: 25 Jul 2014

See all articles by Romano Trabalzini

## Romano Trabalzini

Imperial College London

## William A McGhee

NatWest Markets

Date Written: July 24, 2014

### 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

Suggested Citation

Trabalzini, Romano and McGhee, William A, Deconstructing the Volatility Smile (July 24, 2014). Available at SSRN: https://ssrn.com/abstract=2471052 or http://dx.doi.org/10.2139/ssrn.2471052

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