A Remark on Gatheral's 'Most-Likely Path Approximation' of Implied Volatility

Abstract

We give a rigorous proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we fix the problem of a circular definition in the original derivation of Gatheral, who introduced the implied volatility representation in his book 'The Volatility Surface'.

Keywords: Implied Volatility, Local Volatility, Stochastic Volatility, Volatility Surface, most-likely path