A remark on Gatheral's 'most-likely path approximation' of implied volatility

TL;DR

This paper provides a new proof for the representation of implied volatility as a time-averaged weighted expectation, clarifying the existence of forward implied variance in Gatheral's original work.

Contribution

It introduces a novel proof of implied volatility representation, enhancing understanding of forward implied variance in the context of local and stochastic volatility models.

Findings

New proof of implied volatility representation

Clarification on the existence of forward implied variance

Improved theoretical understanding of volatility surface models

Abstract

We give a new proof of the representation of implied volatility as a time-average of weighted expectations of local or stochastic volatility. With this proof we clarify the question of existence of 'forward implied variance' in the original derivation of Gatheral, who introduced this representation in his book 'The Volatility Surface'.