# Why is the ATM vol kind of an average volatility

In this question I asked about the mathematical rationale of using the ATM vol to price quanto options. One of the reasons pointed (as an answer) was, as expected, that the ATM volatility is kind of an average volatility.

What is missing is to prove this mathematically or at least give some intutiton of why this is true through mathematical equations. Could you please provide the mathematical reasoning for the conclusion that the ATM vol is kind of an average volatility?

Thank you

See Jim Gatheral's Book "The Volatility Surface" (extract here http://janroman.dhis.org/finance/Volatility%20Models/lecture2%20Fitting%20vola%20skew.pdf) where he obtains the squared implied volatility for strike $K$ as a time average of weighted expectation of square local volatility, then argues that the weight density is concentrated around a curve that connects the initial underlying price to the strike $K$, and finally approximates the density as if it was entirely concentrated on this curve, which he calls the most likely path approximation method. Also remember that the squared local volatility is the expectation of the squared instantaneous (stochastic) volatility conditional on the underlying price and you can view the squared implied vol as approximately the time average of squared instantaneous volatility along that most likely path.