ATM Implied Volatility as Average Variance Along a Likely Path
Summary
The response offers an intuition for why at-the-money implied volatility can act like an average volatility, particularly when used in quanto option pricing. It describes a result in which squared implied volatility is represented as a time average of weighted expected squared local volatility. The weighting is concentrated near a likely path from the initial underlying price to the option strike; approximating that weighting as concentrated on the path gives a practical interpretation of implied variance.
For an option on the forward price with an at-the-money strike, the likely path is approximated by a constant forward price. Under that approximation, squared ATM implied volatility is the time average of squared instantaneous volatility along the constant path. A variance swap might better capture expected integrated variance, but the response argues that uncertainty in estimating FX and underlying correlation limits the practical benefit. It emphasizes keeping the quanto adjustment independent of strike so call-put parity yields a consistent quanto forward. The account is an approximation, not a general proof that ATM volatility equals average volatility.
Key ideas
- Squared implied volatility can be viewed as a weighted time average of expected squared local volatility.
- A likely-path approximation concentrates the weighting around a path from the initial price to the strike.
- For an ATM option on a forward, that path can be approximated as constant at the forward price.
- Under this approximation, squared ATM implied volatility averages squared instantaneous volatility along that path.
- Quanto adjustments should avoid strike dependence so call-put parity implies a consistent quanto forward.
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Full text
# Why is the ATM vol kind of an average volatility # 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 ## Answer by Antoine Conze (score 1) https://quant.stackexchange.com/a/34493 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. Now if you use as your underlying the forward price and look at ATM implied vol, the most likely path can also be approximated with the constant path set to the forward price and the squared implied ATM vol becomes the time average of squared instantaneous vol along that constant path. You might argue that the variance swap would give a true integrated expected square instantaneous volatility and thus be a better approximation in the context of computing the quanto adjustment for quanto options, but as I said in my answer to your original question Approximations for Quanto Options pricing there is so much uncertainty in estimating the FX / option underlying correlation that it does not really matter, the one thing you really want to make sure of is that your quanto adjustment is not strike dependent so that the quanto forward obtained by call/put parity is the same at all strikes.
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