# Question on Realized Vol vs Implied Vol

I have heard the following argument- barring transaction fees, if my estimation of future realized vol is 30% and 1-month ATM implied vol is 20%, then I could potentially buy a 1-month ATM call/put and delta hedge it; as time passes and my vol estimate comes true I will make a profit.

The underlying argument is that ATM implied vol is a good proxy for market consensus value of realized vol. So I can compare ATM implied vol against my estimate of upcoming realized vol.

Is this right at all? If yes, why is this the case?

If no, which strike and expiry on the vol surface is the best proxy of realized vol?

## 1 Answer

I have heard the following argument- barring transaction fees, if my estimation of future realized vol is 30% and 1-month ATM implied vol is 20%, then I could potentially buy a 1-month ATM call/put and delta hedge it; as time passes and my vol estimate comes true I will make a profit.

You are not guaranteed to make a profit even in this case since delta hedging profit and loss is path dependent. For instance if in a period of high realised volatility (exceeding your implied vol) your gamma happens to be high because you are at-the-money again then all your past gains could potentially be wiped out.

EDIT: I just realised in your example you are the buyer, but that doesn't change the idea: you might still end up with a loss if realised volatility is lower than your implied when your gamma is high.

The underlying argument is that ATM implied vol is a good proxy for market consensus value of realized vol. So I can compare ATM implied vol against my estimate of upcoming realized vol. Is this right at all? If yes, why is this the case? If no, which strike and expiry on the vol surface is the best proxy of realized vol?

This is a tricky one. First of all, there is a result by Durrleman that states that the very short time to maturity limit of ATM implied volatility is the instantaneous volatility. However, if you are speaking of future realised volatility, i.e. not over an infinitesimal time-step but over a finite time interval, then the ATM implied volatility is not the risk-neutral expectation of future realised volatility.

The risk-neutral expecation of future realised volatility is the volatility swap strike, and under the assumption that the smile is generated by a general stochastic volatility model (possibly driven by fractional noise), that is approximately equal to the implied volatility at the strike where the Black-Scholes vanna of a vanilla call/put is zero. This is a result proved in a paper by Rolloos and Arslan.

Note that I specifically mention risk-neutral expectation of future realised volatility, and I did not say proxy, because I am not sure what you mean with 'proxy'. In most cases I would say market consensus on expected future realised volatility is its risk-neutral expectation which I spoke about.

Then there is of course the risk-neutral expectation of future realised variance, which is the variance swap price, and the square root of the variance swap strike will not be equal to the volatility swap strike.