# About the implied volatility as average volatility over the life of an option

The first time I read about local volatility, implied volatility turned out to be the average volatility from today to the option's expiry date.

Let we have two Call options, $$C_1$$ and $$C_2$$, expiring on $$T_1 = 15$$ days and $$T_2 = 45$$ days and we extract the implied volatilities: $$\sigma_1 = 40\%$$ and $$\sigma_2 = 30\%$$, the term structure exhibits backwardation. Moreover, I add another hypothesis: the implied volatility is the perfect forecast of the realized volatility, so no way to make (or lose) money by hedging the Delta because the underlying will be as much volatile as the implied volatility says.

Now I buy a Delta neutral $$C_2$$ ($$= C_2 - \Delta$$ stocks): I'm paying $$\sigma_2 = 30\%$$ to enter this position that will last $$45$$ days over which I will see average volatility equal to $$30\%$$. However, $$\sigma_1 = 40\%$$, so it will happen that the underlying will be more volatile during the first $$15$$ days than during the following $$30$$ ($$= 45 - 15$$) days (of course, we could talk about forward implied volatility but that wouldn't add much content here).

My question is: given that the underlying is much more volatile during the first $$15$$ days and I'm paying lower average volatility ($$\sigma_2 < \sigma_1$$), what does stop me from making money by systematically closing this trade within $$15$$ days if the implied volatility term structure keeps being in backwardation and the other hypothesis hold true?

Under your hypotheses, the implied volatility at which you close the trade out will be the forward volatility $$\sigma_3$$ where $$\sigma_3<\sigma_2$$, so you will make a loss on that. This loss will offset the theoretical gains you have made for the first 15 days of gamma hedging.
• Maybe I got your point: forward implied volatility from $T_1$ to $T_2$ in my example is $23\%$. So of course a volatility crush is implied, but it's just in the current implied volatility surface. I've never seen a forward implied volatility surface being able to actually forecast the future shape of the implied volatility term structure. It means that, if no volatility crush occurs and the implied volatility stays the same, I get a profit by "rolling up" the curve. – Lisa Ann Jan 31 at 12:00