Implied Volatility, Gamma Hedging, and Backwardated Volatility
Summary
The document examines whether an investor can profit by buying a longer-dated delta-hedged call when implied volatility is lower than that of a shorter-dated option, then closing the position during the initially higher-volatility period. It uses a hypothetical backwardated term structure and assumes implied volatility perfectly forecasts realized volatility. The question highlights the difference between an option’s average volatility over its life and volatility over a shorter interval.
The answer says that closing the trade requires marking the remaining option at its forward volatility, which is below the longer-dated implied volatility under the example’s assumptions. The resulting loss in option value offsets the theoretical gains from gamma hedging during the first interval. This is a concise argument rather than a full derivation; it depends on the perfect-forecast assumption and does not analyze transaction costs, hedge slippage, changing market conditions, or alternative volatility dynamics.
Key ideas
- A longer-dated option’s implied volatility reflects exposure across its full remaining life, not just its initial interval.
- A higher short-dated implied volatility does not by itself establish a profitable gamma-hedging strategy.
- Closing a delta-hedged longer-dated option entails valuing the remaining maturity at the relevant forward volatility.
- Under the stated assumptions, the lower forward volatility at exit offsets theoretical gains from the initial higher-volatility period.
- The argument omits trading frictions and relies on the assumed relationship between implied and realized volatility.
Tags
Full text
# About the implied volatility as average volatility over the life of an option # 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? ## Answer by dm63 (score 1, accepted) https://quant.stackexchange.com/a/60836 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.
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