Hedging Calendar Spreads to Target Forward Implied Variance
Summary
The document explains how to structure a delta-hedged calendar spread to isolate changes in forward implied variance. Rather than targeting forward volatility as the square root of a difference in total variance, it recommends trading the difference between the later and earlier options’ total implied variances. Near-the-money options at both maturities can be combined with weights chosen so the spread’s daily profit and loss approximates changes in that variance difference.
The reasoning uses Black–Scholes gamma and delta to remove much of the underlying price move, and notes that vanna and volga are small near at-the-money, though not zero. The argument is an approximation, not a complete replication: it depends on near-ATM options and on managing residual Greek exposure. The document gives no empirical backtest or performance evidence, and it does not detail transaction costs or the precise operational choice of notionals.
Key ideas
- Target changes in the difference between two maturities’ total implied variances to express a forward variance view.
- A delta-hedged calendar spread can approximate exposure to that variance difference.
- Near-the-money options help limit vanna and volga effects, but those effects remain nonzero.
- Use Black–Scholes gamma and delta relationships to set relative notionals and hedge underlying moves.
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Full text
# How to structure a trade using vanilla equity options to get vega exposure to forward volatility?
# How to structure a trade using vanilla equity options to get vega exposure to forward volatility?
I have been thinking about structuring a trade to get exposure to the forward volatility. For example, let's say SPY ATM 1 month IV is 20 vol and SPY ATM 2 month volatility is 30 vol. Then the forward vol would be SQRT(30^2 * 2 - 20^2 * 1) or approx 37.5 vol. I want vega exposure to this forward vol. All I can think of is a gamma neutral calendar spread. Any other ideas?
## Answer by Frido (score 5, accepted)
https://quant.stackexchange.com/a/75346
Let $I(K_1)$ be the IV of a vanilla option with strike $K_1$ and maturity $T_1$ and similarly $I(K_2)$ corresponds to strike $K_2$ and maturity date $T_2 > T_1$.
What I'd suggest you try to trade is not $\sqrt{I^2(K_2)T_2 - I^2(K_1)T_1}$, but the difference in total implied variance $I^2(K_2)T_2 - I^2(K_1)T_1$ instead. So basically what you want to trade is the change in the difference in total implied variance: $$ \mathrm d[I^2(K_2)T_2 - I^2(K_1)T_1] $$
Since $I^2(K_2)T_2 = (I(K_2)\sqrt{T_2})^2$ and similarly for $I^2(K_1)T_1$, $$ \mathrm dI(K_i)\sqrt{T_i} \approx \frac{1}{2I(K_i)\sqrt{T_i}} \,\mathrm dI^2(K_i)T_i $$
Now for options close to the ATM strike the vanna and volga of the option is quite small (although nonzero, but I won't go into that now). So if $K_1,K_2$ both close to ATM the market change of the options can be written as \begin{align} \frac{1}{\Gamma^{BS}(K_i)S_0^2} \left[ \mathrm dC^{BS}(K_i) - \Delta^{BS}(K_i) \mathrm dS_0 \right] &\approx I(K_i) \sqrt{T_i} \, \mathrm dI(K_i) \sqrt{T_i} + \frac12 \sigma_0^2 \, \mathrm dt\\ &\approx \frac12 \mathrm dI^2(K_i)T_i + \frac12 \sigma_0^2 \, \mathrm dt \end{align} where $\Gamma^{BS}$ is Black-Scholes gamma and $\Delta^{BS}$ is the Black-Scholes delta. I am assuming you know what the BS greeks are (including vega, and the relationship between vega and gamma).
It should be pretty clear now what the notionals are of the delta-hedged calendar spread to have a 1-day p/l equal to $\mathrm dI^2(K_2)T_2 - \mathrm dI^2(K_1)T_1$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.