Basic Points

• Forward implied volatility (IV) is the difference in IV that can be derived from two points on the term structure.
• It tells us how much IV needs to change from one date to the next after being adjusted for DTE (days to expiration).
• Forward IV is calculated as the square root of the difference in variance divided by the difference in time, with time (T) being the DTE.
• There are other ways to calculate this, but arguably the simplest way to express and calculate this is as: √[(T₂*σ²  - T₁*σ²)  /  ( T₂ - T₁)].
• Regarding the notation, σ means volatility (a one standard deviation [68.3% chance] estimation of the percentage move over a year based on options prices).
• And σ² means variance, which is the square of volatility. Data scientists like to square volatility and work with variance because it keeps each deviation positive. This way, deviations in either direction do not cancel each other out when summing up the deviations to average them.
• The DTE is multiplied against the IV% to weight it.
• As a major takeaway, skew edge can simply subtract differences in IV% vertically since the date is the same, but when calculating term structure edge we have a responsibility to calculate the forward implied volatility in order to compare those different IV% values.

Expert: Derivation of Forward Implied Volatility 

The reason why we are using a coefficient (time) when subtracting variance from variance is because that variance has different meaning based on how much time is remaining. One is stronger than the other depending on time. 

• Reference: √[(T₂*σ²  - T₁*σ²)  /  ( T₂ - T₁)]. 

For example: If one option is 80 DTE (with an IV of 17%) and another option is 50 DTE (with an IV of 12%), then we could frame it this way: √[(80 * 0.17²)  - (50 * 0.12²))   /   (80  - 50)]. 

Answer to the example: the Forward IV is 23%.

On a more basic level, the reason why we are subtracting variance from variance instead of volatility from volatility is because it simply does not go that way (since volatility comes after variance given that it is the square root of variance). 

As a minor point, it is redundant to analyze the volatility like this, trying to distinguish between about 256 days for realized volatility (given that weekend gaps get factored into the same moves). This ends up being the same calculation and it is unnecessary to go this extra step: 

√[(80/365 * 0.17²)  -  (50/365 * 0.12²))  /  (80/365 - 50/365)].