Skip to content
All library documents

Interpreting Maturity-Normalized Option Skew in Log-Strike Space

Article Quant Q&A · Author: Neerav
Read the original

Summary

The document explains why an option skew measure based on volatility differences divided by log-strike distance, with a square-root-of-maturity adjustment, can remain similar across listed expiries. The key comparison uses strikes set at a fixed number of volatility-scaled standard deviations around a forward-like level, rather than at fixed percentage distances from spot. Their log-strike separation grows with volatility and the square root of time, which cancels the maturity factor in the skew formula.

This makes the measure roughly comparable across maturities when options are sampled at similar standardized distances, which are closely related to delta. A fixed percentage moneyness does not represent the same degree of out-of-the-money exposure at short and long expiries. The explanation is an intuition and scaling argument, not empirical proof that skew must be constant. Its usefulness depends on the strike-selection convention, the underlying volatility assumptions, and market conditions; it does not establish a universal term-structure law.

Key ideas

  • Log-strike distance expresses relative rather than absolute strike separation.
  • Comparing strikes at fixed volatility-scaled distances helps align option exposure across maturities.
  • The square-root-of-time factor can cancel the maturity scaling of log-strike separation.
  • Fixed percentage moneyness can represent very different degrees of moneyness at different expiries.

Tags

OptionsVolatilityDerivatives pricing
Full text
# Why does Skew measure remain more-or-less constant for Listed Expiries?


# Why does Skew measure remain more-or-less constant for Listed Expiries?












I have looked at the Variance Swap Papers published by GS-VarSwap and JPM-VarSWap where they talk about approximation to VarSwap strike using ATMF vol and Skew (slope of the volatility skew for 90-110 strikes).

But, I have also come across another 'skew measure' which is defined as $$ \mathrm{skew} = \frac{\sqrt{T}(\sigma_1 - \sigma_2)}{\log(K_1/K_2)}. $$ I understand that $\sqrt{T}$ makes the vol difference 'normalized' in maturity-space. And to my surprise, this measure remains almost constant for mid-to-long term maturities (>6M).

My questions are

- What could be the assumption behind taking log-strikes instead of absolute strikes?

- What is the intuition behind this measure being constant for different maturities for a given underlying?

## Answer by joelhoro (score 1, accepted)

https://quant.stackexchange.com/a/3813

This is a common convention. If your spot is $S$ and you're looking at options maturity in $T$, it is natural to look at the the strikes $S_\pm=S.exp^{-\frac12\sigma^2T\pm\alpha\sigma\sqrt T}$ for a fixed $\alpha$. So your skew measure will be something like $$ \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{\log (S_+/S_-)} = \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{2\alpha\sigma\sqrt T} = \frac{\sigma_{S_+} -\sigma_{S_-}}{2\alpha\sigma} $$

Which is as intrinsic as could be.

Think about it this way: 1 week before expiry a 110% call is way more OTM than a 110% call expiring in one year, just because 110% is very far if you've only got one week left. Another way to look at it is that you want to compare the difference in vol for a given change in delta rather than in % of the spot. And clearly the $\alpha$ is strongly tied to the delta.

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.