4
$\begingroup$

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

  1. What could be the assumption behind taking log-strikes instead of absolute strikes?
  2. What is the intuition behind this measure being constant for different maturities for a given underlying?
$\endgroup$
  • $\begingroup$ Links are dead. $\endgroup$ – Robino Nov 7 '14 at 11:07
  • $\begingroup$ Updated the links. $\endgroup$ – Neerav Nov 10 '14 at 16:49
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.