Smile convexity vs ATMvol variance

tourkine · December 11th, 2003, 12:27 pm

Hi all,To my knowledge in most of stochastic volatility models the ATM smile convexity is mainly explained by vvol (volatility of volatilty). On the other hand the variance of the ATM vol is also (quite often) closely related to vvol. Can anyone show a link (or even a vague relation) between the observed smile convexity and ATMvol variance without specifying a model (or with minimal assumptions about the instantaneus vol process)?Cheers,Aleksei.

exotiq · December 11th, 2003, 12:32 pm

I'm not sure why you are averse to specifying a model; the idea of implied vol itself is backed out of a model. The important thing is whether something specified in one model can carry over when the assumptions change, but enough philosophy...The simple SABR that pat Hagan wrote about (lognormal forward and vol, no mean reversion) has a formula for implied vol where the convexity of the smile increases with vol vol. I'd start with that, then change the model...

tourkine · December 11th, 2003, 1:21 pm

Thank you for reply,To my intuition the relation between vvol and the smile convexity holds in general case. At least it holds for both SABR and Heston models.For some other models one can check I through a simple perturbation analysis. The idea was to get some information about (unknown) volatility process basing on the ATMvol variance that is itself implied by the observed smile convexity.

Gusak · December 12th, 2003, 9:51 am

Isn't it just like that:ATM option price is much less convex in vol (~linear), than out of the money option prices. Hence, out of the money options benefit from an increase in vol of vol , while ATM price stays virtually unaffected. This translates into more convex implied volatility curve.

mghiggins · December 12th, 2003, 5:27 pm

tourkine - this is an excellent question, and one that very few people examine. It's a nice head check on the reasonableness of a model: given that you set the model parameters to match the smile, what do those parameters imply about, say, the std dev of daily moves of implied vol?If it doesn't tie out then it suggests that the model is perhaps not a qualitatively good one for your market.