self-consistent parametric form for equity implied volatility

Question

I recall reading a paper, but can't remember where I found it. In short, there was a parametric form for volatility smile/skew that fit both index and single stock vol slices and had intuitive parameters that were consistent in time. It was something like ATM vol + skew + convexity + 2 or 3 parameters to take care of the OTM quirks and the whole thing was based on log(K/S)/sqrt(t) axis so the parameters were more or less consistent in time. Yet, at the same time it was not a stochastic volatility model, simply a parametric form for implied volatility.

Does anyone remember this paper or have heard of a parametric form that fits these requirements?

Answer 1

It's extremely common in the industry to have a parabolic skew of this type with some cutoff parameters. At it's simplest, such a model looks like this

$$ \sigma_{ATM}(t) = \sigma_0 + s(t) $$

where $s(t)$ is a vol term structure function and can be further simplified to

$$ s(t) = \frac{s}{\sqrt{t}} $$

if you are willing to accept the inaccuracies. Usually at-the-money is ATM forward (i.e. for the strike equal forward price $F(t)$).

Then, base vol $\hat\sigma$ can be characterized by

$$ \hat\sigma(K,T) = \sigma_{ATM}(t) + \gamma \frac{\log(K/F(t))}{\sigma\sqrt{t}} + \lambda \left(\frac{\log(K/F(t))}{\sigma\sqrt{t}}\right)^2 $$

and then we window the vol to keep it from going too crazy:

$$ \sigma(K,T) = \max(\min(\hat\sigma(K,T), \sigma_{max}), \sigma_{min})) $$

Zillions of minor variations on this scheme exist, going back to the 1980s.

Answer 2

You might want to look at "If the skew fits" article by Gregory Brown and Curt Randall from Risk magazine (April, 1999).

Their parameterization has the following form:

$$ \sigma(S,t) = \sigma_{ATM}(t) + \\ \sigma_{skew}(t) * tanh(\gamma_{skew} (t) * {\log(S/S_{0})} - \theta_{skew}(t)) + \\ \sigma_{smile}(t) * [1 - sech (\gamma_{smile}(t) * {\log(S/S_{0})-\theta_{smile}(t)})] $$

They also give a brief explanation of the model and a way to calibrate it.

Answer 3

Are you looking for this? Stock return characteristics, skew laws, and the differential pricing of individual equity options, G. Bakshi, N. Kapadia, and D. Madan. Review of Financial Studies 16(1):101--143 (2003)

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.8.1805&rep=rep1&type=pdf

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