# self-consistent parametric form for equity implied volatility

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?

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.

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.

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)