# Volar Higher Order Parametrizations

I came across this presentation from volar.io. The authors show fitting examples for a flexible volatility smile parametrization in 5 to 8 parameters which is also able to fit the locally concave market implied volatility smiles around special events.

Does anybody know the details of their parametrization and can you provide a reference? In particular, is it a simple extension of their C3 parametrization where the Cn curve is given by

$$\sigma^2(z) = \sigma_0^2 \left( 1 + \sum_{i = 1}^{n - 1} \frac{1}{n!} \xi_i z^n \right)$$

with

$$z = \frac{\ln(K / F)}{\sigma_0 \sqrt{T}}.$$

I suppose this is not the case and there is more to it. Some reasons:

• Their examples look very stable on the wings which I would not expect from higher order polynomials. While they do not show too much extrapolation, their C5 and C6 curves on slides 31 and 32 look fairly well behaved in the wings (where they loose some quality of fit though).

• It might be possible that they define a lower and upper cutoff beyond which they use a different tail function (e.g. linear in variance) and impose smoothness in these points. However on slide 10, they explicitly write that they don't like "hacks" in the wings.

• Another "goal" states on slide 10 is for no-arbitrage constraints to be easy to incorporate. In the above setup, absence of butterfly arbitrage at all strikes used for the calibration creates non-linear constraints for an otherwise nice linear problem.

• Interesting question, I was not aware of such developments. I wonder if no (static) arbitrage conditions can easily be accounted for under such a parametrisation. – Quantuple Sep 28 '16 at 13:44
• @Quantuple: That will exactly be my follow up question once we know how the parametrization looks like. It is usually not difficult to compute the density by just differentiating - either by hand (Mathematica) or using AD. The question is if we obtain sth. tractable that can be easily incorporated as a constraint in the fitting. Also - can it be ensured for all strikes or just a selected set? – LocalVolatility Sep 28 '16 at 13:53
• Completely agree with you. – Quantuple Sep 28 '16 at 13:57
• @LocalVolatility There seems more to it, they claim this can be done in real-time so Mathematica or AD is not a likely solution. – pyCthon Sep 30 '16 at 21:19
• It's just a Taylor expansion, given its fitted using data in the wings, it's still going to be stable there. It's only extrapolation where it starts to become bad. – will Oct 29 '16 at 18:12