Higher-Order Volatility Smile Fits and Wing Parameters

# Volar Higher Order Parametrizations


# 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

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

with

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

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.

## Answer by onlyvix.blogspot.com (score 2)

https://quant.stackexchange.com/a/30360

I am not a customer, and not familiar with the details of their parametrization, but on page 25 they imply that next 2 parameters (S5) are independent wing parameters.

Later on page 25 and 26 they imply that other parameters S6,S7,S8 are to introduce W-shaped wiggle for names like SPY, AAPL and GOOG.

## Answer by Misha Fomytskyi (score 1)

https://quant.stackexchange.com/a/40729

The formula for S3 volatility curve is explicitly given in one of the presentations on Vola Dynamics website. In fact, it is functionally equivalent to SSVI curve. BTW, the company name and the website address have changed. See voladynamics.com