# Fitting Function for Skew

I am faced with having to fit skew/smile to option quotes with different strike and same maturity. In order to keep things reasonably simple and to avoid potential artifacts from fitting higher order polynomials, I thought of using a quadratic polynomial.

Is there a consensus on which functions to use, resp. a paper discussing what makes most sense ?

• Don't interpolate implied volatilities, it doesn't assure the absence of arbitrage opportunities. Instead, interpolate prices by using a lognormal mixture: you'll get an implied density always positive (= no arbitrage) and a few parameters to describe the whole chain. This goes straight to the point. Then "skew" is just the fat left tail of your density function, something you can manipulate in every possible way simply by knowing the lognormal mixture density function (which is very straightforward). – Lisa Ann Feb 24 at 15:27
• Many thanks for this ! I am checking that things remain arbitrage-free, but an approach which assures this by definition, is clearly more elegant – ZRH Feb 24 at 16:19
• I would second what Lisa Ann commented. Professional services like Bloomberg fit the prices. I spent more than 3 months on fitting prices vs implied volatility using all sorts of ways, and found fitting prices more stable, and more realistic. Fitting volatility with whatever technique can end up with weird shapes in some corner case (some weird day etc), or in some bad data cases. – uday Feb 25 at 13:28
• I have now implemented this in a clean way. @Lisa Ann, I have done it according to Bahra (1996), seeing that for typical equity shapes (high skew and little smile), the fit is less than perfect if you use just two lognormals. Can this be remedied by using more, or is it generally a weakness of the approach ? – ZRH Mar 5 at 9:19
• @ZRH, using more than two lognormals is the right answer. Usually, for American options (I guess that, if you're working with equity, you're dealing with American ones) a mixture of three is the best. By increasing the number of lognormals you could fit any option chain, but this comes at the expense of the model's thrift. Three standard deviations and two weights for a whole chain seem a reasonable trade-off. – Lisa Ann Mar 5 at 11:02

## 2 Answers

Why don't you just use SSVI (https://arxiv.org/abs/1204.0646) or maybe even eSSVI (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2971502)? With this parametric approaches an arbitrage free volatility surface is guaranteed and you only need a handfull of parameters.

Gatheral and Jacquier even give you the calibration procedure which should be simple to implement.

• Agree with @JohnDoe. SVI (including its arb-free enhancements) is really good enough for most options markets. – ilovevolatility Mar 7 at 12:33
• @JohnDoe: Currently, I am fitting single slices, and I don't have enough strikes to really see the smile well (i.e. the minimum of variance as a function of k), so I find it very hard to get my fitting procedure to work – ZRH Mar 10 at 17:56

@Lisa Ann: Typing an answer to my own post, mostly to share my "findings" for the benefit of anyone coming across this.

Looking at the paper of Brigo, Mercurio and Rapisarda, they fit using a single forward price. This comes at the expense of being able to fit only smiles, where the minimum is ATMF. I asked why, and got the answer that choosing different forward prices for the individual fit functions (as does Bahra) will allow for fitting smiles with non-ATMF minima (which I often find in the markets I am concerned with), however it may result in results that allow for arbitrage. While for $$\lim_{K \to 0}$$ and $$\lim_{K \to \infty}$$ there is no issue. However, I have indeed observed that CDF values <0 resp. >1 do occur.

Looks like shifting the distributions as described in the Brigo paper is called for ...