Fitting Option Skew While Controlling Arbitrage Risk
Summary
The discussion compares ways to fit implied volatility skew across strikes for a single option maturity. A quadratic polynomial is raised as a simple alternative to higher-order polynomial fits, but the responses focus on parametric distribution approaches and the risk of arbitrage violations.
One answer describes fitting with a shared forward price, which constrains the smile’s minimum to at-the-money-forward, and contrasts this with using different forwards to represent off-center minima. The latter flexibility may produce arbitrage-consistent problems, including invalid cumulative distribution values. The responses suggest shifted distributions or SSVI and eSSVI parameterizations; the latter approaches are described as offering arbitrage-free surface construction with relatively few parameters and a calibration procedure. The exchange provides recommendations rather than a full comparison or empirical test, so model suitability and calibration quality still require checking against the relevant market quotes and constraints.
Key ideas
- A quadratic polynomial offers simplicity, but flexibility in fitting skew can create artifacts or arbitrage concerns.
- Using one forward price constrains the fitted smile minimum to the at-the-money-forward point.
- Separate forward prices can accommodate off-center minima but may produce invalid distribution properties.
- Shifted distributions and SSVI-style parameterizations are proposed to address fitting and arbitrage constraints.
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Full text
# Fitting Function for Skew
# 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 ?
## Answer by ZRH (score 2, accepted)
https://quant.stackexchange.com/a/44455
@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 ...
## Answer by JohnDoe (score 4)
https://quant.stackexchange.com/a/44271
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.