Interpolating and Fitting Implied Volatility Curves
Summary
The discussion explains why estimating implied volatility between quoted moneyness points is more involved than applying a high-degree polynomial. A useful volatility curve should be smooth and avoid arbitrage, so a simple linear or polynomial interpolation may produce undesirable shapes. Cubic splines are offered as an interpolation alternative, while parametric models can be calibrated to market quotes.
The appropriate method depends on the intended use and market. The responses mention SABR for interest rates and SVI for stocks or commodities, and distinguish exploratory research from production implementations, where convergence and edge cases require attention. The document gives no comparison of model performance or worked calibration, and it does not specify how to extrapolate beyond the observed moneyness range. Its guidance is therefore a starting point rather than a complete fitting procedure.
Key ideas
- A fitted volatility curve should be smooth and avoid arbitrage.
- High-degree polynomial interpolation may create unsuitable shapes.
- Cubic splines are one alternative for interpolating quoted volatilities.
- Parametric models can be calibrated to market quotes, with model choice depending on the asset class and use case.
- Production fitting requires handling convergence failures and edge cases.
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Full text
# Calculating implied volatility from moneyness/volatility values for date # Calculating implied volatility from moneyness/volatility values for date For an option expiring at a particular date I have ``` Moneyness 0.4,0.7,0.85,0.95,1,1.05,1.15,1.3,2.5 Vol 0.105,0.075,0.045,0.045,0.202,0.045,0.045,0.075,0.085 ``` How do I get the volatility for an option with a particular moneyness? Do I interpolate with an 8th degree polynomial? How would I then handle cases where moneyness is outside the range? Or do I need to do curve fitting? ## Answer by byouness (score 0, accepted) https://quant.stackexchange.com/a/39863 The interpolation of the implied vol surface is no easy task unfortunately and it is subject of extensive research. This is because you want the vol surface to have some nice characteristics, e.g.: be smooth, non arbitrable, etc. Two approaches exist: - Assuming a parametric form for the volatility surface and calibrating it on the quoted implied volatilies. - Interpolation of the quoted implied volatilities. In this case, linear or polynomial might not be a good idea (not smooth), try cubic splines instead. I will just add that everthing depends on what you want to do with this volatility afterwards. ## Answer by Hui (score 1) https://quant.stackexchange.com/a/39919 I would say the best solution will be curve fitting, which is definitely not an easy solution. Depends on whether you are fitting interest rates or stocks/commodities. You might try SABR for the former and SVI for the letter. There are lots of papers about them that you should easily find online. However, it also depends on you are doing research or developing production-level. If the former, you can easily find open source old in R or python to quickly play and visualize. If the latter, you might need to spend more time to handle edge cases solving nin-converging issues. Fittings always have many tricky things you need to figure out.
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