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Cubic-Spline Volatility Skews Can Produce Negative Implied Volatilities

Article Quant Q&A · Author: KaiSqDist
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Summary

The document discusses constructing a model-free implied-volatility measure from a range of traded call and put prices. The described procedure first builds an implied-volatility skew by interpolating and extrapolating market observations with a cubic spline, then uses option prices across strikes in a discretized variance-contract calculation. The question is whether negative volatility values from the spline are acceptable and how to prevent them.

The response says implied volatility should be positive and treats negative interpolated values as evidence that the interpolation method or its implementation is unsuitable. It notes that cubic splines are flexible enough to overshoot in some configurations, while linear interpolation between positive inputs remains positive. This is a qualitative warning, not a worked example or a comparison of alternative constrained interpolation methods. It also does not address extrapolation behavior, arbitrage consistency across strikes, or how interpolation choices affect the final model-free volatility estimate; those issues need separate validation.

Key ideas

  • The model-free volatility calculation uses option prices across a range of strikes.
  • A cubic spline can overshoot and produce negative interpolated volatility in some configurations.
  • Negative implied-volatility outputs should prompt review of the method or implementation.
  • Linear interpolation between positive volatility observations preserves positivity.
  • The response does not evaluate extrapolation or option-price arbitrage consistency.

Tags

EquitiesOptionsVolatilityDerivatives pricing
Full text
# Using Cubic Spline with Vol Skew for Equity Options in R


# Using Cubic Spline with Vol Skew for Equity Options in R












I was recently attempting to replicate a part of the paper - DeMiguel, Plyakha, Uppal and Vilkov (2013), where they compute a model-free implied volatility (MFIV) quantity.

In the paper, the MFIV is computed as the root of the variance contract, which according to Bakshi et al. (2003), is the discretized sum of scaled prices of a continuum of call and put option prices. This continuum of call and put options are calculated with an inter and extrapolated volatility skew that is constructed with a cubic spline from existing implied volatilities of options that are currently traded in the market.

My question is, how can I not get negative implied volatilities from the cubic spline implementation? And is it correct to get negative implied volatilities in the first case?

## Answer by Richi Wa (score 1, accepted)

https://quant.stackexchange.com/a/76772

Posting my comment as an answer :):

Could you post an example of your problem? If you have positive volatilities and interpolate between them, then the interpolated values should be positive too (also using cubic spline interpolation). So: no, your implied vols should always be positive.

EDIT: This is an edit after @jherek 's comment. The following needs to be noted:

- Implied vols should always be positive. If your interpolation gives you negative ones, either there is a general mistake or the method of interpolation does not fit.

- Due to their flexibility, cubic splines can, in special constellations, lead to negative values. Some interpolation algorithms use cubic splines in general and linear interpolation if points lie close to each other. In the linear case, positive inputs always lead to positive interpolation.

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.