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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?

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    $\begingroup$ Hi and welcome. 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 ... $\endgroup$
    – Richi Wa
    Sep 20 at 11:55
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    $\begingroup$ @RichiWa Thank you, I noticed that I was using OTM put and call-implied vols to inter and extrapolate a single volatility skew. When moving across K/S from left to right (put to call-implied volatility) there was a large deviation in IV. This resulted in the cubic spline being fitted very strangely, which led to negative IVs in the extrapolation region. Your answer gave me the confidence to check my modelling again :) $\endgroup$
    – Kai
    Sep 20 at 12:33
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    $\begingroup$ very good. Happy if that helped! $\endgroup$
    – Richi Wa
    Sep 21 at 13:06
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    $\begingroup$ @RichiWa, can you turn your comment into an answer ? $\endgroup$
    – Bob Jansen ♦
    Sep 23 at 19:15
  • $\begingroup$ @BobJansen I just did after your motivation:) $\endgroup$
    – Richi Wa
    Sep 23 at 20:04

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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.
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    $\begingroup$ This is not true. Standard C2 cubic spline interpolation does not preserve positivity. You only need Heaviside like IV quotes to convince yourself. There is an algorithm to preserve positivity at the cost of potential second derivative discontinuity, see chasethedevil.github.io/post/… $\endgroup$
    – jherek
    Sep 25 at 19:47
  • $\begingroup$ I agree. Isn't it usually the case that algorithms for cubic spline interpolation switch to linear interpolation if points are close? Then positivity would be assured. My main point was that implied volas shall always be positive ... I will add this to the answer ... I thought about this too. $\endgroup$
    – Richi Wa
    Sep 26 at 6:44
  • $\begingroup$ @jherek if you make this an answer with some more details, I would vote for it ! :) $\endgroup$
    – Richi Wa
    Sep 26 at 6:49

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