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?