Implied Black Volatility continued

Last week, in When Uncertainty is Good, I wrote on the difficulties of identifying an implied volatility, when vega is close to zero. I recapitulate the results we obtained by solving for the implied volatility from noisy data were quite poor when far away from the money.

The noisy call prices we used were as follows


Noisy call price as function of the strike price
 

This looks OK, but when we zoom in at the right end, we see what happened:


Noise dominates signal

Our eyes (and the brains behind) see clearly, what the "true" call price curve should be.


Does Pre-Smoothing Help?
A naive approach would be to take averages of noisy data, specifically we take the averages of the point itsellf and its 4 left neighbours and its 4 right neighbours. The averagred prices are drawn in the following plot

Averaging leads to smaller variance levels

 

Taking these as inputs we obtain



Implied volatilities from pre-smoothed noisy data

These are already much better results. They could be further improved by applying, say, a Gaussian filter instead of naive averaging and by choosing the filter bandwith depending on the level of vega.