Smoothing Dupire

Last week, I naively applied Dupire's formula for deriving a local volatilty surface from a call price surface (in The ingredients of Dupire's formula) to noisy data. The diificulties arose from the second derivative of the call prices with respect to the strike price in the denominator. We obtained, again by naive differentiation, the following second derivative
and, consequently applying Dupire's formula, the following local vol surface

Could we, similar to the process in Implied Black volatility continued, obtain better results by pre-smoothing?
To try it out, we just replace every call value on the surface by the average over the 5 point stencil

This yields

for the second derivative and
for the local volatiltiy. Not that we used the same vertical scale for the pliots arising from presmoothing as for the original ones.

Hence, presmoothing delivers better results. But we can still do it much better by regularisation techniques.
In the forthcoming week, I will attend an inverse problems conference in Bristol, UK. Therefore, the next local vol blog will appear in two weeks.