For market making in front month vanilla commodity options we need a volatility curve that updates every second or so as the underlying and the options change prices.
If all the strikes have a good two-way market then a simple smoothing spline produces a usable curve. But when the bids disappear in a few strikes, how should we preserve the shape of the curve and fit it to the new market data?
Should we be working with strikes or in log(strike) space?
For simple interpolation, implied variance (i.e. implied vol squared times maturity) vs. log moneyness (i.e. log of strike over forward) is probably the best choice. In these coordinates, Roger Lee results on the asymptotics imply the curve flattens to linear in the high and low moneyness limits, which fits with natural spline boundary conditions.
There is no canonical solution for the full problem you face of interpolating/extrapolating missing data across strikes and time. You can make modelling approaches as complicated as you want; it just is a hard problem.
I would suggest to do the simplest thing you can tolerate. Complex solutions have a way of causing more problems than they solve.
For the first question there are two approaches, the first is you can simply backfill and use the last minutes tick if it exists.
If there is no liquidity, the second way is you can try cubic or other interpolations to see if it creates a better curve.
