Sorry, I confused myself w.r.t. the linear in sigma^2 T interpolation.
As for the objective criterion. Maybe the total variation of the caplet volatility curve should be minimal?
It does. I am setting up an arbitrage-checker on a discrete grid though and want some theoretical foundation for that.And I quote
[$]\frac{\partial C}{\partial K} \le 0[$]
[$]\frac{\partial^2 C}{\partial K^2} \ge 0[$]
[$]\frac{\partial C}{\partial \tau} \ge 0[$]
Does this make sense?