There is a well known conclusion that implied vol cannot be linear interpolated. But I am not sure this interpolation is for strike $K$ or time $T$?
That's definitely true for time. When strike $K$ is fixed, interpolation of implied vols respect to time $T$ can make the Calendar arbitrage. Surely we assume the interpolation points satisfy the no arbitrage condition.
But is it true when time $T$ is fixed, interpolation of implied vols respect to strike $K$ can make a arbitrage.
I really struggled for related problems, can anyone give me a precise answer?
