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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?

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