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For equity and fx vol surfaces you'd generally prefer to interpolate linearly in [$]\sigma^2t[$] over [$]\sigma[$] (keeping the forward moneyness constant) to avoid calendar arbitrage. For swaption or caplet vol interpolation it's not so clear why [$]\sigma^2t[$] is the better choice, since the underlying changes with [$]t[$] and hence there is no arbitrage argument supporting the decision.

Are there other reasons why you'd still interpolate IR Vols in [$]\sigma^2t[$] rather than [$]\sigma[$]. And what do people / systems actually do? Always assuming you do not something completely different like interpolating SABR parameters. 

Not my area, but since no one has answered, one suggestion. Take some market data, holding out some observations. Interpolate with both methods to estimate the hold-outs and compare. If both results are within bid-ask spreads, there's an answer: it doesn't matter. 

Good idea, thanks Alan. I'll try that.

My 2 cents here... for Pricing I see mostly SABR parameters being interpolated rather than actual Vols, for Risk Management I see V2T or V in Moneyness as Nb Scenarios and Performance are a constraint.

This will all probably change once RFRs indices become more prevalent in Swaptions and Cap/Floors.

Bang's paper on SABR is a good read btw.

M