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I'm wondering how can we check free arbitrage conditions in ATM swaptions surfaces since we only have access to Expiry, Tenor and volatility? Can someone help me please, i didn't find any article about it. N.B:I'm working on python

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There are no no-arbitrage conditions on ATM vols of swaptions with different expiries/tenors, because the underlying swaps forward rates are different instruments. There are conditions however for these vols to be compatible with specific IR models. For instance when calibrating a Hull & White model on a set of coterminal swaptions, it sometimes happens that the models fails to fit a specific coterminal, because that would require a negative short rate instantaneous variance on the corresponding time bucket. It does not imply that the coterminal vols are not arbitrage free, but rather that the Hull & White model can only fit a limited set of market configurations.

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See Johnson, Simon and Nonas, Bereshad, Arbitrage-Free Construction of the Swaption Cube (January 5, 2009). Available at SSRN: https://ssrn.com/abstract=1330869 or http://dx.doi.org/10.2139/ssrn.1330869

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