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You don't need an approximation, i.e., if you have the Black's vols, you can simply compute the corresponding price and then invert Bachelier model (normal model) to get implied normal volatility. In the case of the transition from Normal (Bachelier) to Lognormal (Black-Sholes) you need to be more careful if you have negative forwards.


You basically have it. $$Normal Vol= Black Vol * Forward Swap Rate$$. Normal vol is usually quoted as an annual vol , not converted to daily by dividing by sqrt(252). The forward swap rate is the fair market rate for the swap that underlies the swaption. So one might have 1yr 10yr normal vol =70bp, forward swap rate = 1.40% and Black vol = 50%. ...

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