Swaption ATM Vol Quotes and Interpretation: Normal Vol to Black

How do you interpret the time-series of 1m10y black vol vs normal vol? Normal vol would have you believe, that rate vol has since 2000 been low whereas black vol would show you a different picture.

What is the market convention of interpreting normal vol (bps) vs Black vol (lognormal)?

Is there a way to jump between the two? normal vol in bps = swap rate * black vol / sqrt(252)? Can you please show me an example if not too arduous to do so?

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%.

Practitioners generally use Normal Vols nowadays.

• This is approximation, right? That only works for ATM, right?
– emot
Aug 20, 2021 at 9:33
• There exist "extensions" that take into account moneyness as well, see this QSE thread, for instance. The accepted answer therein also points to @Lech's answer provided below (i.e., Normal Vol -> gives Option Price via Bachelier formula -> gives Black Vol via inversion BS-formula). Aug 20, 2021 at 12:31

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.

• Thank you, I'm quite new to this space, if not too arduous, any chance, you can walk off a simple example? If not, totally cool, and I will walk the path myself with your advice above. Thanks again! Aug 20, 2021 at 22:41
• As a simple example please take a look at ComputationalFinanceCourse Lecture 4 and instead Black-Scholes model replace it with Bachelier's model.
– Lech
Aug 22, 2021 at 14:51