Converting Swaption Volatility Between Normal and Black Quotes

Summary

This discussion explains how to interpret at-the-money swaption volatility quoted under normal and Black conventions. Normal volatility is expressed in rate units, commonly basis points, while Black volatility is relative to the forward swap rate. The response gives the approximate conversion as normal volatility equal to Black volatility multiplied by the forward swap rate, and clarifies that normal quotes are typically annualized rather than converted to daily units by dividing by the square root of 252.

The document also describes a more general conversion: calculate the option price using one volatility model, then invert the other model to find the implied volatility that matches that price. The simple relationship is an approximation and needs care in cases such as negative forward rates, where lognormal Black assumptions become problematic. The exchange offers conventions and methods, not a time-series analysis supporting either volatility measure.

Key ideas

• Normal swaption volatility is quoted in absolute rate units, while Black volatility is relative to the forward swap rate.
• A common approximation converts Black volatility to normal volatility by multiplying by the forward swap rate.
• Normal volatilities are generally annualized, so the stated conversion does not call for a daily scaling adjustment.
• A model-consistent conversion can be made by pricing under one model and inverting the other model.
• Negative forward rates require care when converting to a lognormal Black framework.

Tags

# Swaption ATM Vol Quotes and Interpretation: Normal Vol to Black


# 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?

Much appreciated in advance.

## Answer by dm63 (score 3, accepted)

https://quant.stackexchange.com/a/66526

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.

## Answer by Lech (score 0)

https://quant.stackexchange.com/a/66527

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.