Converting Cap Black Volatility to Hull-White Short-Rate Volatility

Summary

The document examines a Bloomberg convention for setting volatility in a Hull-White short-rate model. The described calculation converts a cap volatility into a rate volatility by multiplying the quoted cap volatility by the corresponding swap rate expressed as a decimal. The proposed rationale is a conversion from a lognormal volatility of relative rate changes to a normal volatility of absolute rate changes: approximately, volatility of the rate itself is the rate level times volatility of its percentage change.

The author questions how well this conversion fits a one-factor short-rate model, since a ten-year swap rate is not the instantaneous short rate. Treating that tenor's swap rate and cap volatility as proxies for short-rate inputs is therefore an assumption, not a direct consequence of the unit conversion. The document reports that supporting documentation mentions the lognormal-versus-normal distinction, but it does not establish how Bloomberg justifies tenor selection or the proxy relationship. The method's applicability depends on the volatility convention and modeling calibration context.

Key ideas

• Multiplying a rate by its lognormal volatility approximates a normal volatility in rate units.
• The described convention uses a tenor-matched swap rate and cap volatility to set a Hull-White input.
• A swap rate is not the model's instantaneous short rate, so using it as a proxy requires justification.
• The document gives a tentative explanation but leaves the tenor and calibration rationale unresolved.

Tags

# Reasoning for Bloomberg's short rate volatilty calculation


# Reasoning for Bloomberg's short rate volatilty calculation












Bloomberg, in its documentation, explains that it calculates the short rate volatility for its Hull White implementation by multiplying the e.g. 10y IRS rate (divided by 100) by the 10y cap vol.

Why?

I must be missing something obvious.

---Update---

Finally got a response in terms of more documentation, and a half sentence that mentions lognormal vs normal, which I'm assuming means they are saying that the cap vol is a Black (i.e. lognormal) volatility of dr/r, but we want a normal volatility for the Hull-White model of dr (which they use). So multiply by r both sides.

Which is, I suppose, approximately the right idea? Except why the 10y cap vol and 10y rate, when the Hull-White model is a one-factor model of the short rate; the 10y rate is not the short rate, even if we assume the volatility is constant and equal to the value at 10y.

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