Relating Hull–White Mean Reversion to Bermudan Swaption Tenors

Summary

The document asks how to select the mean-reversion parameter in a one-factor Hull–White model when pricing Bermudan swaptions. It describes a calibration approach based on the ratio of standard deviations of two forward rates with a shared expiry and different maturities, citing an interest-rate modeling text as its source. The question is how those expiry and maturity points should correspond to a Bermudan’s non-call period and swap tenor.

The author suggests using the non-call period as the common expiry and the swap tenor as the gap between the two forward-rate maturities, but expresses uncertainty about that mapping. No answer or empirical evidence is included, so the proposed correspondence remains unvalidated. The note identifies a calibration-design issue rather than giving a complete procedure; it does not specify how to choose the individual forward-rate maturities or assess whether the resulting parameter is appropriate for Bermudan prices.

Key ideas

• The described Hull–White calibration infers mean reversion from the volatility ratio of forward rates with a common expiry.
• The author proposes mapping the Bermudan non-call period to the common expiry.
• The author suggests relating swap tenor to the gap between the forward-rate maturities.
• The document does not resolve or validate this proposed mapping.

Tags

# Term structure of mean-reversion


# Term structure of mean-reversion












I am working with the 1-factor Hull-White model; my final objective is to price Bermuda swaptions. Beforehand, I should fix a value for the mean-reversion parameter. To do so, I am following the calibration methodology of [Andersen, L. B., & Piterbarg, V. V. (2010). Interest Rate Modeling, vol. II] (chap. 13, section 13.1.8). That is, the mean-reversion parameter is deduced from ratios of standard deviations of forward rates: $$ \text{Mean-rev} = f \left( \frac{\text{Std}(F(T;M_1)}{\text{Std}(F(T;M_2)} \right), $$ with $T$ the common expiry of the forward rates, and $M_1 \leq M_2$ their respective maturities.

When pricing Bermuda swaptions, I am able to give a proper value of the mean-reversion parameter per pair of (non-call period $\times$ tenor). My question is: how can I link the quantities $T, M_1, M_2$ to the (non-call period $\times$ tenor) of the Bermudas?

I have though that $T$ could be taken as the non-call period (since there are no cash-flow before that date) and the duration $M_2 - M_1$ could correspond to the tenor of the Berms as $[M_1, M_2]$ would define the cash-flows period. But I am not fully convinced.

Thanks in advance!

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