Calibrating Hull–White Volatility from Swap Rate Volatility
Summary
The document asks how a quoted swap rate volatility relates to the time-dependent volatility in a one-factor Hull–White interest rate model. It presents a formula that maps the model volatility path into swaption volatility using a maturity and tenor dependent factor derived from discount bond prices and the forward swap rate. The questioner attempts to derive swap rate volatility with Itô’s formula and compares it with an approximation to the swap rate.
The post does not provide an answer or verify the proposed expressions. Its main value is to frame the calibration problem and show which quantities appear in the stated mapping. The presented equations and attempted derivations should be treated as unresolved: the document offers no derivation, correction, numerical example, or calibration evidence establishing that they are valid. It also assumes a risk-free measure and a particular model specification, so the formulas are not presented as general across pricing frameworks.
Key ideas
- The model describes short-rate dynamics with mean reversion and time-varying volatility.
- The stated mapping links swap rate volatility to an integral of squared Hull–White volatility weighted by mean reversion.
- The swap-specific factor depends on discount bond prices, maturity, tenor, and the forward swap rate.
- The proposed Itô derivations remain unanswered and are not validated in the document.
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# Calibrating Hull White volatility on swap rate volatility
# Calibrating Hull White volatility on swap rate volatility
I'm strugling with the Hull-White 1F model. I'am trying to calibrate the volatility with the swap rate volatility. Here is the model I'am curently working on : $$ \begin{align} dr_t = a(b-r_t)dt + \sigma_t^{HW} dW_t \end{align} $$ Some one gave me a code that compute de HW1F volatility, with the swap rate volatility for a maturity $T_i$ and tenor $\delta_i$, by inversing the formulae bellow. From my understanding $\sigma_t^{Mkt}$ is the swap rate volatility at time $t$ $$ \begin{align} \sigma^{Mkt}_{T_i} &= VF(T_i,\delta_i) \sqrt{\frac{1}{T_i}\int_0^{T_i}(\sigma^{HW}_s)^2e^{2as} ds} \\ \\ VF(T_i,\delta_i) &= \frac{S(T_i,\delta_i)}{a}\times \bigg[\frac{P(0,T_i)e^{-aT_i}-P(0,T_i+\delta_i)e^{-a(T_i+\delta_i)}}{P(0,T_i)-P(0,T_i+\delta_i)} - \sum_{j=1}^{\delta_i} \frac{P(0,T_i+j)e^{-a(T_i+j)}}{\sum_{k=1}^{\delta_i}P(0,T_i+k)} \bigg] \\ \\ S(T_i,\delta_i) &=\frac{P(0,T_i)-P(0,T_i+\delta_i)}{\sum_{k=1}^{\delta_i}P(0,T_i+k)} \end{align} $$
I have understand the principle but don't know where this formulae for $\sigma^{Mkt}_t$ came from. I have compute the volatility of the swap rate by using Itô formulae on $S(T_0,T_n)$ but came up with this formulae (might be some small mistake): $$ \sigma_t^S = \sigma_t^{HW}\bigg[\frac{S(T_0,T_n)}{a}\bigg(e^{-a(T_n-t)}+\frac{e^{-a(T_0-t)}-e^{-a(T_n-t)}}{a}+\frac{\sum_{i=1}^nP(t,T_i)e^{-a(T_i-t)}}{\sum_{i=1}^nP(t,T_i)}\bigg)+\frac{P(t,T_n)}{\sum_{i=1}^nP(t,T_i)}\frac{e^{-a(T_0-t)}-e^{-a(T_n-t)}}{a}\bigg] $$
I have also compute the volatility for an approximation of the swap rate $\tilde{S}(T_0,T_n) = \frac{P(0,T_n)}{\sum_{i=1}^nP(0,T_i)}\bigg[\frac{P(t,T_0)}{P(t,T_n)}-1\bigg] $ (see calibration-hull-white page 6) and came up with this formulae
$$ \sigma_t^{\tilde{S}} = \sigma^{HW}_t \frac{\tilde{S}(T_0,T_n)}{a} \bigg( \frac{P(0,T_n)e^{-a(T_0-t)}-P(0,T_n)e^{-a(T_n-t)}}{P(t,T_n)-P(t,T_n)} \bigg) $$
If any one know where the formulae for $\sigma^{Mkt}_t$ came from I would be pleased to know.
Every result is expressed using the risk free measure.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.