Hull–White Bond Pricing with Time-Dependent Parameters
Summary
The document asks whether the standard one-factor Hull–White zero-coupon bond formulas assume constant mean reversion and volatility, and what to use when those parameters vary over time. The response identifies the general one-factor Gaussian short-rate model, in which mean reversion, the long-run rate level, and volatility can each depend on time. Its bond-pricing formula is obtained by integrating the time-varying terms, using the model’s relationship to the Heath–Jarrow–Morton framework.
The response points to a textbook discussion for the full derivation, but does not reproduce the integrated formulas or explain how to implement them. The question’s idea of representing parameter curves with piecewise-continuous knots is raised but not addressed, so the document does not establish that approach or give calibration guidance. It is an orientation to the appropriate model class and source, rather than a complete pricing recipe.
Key ideas
- The familiar Hull–White bond formula is presented as a constant-parameter case.
- A general one-factor Gaussian short-rate model allows mean reversion, rate level, and volatility to vary with time.
- Bond pricing in the time-dependent model involves integrating its parameter functions.
- The response links the derivation to the Heath–Jarrow–Morton framework but does not provide explicit integral formulas.
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Full text
# Hull White 1 Factor Formulas with Time Dependent Variables
# Hull White 1 Factor Formulas with Time Dependent Variables
In John Hull's "Options Futures and Other Derivatives" I see that bond prices in Hull White 1 Factor model are specified as the following:
$P(t,T) = A(t,T)e^{-B(t,T)r(t)}$
where
$B(t,T) = \frac{1 - e^{-a(T-t)}}{a}$
$lnA(t,T) = ln\frac{P(0,T)}{P(0,t)} + B(t,T)F(0,t) - \frac{1}{4a^3}\sigma^2(e^{-aT} - e^{-at})^2(a^{2at} - 1)$
Am I correct in thinking that these formulas only work if $a$ and $\sigma$ are constant? If so, what formulae should I be using if $a(t)$ and $\sigma(t)$ are time dependent? I have seen some references to integration, but I can't find a source material (that I trust) which spells out the appropriate core formulae which should be integrated over.
Edit: Alternatively, after looking at my own question... should I create a set of forward rates using $P(t_i,t_j)$ where $t_i$ and $t_j$ are "knots" on a piecewise-continuous linear function for $a(t)$ and $\sigma(t)$?
Then with those forward rates I can discount my zero-coupon bonds etc?
## Answer by mmencke (score 1)
https://quant.stackexchange.com/a/65798
In [Andersen & Piterbarg (2010)] on pages 415-417 the so-called General One-Factor Gaussian Short Rate Model is described, where the short rate is assumed to follow SDE $$ dr(t)=\kappa(t)(\theta(t)-r(t))dt+\sigma_r(t)dW(t) $$ The formula for $P(t,T)$ involves integration over the time-dependent variables as you correctly suggested. They arrive at the formula by utilising that the model is a special case of the Heath-Jarrow-Morton framework.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.