Why Hull–White Matches Initial Bond Prices Regardless of Volatility
Summary
The document explains why the Hull–White short-rate model’s zero-coupon bond price at the initial time is independent of volatility and mean reversion. The model is calibrated to the observed market zero curve, so its initial bond prices reproduce that curve rather than being determined by those parameters. The response demonstrates this with a formulation that models the short rate relative to the initial forward curve: at the starting time, the adjusted state and variance terms are zero, leaving the market bond price.
This initial fit does not mean volatility and mean reversion are irrelevant. Bond prices at future times are random variables whose distributions depend on those model parameters. The explanation concerns initial curve matching and does not claim that future interest-rate risk or bond-price uncertainty disappears. It gives a model-based derivation but no empirical comparison or calibration procedure beyond the stated fit.
Key ideas
- Hull–White is parameterized to match the market zero curve at the initial time.
- Initial zero-coupon bond prices therefore do not depend on volatility or mean reversion in the presented formulation.
- Future bond prices remain random and their distributions depend on volatility and mean reversion.
- The derivation uses a short-rate state measured relative to the initial forward curve.
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Full text
# Hull-White zero-coupon bond price does not depend on the volatility?
# Hull-White zero-coupon bond price does not depend on the volatility?
So, today I started pricing zero-coupon bonds using the Hull-White model. An interesting feature is that when t = 0 the bond price does not actually depend on the volatility since the last term of A(0,T) disappears. I find this weird as I would imagine the volatility has something to say especially if the time horizon is large! Is there an intuitive explanation for this?
(Hull-White zero-coupon bond price below).
Regards!
## Answer by byouness (score 2, accepted)
https://quant.stackexchange.com/a/45920
One of the main features of the Hull-White model is that it matches the market at $t = 0$.
This means that at $t = 0$, not only does the zero coupon bond prices (starting from zero) not depend on the volatility, but neither do they depend on the mean reversion level. These prices depend only on the zero curve observed in the market.
Of course, this is not to be confused with future zero bond prices $P(t,T)$ seen from $t = 0$, which are random variables and as a result have a distribution depending on the volatility and mean reversion as well.
To show that the ZC bond prices at $t = 0$ match the market and don't depend on the model parameters, I will use a different (more convenient) formulation (see e.g. Andersen and Piterbarg, section 10.1.2.2), which uses $x(t) = r(t) - f(0,t)$ instead of the short rate $r(t)$. Which leads to the following SDE (keeping your notations):
$$ \begin{aligned} x(0) &= 0 \\ dx(t) &= \left( y(t) - a x(t) \right) dt + \sigma dW(t) \end{aligned} $$
with: $y(t) = \frac{\sigma^2}{2a} \left(1-e^{-2at} \right)$.
The ZC bond price is given by: $$ P(t,T) = \frac{P^M(0,T)}{P^M(0,t)}\exp \left(-\frac{1}{2}B(t,T)^2y(t) - B(t,T)x(t) \right) $$
In the formula above, I used the superscript $^M$ to denote that the $P^M$ prices come from the zero curve observed in the market.
Taking $t = 0$, as $x(0) = y(0) = 0$ and $P^M(0, 0) = 1$, we have:
$$P(t,T) = P^M(0,T)$$
## Answer by Magic is in the chain (score 1)
https://quant.stackexchange.com/a/45462
The drift term of the short rate or forward rate dynamics has been adjusted so as to make the volatility term, that you see in the Vasicek formula for $P(0,T)$, disappear in a way, and be replaced by the current market price.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.