# 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!

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:

$$P(t,T) = P^M(0,T)$$

• Great answer, thanks! – MC_nonmaster Jun 5 at 16:31

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.