### Bond Price Probability Density

Now we derive the probability density of the zero coupon bond prices at a future time under the Hull White model. The derivation (as was the case in the short rate models) is based on the transformation of the short rate density.

To derive formula for the probability density function (pdf) of zero coupon bond price, we make use of two facts that we established in the previous sections: the short rate is normally distributed with some mean and variance (please see the short rate section for the formulae), and the price of a T-maturity zero coupon bond at a fixed time t is a function of short rate (see bond price section):

$$ P(r)=A e^{-r B} $$

We can thus transform the pdf of the short rate, which we denote by \( f_{r}\), to the pdf of the bond price, which we denote by \( f_{p}\)

$$ f_p(p)=f_r (P^{-1}(p)) \left| \frac{dr}{dp} \right| $$

Where in the present case

$$ P(r)=A e^{-r B}=p \quad \Rightarrow r=-\frac{1}{B}ln \frac{p}{A} =P^{-1}(p), \mbox{ and } \frac{dr}{dp}= -\frac{1}{Bp}$$

Hence

$$ f_p(p)=f_r (P^{-1}(p)) \left| \frac{dr}{dp} \right|= f_r \left( -\frac{1}{B}ln \frac{p}{A} \right) \frac{1}{Bp} $$

Where \(f_r\) is the density of normal distribution with mean and variance of the short rate evaluated at \(-\frac{1}{B}ln \frac{p}{A}\). Note as \(f_r\) is normal, \(f_p\) is log-normal.