## Merton model

We derive the Merton model term structure model formulae.

### Bond Price 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.