## Merton model

We derive the Merton model term structure model formulae.

### Term Structure and Bond Price Dynamics

The short rate under the Merton model was shown to be:

$$r_{t}=r_{s} + \theta \left( t-s\right)+ \sigma\int_{s}^{t}{d w_{u}}$$

Now integrating the short rate from s to T, we get

$$\int_{s}^{T}{r_{t} dt}=\int_{s}^{T}{r_{s} dt} + \theta \int_{s}^{T}{ \left( t-s \right) dt} +\sigma\int_{s}^{T}{\int_{s}^{t}{d w_{u}} dt}$$ $$= r_{s} \left( T-s \right) + \theta \int_{s}^{T}{ t dt}-\theta s\int_{s}^{T}{ dt} +\sigma\int_{s}^{T}{\int_{u}^{T}{dt d w_{u}} }$$ $$= r_{s} \left( T-s \right) + \theta \frac{T^2-s^2}{2}-\theta s \left( T-s\right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$ $$= r_{s} \left( T-s \right) + \theta \frac{T^2-s^2-2 T s +2 s^2}{2}+\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$ $$= r_{s} \left( T-s \right) + \theta \frac{\left( T-s \right)^2}{2}+\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$

Which is Gaussian with mean and variance given by:

$$E \left[ \int_{s}^{T}{r_{t} dt} \mid r_{s} \right]=r_{s} \left( T-s \right) + \theta \frac{\left( T-s \right)^2}{2}$$ $$V \left[ \int_{s}^{T}{r_{t} dt} \mid r_{s} \right]=V \left[\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}} \right]$$ $$= {\sigma}^{2}\int_{s}^{T}{{\left( T-u \right)}^{2}du}$$ $$= \frac{{\sigma}^{2} {\left( T-s \right)}^{3} }{3}$$

Hence the bond price at time s can be represented as:

$$P \left( s, T\right)=E \left[ e^{-\int_{s}^{T}{r_{t} dt}}\right]= e^{-E \left[ \int_{s}^{T}{r_{t} dt} \right]+\frac{1}{2}V \left[ \int_{s}^{T}{r_{t} dt} \right] }$$ $$=e^{ -r_{s} \left( T-s \right) - \theta \frac{\left( T-s \right)^2}{2} + \frac{{\sigma}^{2} {\left( T-s \right)}^{3} }{6} }$$