## Ho Lee model

We derive the Ho Lee term structure model formulae.

### Bond price dynamics

Now we study the behavior of bond prices over time under the Ho Lee model, which, as shown in the previous sub-section, can be represented as:

$$d r_{t}= \left( \frac {\partial} {\partial t} f \left( 0,t \right) + {\sigma}^{2}t \right)dt + \sigma d w_{t}$$

Integrating from s (arbitrary start time) to t, we get

$$\int_{s}^{t}{d r_{u}}= \int_{s}^{t}{\left( \frac {\partial} {\partial u} f \left( 0,u \right) + {\sigma}^{2} u \right)du}+\int_{s}^{t}{\sigma d w_{u}}$$ $$r_{t}-r_{s} = \int_{s}^{t}{ d f \left( 0,u \right)du}+{\sigma}^{2} \int_{s}^{t}{u du} +\sigma\int_{s}^{t}{d w_{u}}$$ $$r_{t}=r_{s}+f \left( 0,t \right)-f \left( 0,s \right)+{\sigma}^{2} \frac{{ {t}^{2}-{s}^{2} }}{2} +\sigma\int_{s}^{t}{d w_{u}}$$

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

$$\int_{s}^{T}{r_{t} dt}=\int_{s}^{T}{r_{s} dt} + \int_{s}^{T}{f \left( 0,t \right) dt}-\int_{s}^{T}{f \left( 0,s\right) dt}+\frac{{\sigma}^{2}}{2}\int_{s}^{T}{\left( {t}^{2}-{s}^{2} \right) dt} +\sigma\int_{s}^{T}{\int_{s}^{t}{d w_{u}} dt}$$ $$= r_{s} \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}-f\left( 0,s \right) \left( T-s \right)+\frac{{\sigma}^{2}}{2} \left( \frac{{T}^{3}-{s}^{3}}{3} -{s}^{2} \left( T-s\right) \right) +\sigma\int_{s}^{T}{\int_{u}^{T}{dt d w_{u}} }$$ $$= \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} \left( \left( T-s\right)\left( {T}^{2}+{s}^{2} +s T\right) -3{s}^{2} \left( T-s\right) \right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$ $$= \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} \left( T-s\right)\left( {T}^{2}-2{s}^{2} +s T\right) +\sigma\int_{s}^{T}{\int_{s}^{t}{d w_{u}} dt}$$ $$= \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} \left( T-s\right)\left( {T}^{2}-{s}^{2}-{s}^{2} +s T\right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$ $$= \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}}$$

Which is Gaussian with mean and variance given by:

$$E \left[ \int_{s}^{T}{r_{t} dt} \mid r_{s} \right]=E \left[ \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}} \right]$$ $$= \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right)$$ $$=\left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) -ln \frac{B \left(0,T \right)}{B\left(0, s\right) }+\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right)$$ $$V \left[ \int_{s}^{T}{r_{t} dt} \mid r_{s} \right]=V \left[ \left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + \int_{s}^{T}{f \left( 0,t \right) dt}+\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right) +\sigma\int_{s}^{T}{\left( T-u \right)d w_{u}} \right]$$ $$= {\sigma}^{2}\int_{s}^{T}{{\left( T-u \right)}^{2}du}$$ $$= \frac{{\sigma}^{2} {\left( T-s \right)}^{3} }{3}$$

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

$$P \left( s, T\right)=E \left[ e^{-\int_{s}^{T}{r_{t} dt}}\right]= e^{-E \left[ \int_{s}^{T}{r_{t} dt} \right]+\frac{1}{2}V \left[ \int_{s}^{T}{r_{t} dt} \right] }$$ $$=e^{ -\left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) }-\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s\right) + \frac{{\sigma}^{2} {\left( T-s \right)}^{3} }{6} }$$ $$=e^{ -\left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) }-\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( T+2 s-T+s\right) }$$ $$=e^{ -\left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) }-\frac{{\sigma}^{2}}{6} {\left( T-s\right)}^{2}\left( 3 s\right) }$$ $$=e^{ -\left( r_{s}-f\left( 0,s \right) \right) \left( T-s \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) }-\frac{{\sigma}^{2}}{2} {s\left( T-s\right)}^{2} }$$ $$=e^{ -r_{s} \left( T-s \right) + f\left( 0,s \right) \left( T-s \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) }-\frac{{\sigma}^{2}}{2} {s\left( T-s\right)}^{2} }$$