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Hull White model

We derive the one factor Hull White term structure model formulae.

Bond Price and Rate Dynamics

Now we study the behavior of the short rate and bond prices over time under the Hull White model, which, as shown in the previous sub-section, can be represented as:

$$ d r_{t} = \kappa \left( f \left( 0,t \right)+\frac{1}{\kappa} \frac{\partial f \left( 0,t \right)} {\partial t} + \frac{{\sigma}^2}{2{\kappa}^2} \left( 1 - e^{-2 \kappa t} \right)- r_{t} \right) dt + \sigma d w_{t} $$

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

$$ \int_{s}^{t} {d \left( e^{k u} r_{u} \right)} = \int_{s}^{t} {d \left(e^{\kappa u} f \left( 0,u \right) \right) } + \frac{{\sigma}^2}{2{\kappa}} \int_{s}^{t} {\left( e^{\kappa u} - e^{- \kappa u} \right) du } + \sigma\int_{s}^{t} {e^{k u} d w_{u}} $$ $$ e^{\kappa t} r_{t} -e^{\kappa s} r_{s} = e^{\kappa t} f \left( 0,t \right) - e^{\kappa s} f \left( 0,s \right)+ \frac{{\sigma}^2}{2{\kappa}^2} \left( e^{\kappa t} - e^{\kappa s}+ e^{- \kappa t}- e^{-\kappa s} \right) + \sigma \int_{s}^{t} {e^{k u} d w_{u}} $$ $$ e^{\kappa t} r_{t} = e^{\kappa s}\left( r_{s} - f \left( 0,s \right) \right) + e^{\kappa t} f \left( 0,t \right) + \frac{{\sigma}^2}{2{\kappa}^2} \left( e^{\kappa t} - e^{\kappa s}+ e^{- \kappa t}- e^{-\kappa s} \right) + \sigma \int_{0}^{t} {e^{k u} d w_{u}} $$ $$ r_{t} = e^{-\kappa \left( t- s \right) }\left( r_{s} - f \left( 0,s \right) \right)+ f \left( 0,t \right) + \frac{{\sigma}^2}{2{\kappa}^2} \left( 1 - e^{-\kappa \left( t-s \right) }+ e^{- 2 \kappa t}- e^{-\kappa \left( t+s \right)} \right) + \sigma \int_{s}^{t} {e^{-\kappa \left(t-u \right)} d w_{u}} $$ $$ r_{t} = e^{-\kappa \left( t- s \right) }\left( r_{s} - f \left( 0,s \right) \right)+ f \left( 0,t \right) + \frac{{\sigma}^2}{2{\kappa}^2} \left( 1 - e^{-\kappa \left( t-s \right) }+ e^{-\kappa \left( t-s \right)-\kappa \left( t+s \right)}- e^{-\kappa \left( t+s \right)} \right) + \sigma \int_{s}^{t} {e^{-\kappa \left(t-u \right)} d w_{u}} $$ $$ r_{t} = e^{-\kappa \left( t- s \right) }\left( r_{s} - f \left( 0,s \right) \right)+ f \left( 0,t \right) + \frac{{\sigma}^2}{2{\kappa}^2} \left( 1 - e^{-\kappa \left( t-s \right) } \right) \left( 1- e^{-\kappa \left( t+s \right)} \right) + \sigma \int_{s}^{t} {e^{-\kappa \left(t-u \right)} d w_{u}} $$

Which is Gaussian with mean and variance given by,

$$ E \left[ r_{t} \mid r_{s} \right]= e^{-\kappa \left( t- s \right) }\left( r_{s} - f \left( 0,s \right) \right)+ f \left( 0,t \right) + \frac{{\sigma}^2}{2{\kappa}^2} \left( 1 - e^{-\kappa \left( t-s \right) } \right) \left( 1- e^{-\kappa \left( t+s \right)} \right) $$ $$ V \left[ r_{t} \mid r_{s} \right]= V \left[ \sigma \int_{s}^{t} {e^{-\kappa \left(t-u \right)} d w_{u}} \right] = {\sigma}^{2} \int_{s}^{t} {e^{-2 \kappa \left(t-u \right)} du}=\frac{{\sigma}^2}{2{\kappa}} {\left( 1- e^{-2 \kappa \left( t -s \right)} \right)} $$

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

$$ \int_{s}^{T}{r_{t} dt}= \left( r_{s} - f \left( 0,s \right) \right) \int_{s}^{T}{e^{-\kappa \left( t- s \right) } dt} + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{2{\kappa}^2} \int_{s}^{T} {\left( 1 - e^{-\kappa \left( t-s \right) } \right) \left( 1- e^{-\kappa \left( t+s \right)} \right) dt} + \sigma \int_{s}^{T}{{\int_{s}^{t} {e^{-\kappa \left(t-u \right)} d w_{u}}} dt} $$ $$ = \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{e^{-\kappa \left( T- s \right)}-1}{-\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{2{\kappa}^2} \left[ \int_{s}^{T}{dt} - \int_{s}^{T}{e^{-\kappa \left( t-s \right)}dt} + \int_{s}^{T}{e^{- 2 \kappa t} dt} - \int_{s}^{T}{e^{-\kappa \left( t+s \right)}dt} \right] + \sigma \int_{s}^{T}{{\int_{u}^{T} {e^{-\kappa \left(t-u \right)} dt}} d w_{u}} $$ $$ = \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{2{\kappa}^2} \left[ \left( T-s \right) - \frac{1}{-\kappa}\left( e^{-\kappa \left( T-s \right)} - 1\right) + \frac{1}{-2\kappa} \left( e^{- 2 \kappa T} - e^{- 2 \kappa s}\right) - \frac{1}{-\kappa} \left( e^{-\kappa \left( T+s \right)}-e^{-2\kappa s} \right) \right] + \frac{\sigma}{-\kappa} \int_{s}^{T}{ \left({e^{-\kappa \left(T-u \right)}}-1 \right) d w_{u}} $$ $$ = \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 - e^{- 2 \kappa T} + e^{- 2 \kappa s} + 2 e^{-\kappa \left( T+s \right)}-2 e^{-2\kappa s}\right] + \frac{\sigma}{\kappa} \int_{s}^{T}{ \left(1-{e^{-\kappa \left(T-u \right)}} \right) d w_{u}} $$ $$ = \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 -\left(e^{- 2 \kappa T}+ e^{- 2 \kappa s}-2 e^{-\kappa \left( T+s \right)}\right) \right] + \frac{\sigma}{\kappa} \int_{s}^{T}{ \left(1-{e^{-\kappa \left(T-u \right)}} \right) d w_{u}} $$ $$ = \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] + \frac{\sigma}{\kappa} \int_{s}^{T}{ \left(1-{e^{-\kappa \left(T-u \right)}} \right) d w_{u}} $$

Which is Gaussian with mean and variance given by:

$$ E \left[ \int_{s}^{T}{r_{t} dt} \right]= \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + \int_{s}^{T}{f \left( 0,t \right) dt} + \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] $$ $$= \left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) -ln \frac{B \left(0,T \right)}{B\left(0, s\right) } + \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] $$ $$ V \left[ \int_{s}^{T}{r_{t} dt} \right]=V \left[ \frac{\sigma}{\kappa} \int_{s}^{T}{ \left( 1- e^{-k \left( T- u \right)} \right) d w_{u}} \right]$$ $$=\frac{{\sigma}^2}{{\kappa}^2} \int_{s}^{T}{ \left( 1- e^{-k \left( T- u \right)} \right)^{2} du} $$ $$=\frac{{\sigma}^2}{{\kappa}^2} \int_{s}^{T}{ \left( 1+e^{-2 k \left( T- u \right)}-2 e^{-k \left( T- u \right)} \right) du} $$ $$= \frac{{\sigma}^2}{{\kappa}^2} \int_{s}^{T}{du}+\frac{{\sigma}^2}{{\kappa}^2} \int_{s}^{T}{ e^{-2 k \left( T- u \right)} du}-\frac{2{\sigma}^2}{{\kappa}^2} \int_{s}^{T}{ e^{-k \left( T- u \right)} du} $$ $$= \frac{{\sigma}^2 \left( T-s \right)}{{\kappa}^2} + \frac{{\sigma}^2}{2{\kappa}^3} \left( 1-e^{-2 \kappa \left( T-s \right)} \right) -\frac{2{\sigma}^2}{{\kappa}^3} \left( 1-e^{-\kappa \left( T-s \right)}\right) $$ $$= \frac{{\sigma}^2}{2{\kappa}^3} \left( 2 \kappa \left( T-s \right)+ 1-e^{-2 \kappa \left( T-s \right)} -4 + 4 e^{-\kappa \left( T-s \right)} \right) $$ $$= \frac{{\sigma}^2}{2{\kappa}^3} \left( 2 \kappa \left( T-s \right) -3 -e^{-2 \kappa \left( T-s \right)} + 4 e^{-\kappa \left( T-s \right)} \right) $$

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

$$ P \left( s, T\right)=E \left[ e^{-\int_{s}^{T}{r_{t} dt}}\right]= e^{-E \left[ \int_{s}^{T}{r_{t} dt} \right]+\frac{1}{2}V \left[ \int_{s}^{T}{r_{t} dt} \right] }$$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) +2 e^{-\kappa \left( T-s \right)} - 2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] + \frac{{\sigma}^2}{4{\kappa}^3} \left( 2 \kappa \left( T-s \right) -3 -e^{-2 \kappa \left( T-s \right)} + 4 e^{-\kappa \left( T-s \right)} \right) } $$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left[ 2 \kappa \left( T-s \right) -2 \kappa \left( T-s \right)+2 e^{-\kappa \left( T-s \right)} -4 e^{-\kappa \left( T-s \right)}+e^{-2 \kappa \left( T-s \right)} - 2 +3 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] } $$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left[ -2 e^{-\kappa \left( T-s \right)} +e^{-2 \kappa \left( T-s \right)} +1 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] } $$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left[ \left(1-e^{-\kappa \left( T-s \right)}\right)^2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] } $$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left[e^{2\kappa s} \left(e^{-\kappa s} -e^{-\kappa T}\right)^2 -\left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \right] } $$ $$=e^{ -\left( r_{s} - f \left( 0,s \right) \right) \left( \frac{1- e^{-\kappa \left( T- s \right)}}{\kappa} \right) + ln \frac{B \left(0,T \right)}{B\left(0, s\right) } - \frac{{\sigma}^2}{4{\kappa}^3} \left(e^{-\kappa T}- e^{- \kappa s}\right)^2 \left(e^{2\kappa s}- 1\right) } $$