Hull-White Model

Abstract

Hull and White’s model is a one-factor model of the short rate r(t), first published in [85] and generalized later [86]. Throughout this appendix, we use the following notation for the model:

$$ dr\left( t \right)=\lambda \left( {\theta \left( t \right)-r\left( t \right)} \right)dt+\sigma \left( t \right)dW\left( t \right), $$

(D.1)

with time-dependent volatility σ(t) and mean reversion θ(t) and constant reversion speed λ. This model has the solution

$$\begin{gathered} r\left( t \right)=\mu \left( t \right)+x\left( t \right) \hfill \\ \mu \left( t \right)={{r}_{0}}{{e}^{{-\lambda t}}}+\lambda \int\nolimits_{0}^{t} {{{e}^{{-\lambda \left( {t-s} \right)}}}\theta \left( s \right)ds} \hfill \\ x\left( t \right)=\int\nolimits_{0}^{t} {{{e}^{{-\lambda \left( {t-s} \right)}}}\sigma \left( s \right)dW\left( s \right)} \hfill \\ \end{gathered}$$

with mean

$$\mathbb{E}\left[ {r\left( t \right)} \right]=\mu \left( t \right)$$

and variance

$$\mathbb{V}\left[ {r\left( t \right)} \right]=\mathbb{V}\left[ {x\left( t \right)} \right]=\int\nolimits_{0}^{t} {{{e}^{{-2\lambda \left( {t-s} \right)}}}{{\sigma }^{2}}\left( s \right)ds} $$