Overview
Extended Vasicek
The Vasicek model was extended by Hull and White in order to create a model that can be calibrated to the entire curve.
{% dr = (b(t) - ar)dt + \sigma dW %}
(
see wilmott 17.3)
Zero Coupon Bond Price
{% Z(r,t,T) = e^{A(t, T) - rB(t,T)} %}
where
{% A(t,T) = -\int_t ^T b(s) B(s,T) ds + \frac{\sigma^2}{2a^2} (T-t + \frac{2}{a} e ^{-a(T-t)} - \frac{1}{2a} e ^{-2a(T-t)} - \frac{3}{2a}) %}
and
{% B(t,T) = \frac{1}{a} (1 - e^{a(T-t)}) %}
Calibrating to the Curve
{% b(t) = \frac{\partial^2{log(Z(t,T))}}{\partial{t}^2} - a \frac{\partial{log(Z(t,T))}}{\partial{t}} + \frac{\sigma^2}{2a}(1-e^{2a(T-t)}) %}
