Overview
The Vasicek short rate model is given by.
{% dr = (b - a r) dt + \sigma dW %}
Zero Coupon Bond Price
Given the Vasicek short rate model above, the arbitrage free price at time t of $1 paid at time T is given by
{% p(t,T) = exp[A(t, T) - B(t,T)r(t)] %}
where
{% B(t,T) = \frac{1}{a} [1 - exp(-a(T-t))] %}
{% A(t,T) = \frac{(B(t,T)-T+t)(ab-0.5 \times \sigma^2)}{a^2} - \frac{\sigma^2 B^2(t,T)}{4a} %}
(see bjork pg 382)( see also affine term structure)