Overview
The Vasicek short rate model, is given by the following equation.
{% dr = (b - a r) dt + \sigma dW %}
Bond Option Price
A call option on a bond with
- Strike price {% K %}
- Bond with maturity {% S %}
- Call option with maturity {% T < S %}
The value of the call option, {% C %} is given by
{% c(t,T,K,S) = p(t,S) N(d) - p(t,T)KN(d-\sigma_p) %}
Here {% p(t,T) %} is the zero coupon bond price, given as in
bond price,
and {% d %} and {% \sigma_p %} are given by
{% d = \frac{1}{\sigma_p} log[\frac{p(t, S)}{p(t, T)K}] + \frac{1}{2} \sigma_p %}
{% \sigma_p = \frac{1}{a}[1-e^{-a(S-T)}] \sqrt{\frac{\sigma^2}{2a} [1-e^{-2a(T-t)}] } %}
(see bjork pg 386)