# Zero-coupon bond Vasicek model

I have to find price of zero-coupon bond in the Vasicek model using partial differential equations, but have no idea how to start it. I have found that rate of Vasicek model is described as

$$dr(t) = k(\theta - r(t) ) \; dt + \sigma \; dW(t)$$ $$r(t) = r(s) e^{-k(t-s)} + \theta (1- e^{-k(t-s)}) + \sigma \int_{s}^{t} e^{-k(t-u)} \;dW(u)$$

and it's zero coupon bond is as follow: $$P(t,T) = A(t,T)e^{-r(t)B(t,T)}$$ $$B(t,T) = \frac{1 - e^{-k(T-t)}}{k}$$ $$A(t,T) = exp\left(\left(\theta - \frac{\sigma^2}{2k^2}\right)(B(t,T)-T+t) -\frac{\sigma^2}{4k}B^2(t,T) \right)$$

but I do not know from what formula we are finding price of zero-coupon bond in those financial models.

• You might also post this kind of questio here : quant.stackexchange.com Commented Apr 26, 2021 at 17:43

The idea is that the price of the zero coupon bond is given by $$P(t;T) = E\left[ \exp\bigg\{-\int_t^T r_s \ ds\bigg\} \bigg\rvert \mathscr{F}_t \right].$$ All of the details/computations to get the formulae you reference can be found in: http://www.kurims.kyoto-u.ac.jp/EMIS/journals/HOA/JAMDS/8/11.pdf.