I'm trying to simulate the risk factor of PFE from the interest rate model.
For example, under Vasicek model : $$dr_t = k(\theta-r_t)dt + {\sigma}dW_t$$ with the analytic solution, we can simulate N scenarios of $r_t$: $$E\{r_t|F_s\}=r_se^{-k(t-s)}+\theta(1-e^{-k(t-s)})$$ $$Var\{r_t|F_s\}=\frac{\sigma^2}{2k}(1-e^{-2k(t-s)})$$
Then we can compute the zero coupon bond by: $$P(t,T)=A(t,T)e^{-B(t,T)r_t} $$
My dump question is: In case of, for example, LIBOR, we have the different tenors of yield curve : 1D, 1W, 1M, 3M, 6M, 12M, ..., I should simulate each tenor of the yield curve as 1 process above ? or I should estimate the $r_0$ from all tenors of yield curve then simulate the short rates $r_t$, then use the zero coupon price to compute all tenors of the yield curve ?
Thank you very much !!
