Timesteps in Vasicek model

Question

When simulating stocks one can easily use GBM with only one random variable per simulation to create a new stock price in say 5 years, you don't need to create the whole asset paths if you don't need that.

Now I wonder if that is also the case for the Vasicek model. Can I use the Vasicek short rate model with only one random variable per simulation to create a new short rate in 5 years (without constructing the whole path to the short rate in 5 years?).

If so, how do you go from the new simulated short rate to the whole new yield curve?

Thanks.

Answer 1

Yes you can! Any SDE that has an analytic solution can be simulated exactly. The vasicek model has dynamics $dr=a(b-r)dt+\sigma dW_t$. By Ito's lemma, $$d\left(e^{at}r\right)=e^{at}\left(a(b-r)dt+\sigma dW_t\right) +a e^{at} r dt$$ Simplifying, $$d\left(e^{at}r\right)=e^{at} ab +e^{at}\sigma dW_t$$ Integrating, $$e^{aT} r_T=r_0+b(e^{aT}-1)+\sigma \int_0 ^ T e^{at} dW_t$$ Solving for $r_T$, $$r_T=r_0 e^{-aT} +b(1-e^{-aT})+\sigma \int_0 ^ T e^{-a(T-t)} dW_t $$ Since the Ito integrand is deterministic, the distribution of the Ito integral is normal with mean zero and variance $$\sigma^2\int_0 ^ T e^{-2a(T-t)} dt =\frac{\sigma^2}{2a}\left(1-e^{-2aT}\right) $$ The distribution of $r_T$ is thus normal with expectation $$r_0 e^{-aT} +b(1-e^{-aT})$$