# Timesteps in Vasicek model

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.

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 $$r_0 e^{-aT} +b(1-e^{-aT})$$
• Note that $r_T=r_0 e^{-aT} +b(1-e^{-aT})+\sqrt{\frac{\sigma^2}{2a}\left(1-e^{-2aT}\right) }Z$ , where $Z$ is standard normal, then it is possible to simulate $r_T$ in one step. – Gordon Jan 24 '16 at 19:43