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?



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 $$r_0 e^{-aT} +b(1-e^{-aT})$$

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    $\begingroup$ 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. $\endgroup$ – Gordon Jan 24 '16 at 19:43
  • $\begingroup$ Thanks. If T=2 then rT will be the short rate in two years right. If so how to construct the whole yield curve from this new rate? $\endgroup$ – Oamriotn Jan 25 '16 at 15:00
  • $\begingroup$ @Oamriotn The yield curve is constructed from the bond price formula, see math.nyu.edu/~benartzi/Slides10.2.pdf $\endgroup$ – user9403 Jan 25 '16 at 17:20
  • $\begingroup$ I'm now calibrating the Vasicek model from 1month euribor rates from 2008 -2015 using maximum likelihood calibration (sitmo.com/article/calibrating-the-ornstein-uhlenbeck-model). The strange result I get is that the mu is negative, and much more negative than the lowest 1 month euribor rate. I wonder if negative rates give some problems when using this calibration method? $\endgroup$ – Oamriotn Feb 1 '16 at 17:32

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