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?



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

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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, 2016 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, 2016 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, 2016 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, 2016 at 17:32

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