For a CIR process, which has SDE $$ dr_t = \alpha (\mu - r_t) dt + \sigma \sqrt{r_t} dW_t $$ how can I derive the increments over the discrete time-interval from $r_t$ to $r_{t+1}$?

  • 1
    $\begingroup$ $dr_t$ is replaced by $\Delta r_t$, $dt$ is replaced by $\Delta t$ and $dW_t$ is replaced by $\xi_i \sqrt{\Delta t}$ where $\xi_i$ is random $N(0,1)$ $\endgroup$
    – nbbo2
    Oct 21, 2020 at 10:52
  • 1
    $\begingroup$ An alternative, "exact" simulation would be via the non-central Chi-Squared distribution, for example explained here: deriscope.com/docs/Andersen_Jaeckel_Kahl_2010.pdf $\endgroup$ Oct 21, 2020 at 11:27
  • 1
    $\begingroup$ And of course quant.stackexchange.com/questions/57790/… and quant.stackexchange.com/questions/8114/… $\endgroup$ Oct 21, 2020 at 11:28
  • $\begingroup$ @noob2, would you be able to explain why that is the case? $\endgroup$
    – John Smith
    Oct 21, 2020 at 12:04
  • $\begingroup$ It is the Euler Maruyama method, the simplest (but not the only, as Kermitfrog pointed out) way to simulate an SDE in discrete time. $dW_t$ is random zero-mean and has a variance proportional to $dt$, so the standard deviation is proportional to $\sqrt{dt}$ $\endgroup$
    – nbbo2
    Oct 21, 2020 at 12:21

1 Answer 1


I am not totally sure I understand what you want to achieve. It seems like you are interested in discretizing CIR SDE. This can be done using the Euler-Murayama scheme for an equidistant decomposition of the time interval $[0, T]$, $\{0=t_0<\dots<t_n=T\}$.

First of all, let us write the model dynamics: $$r_t=r_0+\alpha\int_0^t(\mu-r_s)ds+\sigma\int_0^t\sqrt{r_s}dW_s$$

We need to discretize this process: $$r_{t+\Delta t}=r_t+\alpha(\mu-r_t)\Delta t+\sigma\sqrt{r_t}W_{\Delta t}$$ with $\Delta t=\frac{T}{n}$ and $W_{\Delta t}\sim\mathcal N\left(0,\frac{T}{n}\right)\Rightarrow W_{\Delta t}=\sqrt{\frac{T}{n}}\varepsilon,$ with $\varepsilon$ being a standard normal random variable.

Finally, we can use the trapezoidal rule to numerically integrate the simulated CIR rates and compute what you need (for example, the Monte Carlo zero-coupon bond prices).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.