# simulating from the CIR++

I am looking at the CIR++ model which is described in interest rate models by Brigo et al, and was wondering on how to actually simulate from this model. The model reads

$$r_t=x_t+\phi(t),$$

where $x$ follows a CIR model.

To keep it as simple as possible i thought i would simulate the x process according to a 'truncated' Euler scheme, i.e.

$$x_{t_i}=x_{t_{i-1}}+\kappa(\theta-x_{t_{i-1}})\Delta t+\sigma \sqrt{x_{t_{i-1}}^+} \Delta t Z_t,$$

where $Z_t$ is $N(0,1)$. My question then is, would it make sense do to so and simply add $\phi(t)$ accordingly, meaning

$$r_{t_i}=x_{t_{i}}+\phi(t_{i}).$$

• What is $\phi(t)$ ? – Richard Mar 30 '16 at 11:44
• why not $r_{t_i}=x_{t_i}+\phi(t_i)$? – Gordon Mar 30 '16 at 12:58
• $\phi(t)$ is a determinstic shift function, which is chosen in order to match the initial term structure. You are right Gordon, that is a typo. – Bohlke Mar 30 '16 at 13:12
• Sorry but I find your notations a bit sloppy. What does the first term in the rhs of your Euler discretisation represent? It should be $x_{t_i}$. Plus, this is not exactly the full truncation method, $x_{t_i}$ involved in the mean reversion component should be truncated to $x_{t_i}^+$. The rest seems ok but could you clear things up / fix your notations to be sure? – Quantuple Mar 30 '16 at 16:14
• Sorry, things went a little fast. But the first term should be $x_{t_{i-1}}$. The schemes that I have seen in the literature normally only truncates the component inside the square-root, to avoid taking the square-root of negative numbers. But I see your point, a full truncation could be done by following your method. What I was actually doubting was whether the last equation makes sense, where I add the $\phi$ function – Bohlke Mar 30 '16 at 17:11