# CIR discretization Milstein scheme

The CIR model for spot rate $r_t$ is:

$$dr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$

where $\eta, \gamma, \alpha$ are constants.

How to express this SDE in discrete form using Milstein scheme?

The one I derived is:

$$r_{t+1}=r_t+(\eta-\gamma r_t)\delta t+\sqrt{\alpha r_t}\cdot\sqrt{\delta t}\phi +\frac{1}{2}\sqrt{\alpha r_t}\cdot\left(\frac{1}{2}\frac{\alpha}{\alpha r_t}\right)[\delta t(\phi^2-1)]$$

where $\phi$ is normal RV.

Can anyone help me to identify my error? Or is it correct?

The Milstein scheme for the following CIR model

$$dr_t=(\eta-\gamma r_t)dt+\sqrt{\alpha r_t} dW_t$$

should be

$$r_{t+1}=r_t+(\eta-\gamma r_t)\delta t+\sqrt{\alpha r_t}\cdot\sqrt{\delta t}\phi +\frac{1}{2}\sqrt{\alpha r_t}\cdot\left(\frac{1}{2}\sqrt{\frac{\alpha}{r_t}}\right)[\delta t(\phi^2-1)]$$

$$r_{t+1}=r_t+(\eta-\gamma r_t)\delta t+\sqrt{\alpha r_t}\cdot\sqrt{\delta t}\phi +\frac{1}{4}\alpha(\phi^2-1)\delta t$$ where $\phi$ is normal RV.

I think that you wrongly derived $\frac{\partial\left(\sqrt{\alpha r_t}\right)}{\partial r_t}$ in the last term.

${{r}_{t+\Delta t}}={{r}_{t}}+(\eta-\gamma r_t)\delta t+\sigma \sqrt{{{r}_{t}}}\sqrt{\delta t}\,{{Z}}+\frac{1}{4}{{\sigma }^{2}}\delta t({{Z}}^{2}-1)$

Euler is just bad. Milstein in my experience is not much better. A better scheme in general is Predictor Corrector.

Another approach (in Glasserman) is to simulate the SDE using the non-central Chi^2 distribution which resolves Euler problem.

• It would be nice to have the same notations / names as the one lrh09 used, namely $\eta, \gamma, \alpha$ – JejeBelfort Aug 2 '17 at 8:19
• "euler is just bad" pretty bold statement imo, any context? – will Aug 2 '17 at 8:19
• @JejeBelfort can you help to complete it, so I can select yours as answer? – lrh09 Aug 2 '17 at 8:48