# Cox Ingersoll Ross (1985) Model [closed]

How can I convert the following process to a standard Brownian Motion?

$$\mathrm{d}r_t=(a-br_t)\mathrm{d}t+\sigma\sqrt{r_t}\mathrm{d}W_t$$

## 1 Answer

I am not quite sure I get your question. You cannot solve the model in closed-form. What you get is that \begin{align*} r_t=r_0e^{-a t}+\frac{b}{a}(1- e^{-a t})+\sigma e^{-at}\int_0^te^{a u} \sqrt{r_u}\mathrm{d}W_u. \end{align*} Furthermore, you can get that $$r_t$$ follows a (non-central) chi-squared distribution and can compute the (conditional) moments of $$r_t$$ and obtain closed-form solutions for the prices of zero-coupon bonds and European-style zero-coupon bond options.

You may want to have a look at "Interest rate models" from Brigo and Mercurio which is an excellent reference.

• Thank you for your response. Yes, I am trying to obtain closed-form for zero-coupon bonds but to do that I need to work under P*, probability that converts that given $W_u$ to standard Brownian motion, say, $W*$, I am not sure how to get $r_t$ in the form of $W*$. Thank you for the reference btw. – Stochastic learner Feb 12 at 12:14
• @Stochasticlearner You cannot get $r_t$ in explicit form regardless what measure you use. If you want to price bonds, a common approach is to solve the corresponding bond price PDE. Guessing that bond price is affine-linear in $r_t$, the PDE reduces to a system of ODEs. The solution is given in the book I mentioned above. – KeSchn Feb 12 at 12:22