# CIR model. Is there a closed-form solution or even a good proxy of analytical solution?

Is there a closed-form (analytical) solution for the Cox-Ingersoll-Ross SDE $$$$dr_t=k_r(\theta_r-r_t)dt+\sigma_r\sqrt{r_t}dW_t\tag{1}$$$$ ?

Notice that $$\{r_t\}$$ is our process of interest, $$k_r$$ and $$\theta_r$$ are constant parameters and $$dW_t$$ denotes Wiener increment.

I would need a closed-form solution of $$(1)$$ (or even a good analytical proxy) so as to compare it with solution of my numerical simulation of $$(1)$$. I am aiming at determining the order of accuracy of my numerical solution, so I would need an analytical solution (or something like that) of $$(1)$$ so as to compute the average error of my simulation code, that is $$\mathbb{E}\left(r_t^{TRUE}-r_t^{SIMULATED}\right)$$.

Any suggestion or good source?

• There's no closed form solution for the square root diffusion $r_t$ that I know of. You can however write down its (conditional) moments in closed-form. Similarly, you can write down closed-form solutions for the prices of zero coupon bonds and bond options under the CIR model. You can compare both the moments and the prices with your simulation results. – Kevin Aug 6 '20 at 20:51
• Also asked here: math.stackexchange.com/questions/3782360/… – fesman Aug 7 '20 at 5:34