# How can the increments of a CIR process be derived?

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 $$r_{t+1}$$?

Quantitative Finance Asked by John Smith on November 19, 2020

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.

First of all, let us write the model dynamics: $$r_t=r_0+alphaint_0^t(mu-r_s)ds+sigmaint_0^tsqrt{r_s}dW_s$$

We need to discretize this process: $$r_{t+Delta t}=r_t+alpha(mu-r_t)Delta t+sigmasqrt{r_t}W_{Delta t}$$ with $$Delta t=frac{T}{n}$$ and $$W_{Delta t}simmathcal Nleft(0,frac{T}{n}right)Rightarrow W_{Delta t}=sqrt{frac{T}{n}}varepsilon,$$ 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).

Correct answer by FunnyBuzer on November 19, 2020

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