Discretizing the CIR Process with Euler–Maruyama

Summary

This answer shows how to approximate a Cox–Ingersoll–Ross short-rate process over a discrete time step using the Euler–Maruyama scheme. It evaluates the drift and diffusion at the current rate, multiplies them by the step length and the corresponding Brownian increment, and represents that increment as the square root of the step length times a standard normal draw. Repeating the update on an evenly spaced time grid gives simulated rate paths.

The response also suggests using the trapezoidal rule to integrate simulated rates, for example when estimating zero-coupon bond prices by Monte Carlo. This is a basic numerical approximation, not an exact transition method. In particular, the simple Euler update does not explain how to handle the CIR process’s nonnegative rate constraint, so discretization error and possible negative simulated values are limitations to consider. No numerical comparison or convergence evidence is supplied.

Key ideas

• Euler–Maruyama approximates the CIR drift and diffusion using the rate at the start of each time step.
• A Brownian increment over a step is scaled by the square root of the step length.
• Repeated updates on an evenly spaced grid produce simulated short-rate paths.
• Numerical integration of simulated rates can support Monte Carlo bond valuation.
• The basic update does not address preservation of nonnegative rates.

Tags

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


# 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 $r_{t+1}$?

## Answer by FunnyBuzer (score 3, accepted)

https://quant.stackexchange.com/a/58840

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 $[0, T]$, $\{0=t_0<\dots<t_n=T\}$.

First of all, let us write the model dynamics: $$r_t=r_0+\alpha\int_0^t(\mu-r_s)ds+\sigma\int_0^t\sqrt{r_s}dW_s$$

We need to discretize this process: $$r_{t+\Delta t}=r_t+\alpha(\mu-r_t)\Delta t+\sigma\sqrt{r_t}W_{\Delta t}$$ with $\Delta t=\frac{T}{n}$ and $W_{\Delta t}\sim\mathcal N\left(0,\frac{T}{n}\right)\Rightarrow W_{\Delta t}=\sqrt{\frac{T}{n}}\varepsilon,$ 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).

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