Simulating CIR++ Rates with a Truncated Euler Scheme

Summary

The document asks how to simulate the CIR++ short-rate model, in which the short rate is the sum of a CIR process and a deterministic time-dependent shift. It proposes advancing the CIR component with a truncated Euler step: the drift pulls the process toward its long-run level, while the diffusion uses the square root of the nonnegative part of the current state. The shifted short rate is then formed by adding the shift at each simulation date.

The text raises the modeling question but does not give an answer, numerical comparison, or validation. In particular, it does not discuss time-step bias, preservation of nonnegativity, exact or alternative CIR simulation schemes, or how the shift is calibrated. Treat the proposed procedure as a question to investigate rather than as an endorsed method; its practical accuracy and suitability are not established here.

Key ideas

• CIR++ defines the short rate as a CIR state process plus a deterministic time-dependent shift.
• The proposed simulation advances the CIR state with a truncated Euler discretization.
• The rate at each time point is obtained by adding the shift evaluated at that date.
• The document presents this as a question and supplies no evidence that the discretization is accurate or appropriate.

Tags

# simulating from the CIR++


# simulating from the CIR++












I am looking at the CIR++ model which is described in interest rate models by Brigo et al, and was wondering on how to actually simulate from this model. The model reads

$$r_t=x_t+\phi(t),$$

where $x$ follows a CIR model.

To keep it as simple as possible i thought i would simulate the x process according to a 'truncated' Euler scheme, i.e.

$$x_{t_i}=x_{t_{i-1}}+\kappa(\theta-x_{t_{i-1}})\Delta t+\sigma \sqrt{x_{t_{i-1}}^+} \Delta t Z_t,$$

where $Z_t$ is $N(0,1)$. My question then is, would it make sense do to so and simply add $\phi(t)$ accordingly, meaning

$$r_{t_i}=x_{t_{i}}+\phi(t_{i}).$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.