Allowing Negative Short Rates with Shifted CIR and Gaussian Models

Summary

The document addresses how to model short-term interest rates that can fall below zero. The Cox–Ingersoll–Ross model keeps its short rate nonnegative, which can conflict with observed conditions in markets where negative rates occur. One suggested extension, CIR++, adds a deterministic time-dependent shift to the CIR process. The shifted rate can become negative, while the shift is chosen to fit observed discount factors.

The response points to CIR++ as a treatment of the positivity issue and notes that models with normally distributed short rates, including Vasicek, Ho–Lee, and Hull–White, allow negative values directly. The material gives a brief model overview rather than a comparison of calibration quality, pricing performance, or the practical tradeoffs among these approaches. It does not endorse a particular model or specify how to set a lower bound on rates; the central point is that a shifted CIR process and Gaussian short-rate models can accommodate negative rates.

Key ideas

• The standard CIR short-rate process does not allow negative rates.
• CIR++ adds a deterministic time-dependent shift to the CIR process.
• The shift can allow negative short rates while fitting observed discount factors.
• Vasicek, Ho–Lee, and Hull–White models allow negative rates directly through normally distributed short rates.

Tags

# Stochastic process for interest rates allowing negative values


# Stochastic process for interest rates allowing negative values












The Cox-Ingersoll-Ross process for the short term interest rate r(t) does not allow r(t) to become negative, but short-term rates are negative in much of the developed world. To account for this, do you use a CIR process for a shadow rate r'(t) that equals r(t) + c, where c = 0.01 if you think short-term rates cannot get more negative than 1%? Has there been research on this?

## Answer by Kevin (score 5, accepted)

https://quant.stackexchange.com/a/46809

Yes, people have looked into that alreaday, for instance here and here (for lognormal models). Brigo and Mercurio took the CIR short rate $(x_t)$ and added a deterministic shift $\vartheta(t)$ to it in order to obtain the short rate process $(r_t)$ via $r_t=x_t+\vartheta(t)$. The function $\vartheta$ serves to guarantee a perfect fit with observed discount factors and hence, can lead to negative short rate. In their book, you can read about their CIR extension (named CIR++) in Section 3.9 which addresses the issue of positivity explicitly in 3.9.3.

Note that normally distributed short rate models like the models from Vasicek, Ho-Lee and Hull-White allow directly for negative short rates.

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.