Cox-Ingersoll-Ross (CIR)

Watan First Institute has the skills and know-how for advanced modeling and consulting on asset allocation, portfolio optimization, Asset & Liability Management, risk management, ICAAP, Basel III, MiFID, Solvency II, actuarial risk, etc.

Regarding interest rates, it is well known that the Cox-Ingersoll-Ross (CIR) stochastic model to study the term structure of interest rates, as introduced in 1985, is inadequate for modeling the current market environment with negative short interest rates. Moreover, the diffusion term in the rate dynamics goes to zero when short rates are small; both volatility and long-run mean do not change with time; they do not fit with the skewed (fat tails) distribution of the interest rates, etc. Several different extensions of the original model have been proposed to date, with the aim of overcoming the limitations of the CIR model: from one-factor models including time-varying coefficients or jump diffusions to multi-factor models. All these extensions preserve the positivity of interest rates but, in some cases, the analytical tractability of the basic model is violated. Our approach, instead, is based on a proper translation of interest rates such that the market volatility structure is maintained as well as the analytical tractability of the original CIR model. Thus the suggested CIR# model is quite powerful for the following reasons. First, all the improvements are obtained within the CIR framework in order to preserve the single-factor property and the analytical tractability of the original model. Second, market interest rates are properly translated away from zero and/or negative values. The market data sample is partitioned into sub-groups in order to capture all the statistically significant changes of variance in real spot rates and so to give an account of jumps. Third, we have introduced a new way of calibration of the CIR model parameters to actual data.

See below a figure on how real interest rates are calibrated and then simulated over a period of 5.5 years. The noticeable result is that the simulation works also in presence of negative interest rates. This is the outcome of the so-called CIR# model and it is described here  https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2975108.