Cox-Ingersoll-Ross (CIR)

Watan First Institute has the skills and know-how for advanced modelling 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 modelling 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