Dear Experts -- Happy Sunday  

Typically when calibrating an interest rate model (e.g. Cox-Ingersoll-Ross) for multiple short rates (e.g. 3-month, 6-month, 1 year etc.), do practitioners use Cholesky decomposition and take the product of the lower triangular from the Cholesky and the vector of random values in order to induce (historical) covariances into that vector of random values?

If no, thanks for correcting me. If the answer is a yes, please can you advise if the covariance matrix should contain variance-covariances of the raw short rates or the percentage changes in the raw short rates?

Thanks in advance.

What do you mean “multiple short rates”? In a given term structure model, including CIR, the short rate is a somewhat abstract notion that is best operationalized as the overnight rate. The term “calibration” in this context is usually applied to the process of trying to fit model parameters to a cross-section of contemporaneous prices, e.g. of a set of swaps and options, but it’s certainly fair game to also attempt to fit historical covariance structures. In the latter case, people often reduce the dimension of the space of observations by focusing on the first couple of principal components of a set of rates.