*What is the appropriate time step to use in **estimating parameters for, and projecting short rates using, the CIR model?** Can estimates based on daily time series data be used in an interest rate model that projects short rates at a monthly time interval?*

The parameter estimates for instantaneous drift and volatility for the CIR interest rate model obtained from historical short rates data are dependent on the time series of rates used for the exercise. If a daily short rate series is used the resulting parameters are daily estimates of the parameters; if a monthly series is used the parameters will be monthly estimates.

This follows from the general recursive expression that is derived for r(t_{i}) = γ(1 – e^{-к}^{∆}^{t}) + e^{-к}^{∆}^{t}r(t_{i}_{–}_{1}) + ɛ(t_{i}) using a discretization process of the CIR interest rate model. The interval [0; T] is evenly divided into N subintervals with ti = i T/N for i = 1, 2, …, n. Each time-step is calculated as ∆t = t_{i} – t_{i}_{–}_{1}.

Parameters estimated using a time step of ∆t are the:

- Instantaneous drift = к∆t
- Instantaneous volatility = σ√∆t

This time step is ** already implied** in the drift and volatility parameters estimated from the time series data.

For example, the daily time step is inferred in the estimates when you use daily rates for the exercise. For daily rates (i.e. interval of [0,1], subdivided in 360 subintervals, assuming a 360 day year, the time step ∆t =1/360), the drift and volatility estimates are for к(1/360) & σ√(1/360) respectively.

The daily parameter estimates therefore cannot be used to project or simulate interest rates series other than a daily rates series. For example, the daily estimates cannot be used as is, to project short rates at monthly time steps (dt = 1/12) going forward.

For monthly projections of short rates (i.e. interval of [0,1] subdivided in 12 or alternatively an interval of [0,30 (years)] subdivided in 360 subintervals, the time step ∆t =1/12), the daily parameter estimates could possibly only be used if:

- Drift and volatility estimates using a monthly time series of rates equal the drift and volatility estimates using daily rates series, based on the peculiarities of the time series data available,

OR

- Monthly estimates are approximated from daily estimates as follows:

- к(1/12) = к(1/360) * 360/12
- σ√(1/12) = σ√(1/360) * √(360/12)

Ideally, the parameter estimation should be based on the historical time series that is consistent with the time step, dt, used in the interest rate model. Use historical monthly series of short rates to estimate parameters for the interest rate model to project rates on a monthly time step (dt =1/12); use historical daily series of short rates to estimate parameters for a model that uses a daily time step (dt = 1/360).

References:

- A covariance equivalent discretisation of the CIR model by Deelstra & Parker, and
- “Affine Term-Structure Models: Theory and Implementation by Bolder