In this post we explore how the Cox-Ingersoll-Ross (CIR) parameters are estimated, in other words we consider how the CIR model is calibrated to real world data.
The existing literature contains a number of papers that seek to estimate the parameters of the short rate models. Different authors use different data sets, time periods, sampling frequencies, and empirical methodologies.
The empirical methodology we have considered for estimating parameters in the CIR model is the LSE (Least Square Estimate). Various researches on the subject have revealed that the parameter estimates are sensitive to the choice of empirical methodology.
When using discrete data for estimating the parameters of a continuous model, one needs a discrete representation of the process. We will consider two representations below.
The first is a simple discretisation process with function:
Under both processes the principle involved is that the first two moments of both the discrete and the continuous models should be equal.
The process is as follows:
1. The interest rate series (rt*) is obtained for the given benchmark yield curve (e.g. for 3-month rates).
2. A centred CIR model is obtained. This is done by subtracting the rates at each data point with their long run average (?) across the entire series to obtain a transformed series (rt).
3. The least squares method is used to estimate the parameters for the discrete representation of the CIR model. This is done as follows:
a. At each data point the following residual term is calculated:
We have considered the least squared method of estimating parameters for the continuous CIR model using two discretisation techniques; Simple Discretization and Covariance Equivalent Discretization. The next post will look at how these estimated parameters are used in the model to simulate future short rates and consequently the term structure of interest rates.
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