Parameter estimation of interest rate models

When using the interest rate models for pricing or simulation purposes, it is important to calibrate their parameters to real data properly. Here, we present a possible method to estimate the parameters. This method was developed by Chan et al, 1992, and is often referred to as the CKLS method. The procedure was elaborated to estimate the parameters of the following interest rate model with the help of the econometric procedure called Generalized Method of Moments (GMM; see Hansen, 1982, for more details):

Parameter estimation of interest rate models

It is easy to see that this process gives the Vasicek model when γ=0, and the CIR model when γ =0.5. As the first step of the parameter estimation, we discretize this equation with the Euler approximation (see Atkinson, 1989):

Parameter estimation of interest rate models

Here, δt is the time interval between two observations of the interest rate and et is independent, standard normal random variables. The parameters are estimated with the following null hypothesis:

Parameter estimation of interest rate models
Parameter estimation of interest rate models
Parameter estimation of interest rate models

Let Θ be the vector of the parameters to be estimated, that is, Parameter estimation of interest rate models.

We consider the following function of the parameter vector:

Parameter estimation of interest rate models

It is easy to see that under the null hypothesis, Parameter estimation of interest rate models=0.

The first step of GMM is that we consider the sample corresponding to Parameter estimation of interest rate models, which is Parameter estimation of interest rate models:

Parameter estimation of interest rate models

Here, n is the number of observations.

Finally, GMM determines the parameters by minimizing the following quadratic form:

Parameter estimation of interest rate models

Here, Parameter estimation of interest rate models is a symmetric, positive definite weight matrix.

There is a quadprog package in R for these kinds of problems, or we can use general methods for optimization with the optim function.

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