Arbitrary Initial Term Structure within the CIR Model: A Perturbative Solution

Abstract

Single‐factor interest rate models with constant coefficients are not consistent with arbitrary initial term structures. An extension which allows both arbitrary initial term structure and analytical tractability has been provided only in the Gaussian case. In this paper, within the context of the HJM methodology, an extension of the CIR model is provided which admits arbitrary initial term structure. It is shown how to calculate bond prices via a perturbative approach, and closed formulas are provided at every order. Since the parameter selected for the expansion is typically estimated to be small, the perturbative approach turns out to be adequate to our purpose. Using results on affine models, the extended CIR model is estimated via maximum likelihood on a time series of daily interest rate yields. Results show that the CIR model has to be rejected with respect to the proposed extension, and it is pointed out that the extended CIR model provides a more flexible characterization of the link between risk neutral and natural probability.

Notes

1. The term smooth is used here to mean that the discount function P(r, t; T) or equivalently the forward rate f(r, t; T), are continuous and twice differentiable with respect to all their arguments.

2. P∗(0, T) is also assumed to be continuous and twice differentiable with respect to maturity.

3. The coefficients H_j given by (18) are bounded and the solution of the Volterra equation expressed as a power series in the parameters k is well behaved.