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, we provide an extension of the CIR model which admits arbitrary initial term structure. We show how to calculate bond prices via a perturbative approach, and we provide closed formulas at every order. Since the parameter we select 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, we estimate the extended CIR model via maximum likelihood on a time series of daily interest rate yields. Our results show that the CIR model has to be rejected with respect to the proposed extension, and point out that the extended CIR model provides a more flexible characterization of the link between risk neutral and natural probability.