An Orthogonal Polynomial Approach to Estimate the Term Structure of Interest Rates

In this paper, we introduce a new algorithm to estimate the term structure of interest rates. It is obtained from a constrained optimization, where the objective is to minimize the integral of squared first derivatives of the instantaneous forward interest rate subject to the condition that the estimated bond prices lie within the range of observed bid and ask prices. We use a finite series of ordinary Laguerre polynomials to approximate the unknown function of the instantaneous forward interest rate. The objective function can be written explicitly as a quadratic form of the Laguerre constants and the nonlinear constraints can be obtained from a recurrence relationship. The estimation error is less than one basis point, given a sufficient number of bonds.