A General Framework for the Construction and the Smoothing of Forward Rate Curves

This paper establishes a general theoretical and numerical framework for the construction and the smoothing of instantaneous forward rate curves. It is shown that if the smoothness of a curve is defined as an integral of a function in the derivatives of the curve, then the optimal curves are splines that satisfy certain ordinary differential equations. For such curves, and efficient numerical method is given for the determination of the spline parameters subject to mild assumptions. The resulting forward rate curves do not generally possess the desired degree of smoothness due mainly to the constraints imposed on the curves by the various market observed prices. A Partial solution to this problem is then introduced which achieves additional smoothing by taking into account the bid-ask ranges of each market rate. This eliminates much of the oscillatory patterns and the points of high curvature, and produces curves that are ideal for applications such as the estimation of interest rate models, and the pricing and risk management of interest rate derivatives, which are sensitive to forward rate curves.