A unique linear solution to the term structure of interest rates

Abstract

An arbitrage-free term structure model can be regarded as a solution to a fundamental partial differential equation (PDE) and the boundary condition. Traditionally, many authors have designated the instantaneous interest rate as the sole factor or one of the factors, which have severely restricted the domain of the solution. Since none of the existing equilibrium solutions is able to fit the observed term structure of interest rates, and since the new methodology proposed by Ho and Lee (1986) and Heath, Jarrow, and Morton (1992) simply takes the current term structure as given, the observable term structure can only be represented/estimated by the traditional curve-fitting models. Can any of the curve-fitting models be consistent with the notion of no-arbitrage? This paper has proved that a special case of the exponential spline model of Vasicek and Fong (1982), defined in the current paper as an Exponential Polynomial (EP) discount function, is a unique linear solution to the fundamental PDE and the boundary condition for discount bonds. Like all the curve-fitting models, the EP model is assumption-free but theoretically meaningful. It differs from all the existing non-linear solutions by not requiring any specification of the stochastic processes of the surrogate state factors. The variances and covariances of the underlying processes are irrelevant because all the second-order partial derivatives with respect to the state factors are zero. The drifts of the state factors are also irrelevant because the uniqueness of the linear solution dictates a unique specification of the first-order coefficients of the PDE. The preliminary empirical investigation has confirmed that the EP solution can fit the observed term structure consistently over time with a set of constant parameters, i.e., without relying on the "time-varying parameters". Because the exponential basis is time-invariant, the surrogate state factors are well-defined and consistently estimated from cross-sectional samples. The preliminary time-series analysis shows that the first differences of the state factors all appear to exhibit zero-mean increments, which are consistent with the martingale implication of the EP model.