The Simple Yield Curve Models

With Pₜ the price in current dollars of a dollar delivered t time units from now, we assume that P is a decreasing function defined for t ∈ R₊ with P₀ = 1. The negative logarithmic derivative, - •Pₜ/Pₜ defines the yield curve function Yₜ. An n parameter linear yield curve model selects as allowable yield curves Yₜ(r) = ∑ᵢ₌₁ⁿ rᵢ Yⁱₜ with the functions Yⁱ fixed and with r varying over an open subset of Rⁿ on which Yₜ(r) ≥ 0 for all t ∈ R₊. For example, the flat yield curve model with Pₜ(r) = e⁻ʳᵗ is a one parameter linear model with Y¹ₜ(r) = r > 0. We impose two natural economic requirements on the model: (SPA) static prices allowed, i.e. it is always possible that as time moves forward, relative prices do not change, and (NLA) no local arbitrage, i.e. there does not exist a self-financing bundle of futures such that the zero present value is a local minimum with respect to small changes in the space of admissible yield curves. In that case the model always contains one of four simple models. If we impose the additional requirement (LRE) long rates exist, i.e. for every r Limt → ∞ Yₜ(r) exists as a finite limit, then the number of simple models is reduced to two.