The Simple Yield Curve Models

With $P_t$ the price in current dollars of a dollar delivered $t$ time units from now, we assume that $P$ is a decreasing function defined for $t \in \mathbb{R}_+$ with $P_0 = 1$. The negative logarithmic derivative, $- \stackrel{\bullet}{P}_t/P_t$ defines the yield curve function $Y_t$. An $n$ parameter linear yield curve model selects as allowable yield curves $Y_t(r) = \sum_{i=1}^n r_i Y^i_t$ with the functions $Y^i$ fixed and with $r$ varying over an open subset of $\mathbb{R}^n$ on which $Y_t(r) \ge 0$ for all $t \in \mathbb{R}_+$. For example, the flat yield curve model with $P_t(r) = e^{-rt}$ is a one parameter linear model with $Y^1_t(r) = r > 0$. 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 $r$ $Lim_{t \to \infty} Y_t(r)$ exists as a finite limit, then the number of simple models is reduced to two.