Models For Estimating The Structure Of Interest Rates From Observations Of Yield Curves

We present a dynamical systems approach for modelling the term structure of interest rates based on a linear differential equation under uncertainty. In contrast to a stochastic process we introduce impulse or point-impulse perturbations on either (a), the spot (shortest-term, risk neutral) interest rate as the unknown function, or (b), its integral, namely the yield function, or both simultaneously. Parameters are estimated by minimizing the maximum absolute value of the measurement errors. Termed the Optimal Observation Problem, OOP, it defines our norm of uncertainty, in contrast to the expectation operator for a stochastic process. Beyond the learning period (the current time), the solved-for spot rate function becomes the forecast of the unobservable function in a future period, while its integral should approximate the yield function well. Non-arbitrage is addressed by providing a necessary condition expressed as constraints in the OOP, under which non-arbitrage is guaranteed. The property of mean-reversion is also preserved, and functional estimates are provided for the market price of risk. Analogous concepts to “drift” and “volatility” are treated in a manner that provides a criterion for the choice of perturbation to employ in a given real situation. Additional constraints, if necessary, guarantee non-negative short and forward rates, a property not automatically fulfilled in the stochastic case. We test the approach empirically with daily Treasury yield curve rates data, mainly for discount bonds having 3- to 6-month maturities over observation periods of up to one year. Computational results are reported for many numerical experiments together with some financial interpretations, see http://kwel.biz.uiowa.edu/