Shapes of the Term Structure of Interest Rates

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The term structure of interest rates -- summarized in the form of the yield curve or the forward curve -- is one of the most fundamental economic indicators. Its shape encodes important information on the preferences for short- vs. long-term investments, the desire for liquidity and on expectations of central bank decisions and the general economic outlook. It is therefore a natural question -- to be asked of any mathematical model of the term structure -- which shapes of yield curves and forward curves the model is able to reproduce and with which frequency they appear. While this question has been answered earlier and conclusively for one-dimensional affine term structure models, systematic results on the more relevant two-dimensional case have appeared only in the last few years. In a recent break-through, the geometric approach of envelopes has been introduced to completely analyze and classify all possible term structure shapes in the two-dimensional Vasicek model. In this project, we will leverage the method of envelopes to solve the problems of shape classification, state-space segmentation and calculation of long-run frequencies of different shapes in a large number of relevant interest rate models, well beyond the Vasicek model. While the Vasicek model is relatively simple due to its Gaussian nature and linearization of Riccati ODEs, substantial challenges will have to be overcome when extending and adapting the envelope approach to other affine term structure models with and without jumps. Additional challenges appear when non-affine or time-inhomogeneous affine models are considered. Instead of envelopes of famillies of lines, families of one-manifold immersed in the models state space have to be considered. When classifying singular points and self-intersections of the envelope theory of total positivity (pioneered by Samuel Karlin in the 1960ies) will be used.