On the Geometry of the Term Structure of Interest Rates

We present recently developed geometric methods for the analysis of finite dimensional term structure models of the interest rates. This includes an extension of the Frobenius theorem for Fréchet spaces in particular. This approach puts new light on many of the classical models, such as the Hull-White extended Vasicek and Cox-Ingersoll-Ross short rate models. The notion of a finite dimensional realization (FDR) is central for this analysis: we motivate it, classify all generic FDRs and provide some new results for the corresponding factor processes, such as hypoellipticity of its generators and the existence of smooth densities. Furthermore we include finite dimensional external factors, thus admitting a stochastic volatility structure.