Term structure extrapolation and asymptotic forward rates

We investigate different inter- and extrapolation methods for term structures under different constraints in order to generate market-consistent estimates which describe the asymptotic behavior of forward rates. Our starting point is the method proposed by Smith and Wilson, which is used by the European insurance supervisor EIOPA. We use the characterization of the Smith–Wilson class of interpolating functions as the solution to a functional optimization problem to extend their approach in such a way that forward rates will converge to a value which is an outcome of the optimization process. Precise conditions are stated which guarantee that the optimization problems involved are well-posed on appropriately chosen function spaces. As a result, a well-defined optimal asymptotic forward rate can be derived directly from prices and cashflows of traded instruments. This allows practitioners to use raw market data to extract information about long term forward rates, as we will show in a study which analyzes historical EURIBOR swap data.