Consistent Initial Curves for Interest Rate Models

Abstract

It was a significant step forward in modeling the term structure of interest rates to require the model to be arbitrage-free, in the sense that applying the model’s dynamics to the current market yield curve did not produce any arbitrage opportunities, and therefore, internal inconsistency. The Heath-Jarrow-Morton (HJM) class of interest rate models leads to a large family of arbitrage-free term structure processes. Fitting HJM models empirically requires estimating a continuous forward rate curve from the discrete set of bond prices observed in the market. But, as the authors point out, unless this is done with care, the function that is fitted to market rates may not be consistent with the HJM dynamics for the forward curve. That is, current forward rates would be fitted to a functional form that cannot evolve to another forward curve of the same type in the next period under the assumed HJM model specification. This problem is generally overlooked in practice, since the forward rate curve is refitted every period anyway, but it may well show up in the form of parameter instability in the model. In this article, Angelini and Herzel examine this issue both theoretically and empirically, and show that requiring the forward curve to be fitted to a functional form that is consistent with the interest rate process can make a substantial difference in model performance and in parameter stability.