Introduction

The case of the interest rate curve is particularly complex and interesting, since it is not the random motion of a point, but rather the consistent history of a whole curve (corresponding to different loan maturities) which is at stake. Indeed, interest rates corresponding to all possible maturities (from one week to thirty years) are ‘floating’, that is, fixed by the market. When money lenders are more numerous, money is cheaper to borrow, and the corresponding rate goes down. Conversely, the rates are high whenever money lenders are uncertain about the future and ask for a substantial yield on their loans.

The need for a consistent description of the whole interest rate curve is driven by the importance, for large financial institutions, of asset liability management and by the rapid development of interest rate derivatives (options, swaps, options on swaps, etc.). Present models of the interest rate curve fall into two categories: the first one is the Vasicek model and its variants, which focuses on the dynamics of the short-term interest rate, from which the whole curve is reconstructed. The second one, initiated by Heath, Jarrow and Morton takes the full forward rate curve as dynamic variables, driven by (one or several) continuoustime Brownian motion, multiplied by a maturity-dependent scale factor. Most models are however primarily motivated by their mathematical tractability rather than by their ability to describe the data.