Introduction

Before the work of Ho and Lee [39] and Heath, Jarrow and Morton [35] the point of view in the literature was explaining the term structure of interest rates or respectively the cross section of bond prices. The new Heath, Jarrow and Morton (HJM) models perfectly fit to an observed initial term structure by focussing on the arbitrage-free pricing of related derivatives. Given a specification of the volatility for the forward rates or bond prices together with the initial term structure completely determines the risk-neutral bond price dynamics or equivalently the short rate process (see e.g. de Jong and Santa Clara [24], Casassus, Collin-Dufresne and Goldstein [14]). The volatility structure in general can be computed by inverting the option prices similar to the calculation of implied volatilities that are extracted from stock option prices. One drawback of these models lies in the non-Markovian structure of the short rate dynamics resulting in a computationally low tractability. Hence, most of the HJM-models in the literature are restricted to a deterministic volatility structure leading to a Markovian short rate process. It is well known that a deterministic volatility function always leads to Gaussian interest rates and therefore we have to deal with negative interest rates.