Rational Term Structure Models with Geometric Lévy Martingales

In the ‘positive interest’ models of Flesaker-Hughston, the nominal discount bond system is determined by a one-parameter family of positive martingales. In this paper, we extend this analysis to include a variety of distributions for the martingale family, parameterized by a function that determines the behaviour of the market risk premium. These distributions include jump and diffusion characteristics that generate various properties for discount bond returns. For example, one can choose the martingale family to be given by exponential gamma processes or by exponential variance-gamma processes. The models are ‘rational’ in the sense that the discount bond price is given by a ratio of weighted sums of positive martingales. Our findings lead to semi-analytical formulae for the prices of options on discount bonds. A number of general results concerning Lévy models for interest rates are presented as well.