  In the "positive interest" models of Flesaker-Hughston, the nominal discount
bond system is determined by a one-parameter family of positive martingales. In
the present paper we extend this analysis to include a variety of distributions
for the martingale family, parameterised 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 generate skewness and excess kurtosis in the bond
returns by choosing the martingale family to be given by (a) exponential gamma
processes, or (b) 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 L\'evy interest rate models are presented as well.
