Rational term structure models with geometric Levy martingales

TL;DR

This paper extends positive interest rate models by incorporating Levy processes, allowing for skewness and kurtosis in bond returns, and provides semi-analytical pricing formulas for bond options.

Contribution

It introduces a flexible class of rational term structure models using Levy martingales, broadening the modeling of market risk premiums and bond return characteristics.

Findings

Inclusion of jump and diffusion processes in models

Derivation of semi-analytical bond option pricing formulas

General results on Levy interest rate models

Abstract

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.