Rational term structure models with geometric Lévy martingales

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 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.

Keywords:

• interest rate modelling
• asset pricing
• Lévy processes

Acknowledgements

The authors would like to thank seminar participants at the Imperial College Workshop on Stochastics, Control and Finance, London, April 2010, at the Fifth General Conference on Advanced Mathematical Methods in Finance, Bled, Slovenia, May 2010, at the Fields Institute Workshop on Financial Derivatives and Risk Management, Toronto, May 2010, at the Sixth World Congress of the Bachelier Finance Society, Toronto, June 2010, at the Workshop on Mathematical Finance and Related Issues, Kyoto, September 2010, at the Bank of Japan, Tokyo, September 2010 and at the Department of Economics, Hitotsubashi University, Tokyo, September 2010, for useful comments. L. P. Hughston thanks R. Miura, J. Sekine and H. Sugita for hospitality. E. Mackie acknowledges support by EPSRC, and thanks A. S. Iqbal and S. Lyons for helpful discussions.