2010/12/31 by Dorje C. Brody, Lane P. Hughston, Ewan Mackie
Economics, Econometrics and Finance · Mathematics · #Bond #Bond valuation #Capital Investment and Risk Analysis #Credit Risk and Financial Regulations #Exponential function #Jump #Martingale (probability theory) #Parameterized complexity #Short rate #Stochastic processes and financial applications #Term (time) #Yield curve #math.PR #q-fin.PR
paper · pdf · doi:10.1080/17442508.2012.689835
published as Stochastics 84, 719-740 (2012) · expanded version, including general discussion on Lévy interest rate models
arxiv created 2011/11/16 · openalex publication_date 2012/06/25 · arxiv updated 2015/03/17 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.