Year: 2025
Type: article
Abstract: The paper is devoted to the study of the short rate equation of the form \[ \text{d}R(t)=F(R(t))\text{d}t+{\sum \limits_{i=1}^{d}}{G_{i}}(R(t-))\text{d}{Z_{i}}(t),\hspace{1em}R(0)={R_{0}}\ge 0,\hspace{3.33333pt}t\gt 0,\] with deterministic functions $F,{G_{1}},\dots ,{G_{d}}$ and a multivariate Lévy process $Z=({Z_{1}},\dots ,{Z_{d}})$ with possibly dependent coordinates. This equation is assumed to have a nonnegative solution which generates an affine term structure model. Under some mild assumptions on the Lévy measure of Z it is shown that the same term structure is generated by an equation with affine drift term and noise being a one-dimensional α-stable process with index of stability $\alpha \in (1,2)$. For this case the shape of possible simple forward curves is characterized. A precise description of normal, inverse and humped profiles in terms of the equation coefficients and the stability index α is provided. The paper generalizes the classical results on the Cox–Ingersoll–Ross (CIR) model [Econometrica 53 (1985), 385–408], as well as on its extended version where Z is a one-dimensional Lévy process [SIAM J. Financ. Math. 11(1) (2020), 131–147, Bond Markets with Lévy Factors, Cambridge University Press, 2020]. It is the starting point for the classification of affine models with dependent Lévy processes, in the spirit of [J. Finance 5 (2000), 1943–1978] and [Classification and calibration of affine models driven by independent Lévy processes, https://arxiv.org/abs/2303.08477].
DOI: 10.15559/25-vmsta280
Authors: Michał Barski, Rafał Łochowski
Institutions: University of Warsaw, SGH Warsaw School of Economics
Language: English
Subfield: Finance
Domain: Social Sciences
Open Access status: diamond
OpenAlex ID: W4411387967