On the reducibility of affine models with dependent Lévy factors
Pub. online: 17 June 2025
Type: Research Article
Open Access
Open Access
Received
14 November 2024
14 November 2024
Revised
21 March 2025
21 March 2025
Accepted
19 May 2025
19 May 2025
Published
17 June 2025
17 June 2025
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].
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