Affine term structure models driven by independent L\'evy processes

Abstract

We characterize affine term structure models of non-negative short rate $R$ which may be obtained as solutions of autonomous SDEs driven by independent, one-dimensional L\'evy martingales, that is equations of the form $$ dR(r)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=r_0\geq 0, \quad t>0, \quad (1)$$ with deterministic real functions $F,G_1,...,G_d$ and independent one-dimensional L\'evy martingales $Z_1,...,Z_d$. Using a general result on the form of the generators of affine term structure models due to Filipovi\'c, it is shown, under the assumption that the Laplace transforms of the driving noises are regularly varying, that all possible solutions $R$ of (1) may be obtained also as solutions of autonomous SDEs driven by independent stable processes with stability indices in the range $(1,2]$. The obtained models include in particular the $\alpha$-CIR model, introduced by Jiao et al., which proved to be still simple yet more reliable than the classical CIR model. Results on heavy tails of $R$ and its limit distribution in terms of the stability indices are proven. Finally, results of numerical calibration of the obtained models to the market term structure of interest rates are presented and compared with the CIR and $\alpha$-CIR models.