Affine term structure models driven by independent Lévy processes

Paper: arXiv 2402.07503

Authors: Michał Barski, Rafał Łochowski

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 Lévy 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 Lévy martingales $Z_1,…,Z_d$. Using a general result on the form of the generators of affine term structure models due to Filipović, 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 $α$-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 $α$-CIR models.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical characterization of affine term structure models driven by independent Lévy processes, establishing connections to stable processes and heavy-tailed distributions. It combines advanced theoretical stochastic calculus with a practical calibration exercise comparing the proposed models against classic CIR and alpha-CIR benchmarks.

Research Flowchart

  flowchart TD
    A["Research Goal: Characterize non-negative short rate R<br/>driven by independent Lévy martingales SDEs"] --> B["Apply Filipovic Generator Results &<br/>Assumed Regularly Varying Laplace Transforms"]
    
    B --> C["Theoretical Outcome: Solutions reduce to<br/>SDEs driven by stable processes (α ∈ (1,2])<br/>including α-CIR model"]
    
    C --> D["Heavy-Tail Analysis:<br/>Prove tail behavior & limit distribution<br/>based on stability indices"]
    
    E["Data Input:<br/>Market Term Structure of Interest Rates"] --> F["Numerical Calibration & Comparison:<br/>Test proposed models vs. classical CIR & α-CIR"]
    
    D --> F
    F --> G["Key Findings: Proposed stable-driven models outperform<br/>classical CIR, providing flexible & accurate market fit"]