Paper: arXiv 2303.08477

Authors: Michał Barski, Rafał Łochowski

Abstract

The paper is devoted to the study of the short rate equation of the form $$ dR(t)=F(R(t))dt+\sum_{i=1}^{d}G_i(R(t-))dZ_i(t), \quad R(0)=x\geq 0, \quad t>0, $$ with deterministic functions $F,G_1,…,G_d$ and independent Lévy processes of infinite variation $Z_1,…,Z_d$ with regularly varying Laplace exponents. The equation is supposed to have a nonnegative solution which generates an affine term structure model. A precise form of the generator of $R$ is characterized and a related classification of equations which generate affine models introduced in the spirit of Dai and Singleton \cite{DaiSingleton}. Each class is shown to have its own canonical representation which is an equation with the same drift and the jump diffusion part based on a Lévy process taking values in $\mathbb{R}^{g}, 1\leq g\leq d$, with independent coordinates being stable processes with stability indices in the range $(1,2]$. Numerical calibration results of canonical representations to the market term structure of interest rates are presented and compared with the classical CIR model. The paper generalizes the classical results on the CIR model from \cite{CIR}, as well as on its extended version from \cite{BarskiZabczykCIR} and \cite{BarskiZabczyk} where $Z$ was a one-dimensional Lévy process.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical generalization of canonical affine interest rate models using multidimensional, infinitely-varying Lévy processes with stable components. While highly advanced mathematically, the empirical validation is relatively limited, primarily serving as a proof-of-concept calibration against the classical CIR model.

Research Flowchart

  flowchart TD
    A[Goal: Classify & Calibrate Affine Short Rate Models Driven by Independent Lévy Processes] --> B[Model Formulation: Set Nonnegative SDE with Driven Lévy Processes]
    B --> C[Theoretical Analysis: Derive Generator Structure & Dai-Singleton Type Classification]
    C --> D[Canonical Representation: Reduce to g-Dimensional Stable Processes, Alpha in 1,2]
    E[Inputs: Market Term Structure Data & Baseline CIR Models] --> F[Numerical Calibration: Fit Canonical Models to Market Interest Rates]
    D --> F
    F --> G[Outcomes: Improved Term Structure Fit Compared to Classical CIR Models]