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Affine term structure models driven by independent Lévy processes

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.

Affine term structure models driven by independent Lévy processes

Abstract

We characterize affine term structure models of non-negative short rate which may be obtained as solutions of autonomous SDEs driven by independent, one-dimensional Lévy martingales, that is equations of the form with deterministic real functions and independent one-dimensional Lévy martingales . 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 of (1) may be obtained also as solutions of autonomous SDEs driven by independent stable processes with stability indices in the range . The obtained models include in particular the -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 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 -CIR models.
Paper Structure (24 sections, 13 theorems, 232 equations, 8 figures, 5 tables)

This paper contains 24 sections, 13 theorems, 232 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Laplace exponents of Lévy processes
  4. Projections of the noise
  5. Preliminary characterization of generating equations
  6. Problem formulation
  7. One-dimensional generating equations
  8. Non-uniqueness in the multidimensional case
  9. Classification of generating equations
  10. Main results
  11. Proofs
  12. Moments and tails of short rates from the class $\mathbb{A}_g(a,b;\alpha_1,...,\alpha_g; \eta_1,...,\eta_g)$
  13. Moments of the rates $R(t)$, $t \in (0, +\infty)$
  14. Limit distributions of the rates $R(t)$ as $t \rightarrow +\infty$ and their tails
  15. Generating equations on a plane
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $Z$ be a Lévy martingale with characteristic triplet chrakterystyki Z and $Z^{G(x)}$ be its projection projection of Z with the Lev́y measure $\nu_{G(x)}(\textnormal{d}v)$ given by nu_G.

Figures (8)

  • Figure 1: Calibration errors of CIR and $\alpha$-CIR models for the ECB rates at randomly chosen dates.
  • Figure 1: Calibration to the ECB rates from 6.10.2010. View for all/small/large maturities.
  • Figure 2: Parameters of CIR and $\alpha$-CIR models for dates with the greatest error reduction.
  • Figure 2: Calibration to the ECB rates from 3.12.2014. View for all/small/large maturities.
  • Figure 3: Calibration to the ECB rates from 8.04.2022. View for all/small/large maturities.
  • ...and 3 more figures

Theorems & Definitions (18)

  • Proposition 2.1
  • Remark 2.2
  • Example 2.3
  • Theorem 3.1
  • Corollary 3.2
  • Proposition 3.3: Canonical representation of $\mathbb{A}_g(a,b;\alpha_1,...,\alpha_g; \eta_1,...,\eta_g)$
  • Proposition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Remark 3.7
  • ...and 8 more