Probability of Default modelling with Lévy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9

Paper: arXiv 2309.12384

Authors: Kyriakos Georgiou, Athanasios N. Yannacopoulos

Abstract

In this paper we develop a framework for estimating Probability of Default (PD) based on stochastic models governing an appropriate asset value processes. In particular, we build upon a Lévy-driven Ornstein-Uhlenbeck process and consider a generalized model that incorporates multiple latent variables affecting the evolution of the process. We obtain an Integral Equation (IE) formulation for the corresponding PD as a function of the initial position of the asset value process and the time until maturity, from which we then prove that the PD function satisfies an appropriate Partial Integro-Differential Equation (PIDE). These representations allow us to show that appropriate weak (viscosity) as well as strong solutions exist, and develop subsequent numerical schemes for the estimation of the PD function. Such a framework is necessary under the newly introduced International Financial Reporting Standards (IFRS) 9 regulation, which has imposed further requirements on the sophistication and rigor underlying credit modelling methodologies. We consider special cases of the generalized model that can be used for applications to credit risk modelling and provide examples specific to provisioning under IFRS 9, and more.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly rigorous theoretical framework using Lévy-driven Ornstein-Uhlenbeck processes, Partial Integro-Differential Equations (PIDEs), and viscosity solutions to model Probability of Default under IFRS 9. However, it leans heavily on mathematical derivations and numerical schemes rather than extensive empirical backtesting on real-world market datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Estimate PD under IFRS 9 using Lévy-driven OU processes] --> B[Model Formulation: Multi-latent variable asset value process]
    B --> C[Mathematical Framework: Integral Equation & PIDE Representations]
    C --> D[Theoretical Analysis: Prove Existence of Weak/Viscosity & Strong Solutions]
    D --> E[Numerical Schemes: Develop Algorithms for PD Estimation]
    E --> F[Outcomes: Rigorous Credit Risk & IFRS 9 Provisioning Framework]