Probability of Default modelling with L\'evy-driven Ornstein-Uhlenbeck processes and applications in credit risk under the IFRS 9

TL;DR

This paper introduces a sophisticated stochastic modeling framework using Levy-driven Ornstein-Uhlenbeck processes to estimate Probability of Default, addressing new IFRS 9 standards with theoretical and numerical methods.

Contribution

It develops a novel PD estimation approach based on Levy-driven Ornstein-Uhlenbeck processes, including integral and differential equations, with solutions suitable for IFRS 9 credit risk applications.

Findings

Derived integral and PIDE formulations for PD

Proved existence of weak and strong solutions

Developed numerical schemes for PD estimation

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 L\'evy-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.