Managing counterparty credit risk via BSDEs

TL;DR

This paper introduces a dynamic approach using BSDEs to model counterparty credit risk, providing explicit or approximate solutions and practical numerical methods for industry applications.

Contribution

It develops a novel BSDE-based framework for credit risk modeling, including reduction techniques and efficient numerical algorithms for practical implementation.

Findings

Explicit solutions for reduced BSDEs depending on close out conventions

Efficient numerical methods for solving BSDEs with moderate Brownian motions

Factor reduction techniques for portfolios with many risk factors

Abstract

We discuss a general dynamic replication approach to counterparty credit risk modeling. This leads to a fundamental jump-process backward stochastic differential equation (BSDE) for the credit risk adjusted portfolio value. We then reduce the fundamental BSDE to a continuous BSDE. Depending on the close out value convention, the reduced fundamental BSDE's solution can be represented explicitly or through an accurate approximate expression. Furthermore, we discuss practical aspects of the approach, important for the its industry applications: (i) efficient numerical methodology for solving a BSDE driven by a moderate number of Brownian motions, and (ii) factor reduction methodology that allows one to approximately replace a portfolio driven by a large number of risk factors with a portfolio driven by a moderate number of risk factors.