Rating Triggers for Collateral-Inclusive XVA via Machine Learning and SDEs on Lie Groups

TL;DR

This paper introduces a geometrical SDE-based model on Lie groups for rating transitions, integrating machine learning and market data to improve XVA calculations and address rating matrix imperfections.

Contribution

It presents a novel Lie group SDE framework for rating modeling, combined with deep learning for data calibration, enhancing robustness and accuracy in credit valuation adjustments.

Findings

Model accurately calibrates to historical and market data.

Deep learning improves rating matrix robustness.

Application to bilateral CVA under rating-dependent thresholds.

Abstract

In this paper, we model the rating process of an entity by using a geometrical approach. We model rating transitions as an SDE on a Lie group. Specifically, we focus on calibrating the model to both historical data (rating transition matrices) and market data (CDS quotes) and compare the most popular choices of changes of measure to switch from the historical probability to the risk-neutral one. For this, we show how the classical Girsanov theorem can be applied in the Lie group setting. Moreover, we overcome some of the imperfections of rating matrices published by rating agencies, which are computed with the cohort method, by using a novel Deep Learning approach. This leads to an improvement of the entire scheme and makes the model more robust for applications. We apply our model to compute bilateral credit and debit valuation adjustments of a netting set under a CSA with thresholds depending on ratings of the two parties.