Multi-Layer Deep xVA: Structural Credit Models, Measure Changes and Convergence Analysis

TL;DR

This paper introduces a multi-layer deep BSDE framework for portfolio-wide xVA valuation, efficiently capturing complex default and collateral effects with reduced computational cost compared to nested Monte Carlo methods.

Contribution

It develops a novel iterative deep BSDE approach with measure change techniques for accurate, scalable multi-layer xVA modeling, addressing computational intractability of nested simulations.

Findings

Method reduces computational demands significantly.

Successfully scales to high-dimensional portfolios.

Achieves accurate default and collateral effect modeling.

Abstract

We propose a structural default model for portfolio-wide valuation adjustments (xVAs) and represent it as a system of coupled backward stochastic differential equations. The framework is divided into four layers, each capturing a key component: (i) clean values, (ii) initial margin and Collateral Valuation Adjustment (ColVA), (iii) Credit/Debit Valuation Adjustments (CVA/DVA) together with Margin Valuation Adjustment (MVA), and (iv) Funding Valuation Adjustment (FVA). Because these layers depend on one another through collateral and default effects, a naive Monte Carlo approach would require deeply nested simulations, making the problem computationally intractable. To address this challenge, we use an iterative deep BSDE approach, handling each layer sequentially so that earlier outputs serve as inputs to the subsequent layers. Initial margin is computed via deep quantile regression to reflect margin requirements over the Margin Period of Risk. We also adopt a change-of-measure method that highlights rare but significant defaults of the bank or counterparty, ensuring that these events are accurately captured in the training process. We further extend Han and Long's (2020) a posteriori error analysis to BSDEs on bounded domains. Due to the random exit from the domain, we obtain an order of convergence of $\mathcal{O}(h^{1/4-\epsilon})$ rather than the usual $\mathcal{O}(h^{1/2})$. Numerical experiments illustrate that this method drastically reduces computational demands and successfully scales to high-dimensional, non-symmetric portfolios. The results confirm its effectiveness and accuracy, offering a practical alternative to nested Monte Carlo simulations in multi-counterparty xVA analyses.