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

ArXiv ID: 2502.14766 “View on arXiv”

Authors: Unknown

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 $\mathcal{“O”}(h^{“1/4-ε”})$ 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.

Keywords: xVA (Valuation Adjustments), Backward Stochastic Differential Equations (BSDE), Counterparty Credit Risk, Monte Carlo Simulation, Financial Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail
• Why: The paper presents advanced mathematics including deep BSDEs, measure changes, and convergence analysis with explicit O(h^{“1/4-ε”}) rates, indicating high complexity. It also demonstrates practical numerical experiments on high-dimensional portfolios, showing implementation-heavy empirical work aimed at solving real-world xVA problems.

  flowchart TD
    A["Research Goal:<br/>Multi-Layer xVA<br/>Structural Model"] --> B["Methodology:<br/>Deep BSDE System<br/>4-Layer Architecture"]
    B --> C["Data/Inputs:<br/>Portfolio &<br/>Market Parameters"]
    C --> D["Process:<br/>Iterative Deep BSDE<br/>+ Change of Measure"]
    D --> E["Process:<br/>Deep Quantile<br/>Regression for MVA"]
    E --> F["Process:<br/>A Posteriori<br/>Error Analysis"]
    F --> G["Outcomes:<br/>Scalable High-Dim<br/>Accurate Convergence<br/>Alternative to Nested MC"]