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

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Summary

The paper formulates portfolio-wide xVA computation as a stochastic control problem via a system of coupled BSDEs within a structural credit model framework. The bank and counterparty asset values follow diffusion processes, and default is triggered when these processes hit pre-specified barriers, making default times stopping times of the filtration. The adjusted portfolio value is defined implicitly as a conditional expectation of five types of discounted cashflows (contractual, collateral, initial margin, default, and funding), yielding a fixed-point equation that is well-posed by Banach’s fixed-point theorem. The key insight is to decompose this single high-dimensional BSDE into a layered system: Layer 1 solves clean values, Layer 2 computes initial margin via deep quantile regression, Layer 3 handles ColVA/CVA/DVA/MVA, and Layer 4 solves FVA. Each layer’s output feeds forward as input to subsequent layers.

The numerical solution employs a multi-layer deep BSDE solver that extends the original deep BSDE method of Han, Jentzen, and E (2018) in three directions: simultaneous solution of multiple coupled BSDEs sharing a single neural network function approximator, a Girsanov-based change of measure that tilts the forward SDE drift to increase default probabilities during training without altering the PDE solution, and incorporation of McKean-Vlasov BSDE features arising from the risk measure in the driver. The measure-change technique is particularly important because, under the original measure, default events are so rare that neural networks cannot learn the correct control behaviour near default boundaries. By simulating under a tilted measure and absorbing the reweighting into the BSDE generator, the algorithm avoids the large likelihood-ratio variance issues of traditional importance sampling.

On the theoretical side, the authors extend the a posteriori error analysis of Han and Long (2020) to BSDEs on bounded domains with random stopping times, obtaining a reduced convergence rate of O(h^{1/4-epsilon}) instead of the standard O(h^{1/2}). This is the first such result for deep BSDE methods under domain restrictions. Numerical experiments on a portfolio of 33 European basket call options on 5 underlying assets (a 93-dimensional problem in total) demonstrate that the method scales effectively and produces accurate results compared to both closed-form solutions and nested Monte Carlo benchmarks.

Key Contributions

  • Formulation of the full xVA problem (CVA, DVA, FVA, ColVA, MVA) as a four-layer system of coupled BSDEs within a structural credit model, enabling Brownian-driver-only numerics
  • Extension of the deep BSDE method to solve multiple BSDEs simultaneously with a shared neural network, reducing computational cost
  • A Girsanov-based measure change that tilts forward SDE drift to increase default frequency during training, incorporated directly into the BSDE generator rather than via importance sampling reweighting
  • Computation of initial margin via deep quantile regression using the check-function loss, producing VaR-based margin estimates at each time step
  • First a posteriori error analysis for deep BSDE methods on bounded domains with stopping times, yielding O(h^{1/4-epsilon}) convergence
  • Numerical demonstration on a 93-dimensional, non-symmetric portfolio of 33 basket options with 5 risk factors plus 2 defaultable entities

Methodology

The methodology combines several components arranged in a hierarchical pipeline:

  • Forward SDE: A (d+2)-dimensional diffusion process X merging non-defaultable risk factors with bank and counterparty asset processes, with correlated Brownian drivers capturing wrong-way risk
  • BSDE decomposition: The adjusted portfolio value BSDE is decomposed into individual xVA BSDEs (one per adjustment type) using the discounting cashflow approach of Brigo et al., enabling separate computation of each adjustment
  • Layer 1: Clean value BSDEs solved simultaneously for all P derivatives via the deep BSDE method, with a single shared network Z: [0,T] x R^d → R^{sum d_j}
  • Layer 2: Deep quantile regression for initial margin, training separate networks at each time step to estimate conditional VaR of portfolio value changes over the margin period of risk
  • Layer 3: ColVA, CVA, DVA, MVA BSDEs solved with measure-tilted forward processes; the drift shift q is chosen to increase the default probability of the relevant party
  • Layer 4: FVA BSDE, depending on all preceding layers
  • Temporal discretization: Euler-Maruyama for the forward SDE, forward-propagated discretization for the BSDEs, with the unknown initial condition and control process Z approximated by neural networks
  • Error analysis: Extension of Han-Long a posteriori bounds using Bouchard-Menozzi results for BSDEs on domains, yielding recursive error propagation across layers

Key Findings

  • The multi-layer approach successfully scales to a 93-dimensional problem (33 basket options on 5 underlyings + 2 default processes) with 201 time steps
  • The measure-change technique is essential: without it, the neural network learns Z approximately equal to zero near default boundaries, producing poor path-wise accuracy even when terminal condition errors appear comparable
  • Clean values (Layer 1) match closed-form solutions with high accuracy across mean, 1st, and 99th percentiles
  • Initial margin (Layer 2) approximations closely track nested Monte Carlo VaR benchmarks on representative paths
  • ColVA, MVA, and CVA (Layer 3) approximations match nested Monte Carlo reference solutions on representative trajectories and show tight empirical distributions of terminal condition errors
  • Under a stress test reducing counterparty default probability to approximately 0.6%, the measure-change technique maintains accuracy while the no-measure-change variant fails to capture fluctuations near default
  • The O(h^{1/4-epsilon}) convergence rate for stopping-time BSDEs is slower than the standard O(h^{1/2}) rate, reflecting the inherent difficulty of accurately handling boundary exits

Important References

  1. Han, Jentzen, and E (2018) - “Solving high-dimensional partial differential equations using deep learning” (1940 citations). Foundational deep BSDE solver that this paper extends.
  2. Gnoatto, Reisinger, and Picarelli (2023) - “Deep xVA solver - A neural network based counterparty credit risk management framework”. The direct predecessor to this paper, which introduced the deep xVA solver in a reduced-form credit model setting.
  3. Han and Long (2020) - “Convergence of the deep BSDE method for coupled FBSDEs” (181 citations). Provides the a posteriori error analysis framework that this paper extends to bounded domains with stopping times.

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