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An Explicit Scheme for Pathwise XVA Computations

Lokman Abbas-Turki, Stéphane Crépey, Botao Li, Bouazza Saadeddine

TL;DR

The paper develops an explicit-time, regression-based scheme for solving Markovian anticipated BSDEs arising in cross-valuation adjustments (XVAs), enabling pathwise XVA computations without Picard iterations. It combines neural network least-squares and quantile regressions to approximate embedded conditional expectations and conditional shortfalls, and it includes an a posteriori twin Monte Carlo validation to control regression error. The authors demonstrate scalability in a high-dimensional (36+ dimensional) XVA benchmark and show that the explicit scheme outperforms the traditional implicit/Picard approach in both accuracy and speed. This provides a practical, verification-enabled framework for computing CVA, FVA, KVA, and EC in complex portfolios with multiple risk factors and default events. The work contributes to the numerical analysis of ABSDEs, offers tools for high-dimensional risk measurement, and points to future extensions in sensitivity analysis and time-consistent representations.

Abstract

Motivated by the equations of cross valuation adjustments (XVAs) in the realistic case where capital is deemed fungible as a source of funding for variation margin, we introduce a simulation/regression scheme for a class of anticipated BSDEs, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessing the regression error of the scheme at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in a high-dimensional and hybrid market/default risks XVA use-case.

An Explicit Scheme for Pathwise XVA Computations

TL;DR

The paper develops an explicit-time, regression-based scheme for solving Markovian anticipated BSDEs arising in cross-valuation adjustments (XVAs), enabling pathwise XVA computations without Picard iterations. It combines neural network least-squares and quantile regressions to approximate embedded conditional expectations and conditional shortfalls, and it includes an a posteriori twin Monte Carlo validation to control regression error. The authors demonstrate scalability in a high-dimensional (36+ dimensional) XVA benchmark and show that the explicit scheme outperforms the traditional implicit/Picard approach in both accuracy and speed. This provides a practical, verification-enabled framework for computing CVA, FVA, KVA, and EC in complex portfolios with multiple risk factors and default events. The work contributes to the numerical analysis of ABSDEs, offers tools for high-dimensional risk measurement, and points to future extensions in sensitivity analysis and time-consistent representations.

Abstract

Motivated by the equations of cross valuation adjustments (XVAs) in the realistic case where capital is deemed fungible as a source of funding for variation margin, we introduce a simulation/regression scheme for a class of anticipated BSDEs, where the coefficient entails a conditional expected shortfall of the martingale part of the solution. The scheme is explicit in time and uses neural network least-squares and quantile regressions for the embedded conditional expectations and expected shortfall computations. An a posteriori Monte Carlo validation procedure allows assessing the regression error of the scheme at each time step. The superiority of this scheme with respect to Picard iterations is illustrated in a high-dimensional and hybrid market/default risks XVA use-case.
Paper Structure (23 sections, 5 theorems, 70 equations, 2 algorithms)

This paper contains 23 sections, 5 theorems, 70 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Standing Notation
  3. Limiting Equations
  4. Spaces and Martingale Representation
  5. The Markovian Anticipated BSDE
  6. The Explicit Simulation/Regression Scheme
  7. Time Discretization
  8. Fully Discrete Algorithm
  9. A Posteriori Analysis of the Regression Error
  10. A Posteriori Validation of the Local Regression Error
  11. Accumulation in Time of the Regression Errors
  12. XVA Framework
  13. Continuous-Time Setup
  14. Explicit Simulation/Regression XVA Scheme
  15. A Posteriori Twin Monte Carlo Validation Procedure
  16. ...and 8 more sections

Key Result

Lemma 2.1

There exists a positive constant $\Lambda_\rho$ such that holds for any $M, M'\in {\cal S}_2^l$, where $( Z,U)$ and $( Z',U' )$ in ${\cal H}_2^l \times \widetilde{{\cal H}}_2^l$ are the integrands in the martingale representations e:mrp of $M-M_0$ and $M'-M'_0$.

Theorems & Definitions (8)

  • Remark 2.1
  • Lemma 2.1
  • Proposition 2.1
  • Lemma 4.1
  • Theorem 4.1
  • Remark 5.1
  • Remark 5.2
  • Lemma A.1