An Explicit Scheme for Pathwise XVA Computations

ArXiv ID: 2401.13314 “View on arXiv”

Authors: Unknown

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.

Keywords: XVAs (Cross Valuation Adjustments), Backward Stochastic Differential Equations (BSDEs), neural network least-squares regression, conditional expected shortfall, Monte Carlo simulation, Derivatives / Counterparty Risk

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.0/10
• Quadrant: Holy Grail
• Why: The paper introduces an explicit scheme for complex anticipated BSDEs with heavy mathematical derivations and advanced concepts, demonstrating high mathematical complexity. It provides a simulation/regression scheme with a posteriori Monte Carlo validation and an empirical benchmark in a 36-dimensional XVA use-case, indicating substantial empirical rigor.

  flowchart TD
    Goal["Research Goal<br>Pathwise XVA via Fungible Capital"]
    Input["Data Inputs<br>Market Models & Counterparty Data"]
    
    Method["Methodology<br>Explicit Scheme via Neural Networks"]
    Process["Computational Process<br>Monte Carlo + Regression Steps"]
    
    Residual["A Posteriori Validation<br>Regression Error Assessment"]
    Outcome["Findings<br>Superiority over Picard Iterations"]
    
    Goal --> Input
    Input --> Method
    Method --> Process
    Process --> Residual
    Process --> Outcome