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Deep xVA Solver - A Neural Network Based Counterparty Credit Risk Management Framework

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Summary

This paper formulates the portfolio-wide computation of valuation adjustments (xVA) — including CVA, DVA, FVA, and ColVA — as a backward stochastic differential equation (BSDE) problem under a rigorous financial framework following Biagini et al. (2021). The key insight is that, in the Markovian setting, solving the xVA BSDE is equivalent to identifying an optimal hedging (control) strategy, where the terminal condition of the BSDE encodes the target payoff and the control process Z represents the hedge ratios. The full portfolio value V decomposes into clean values minus the XVA process, and the XVA satisfies an F-BSDE whose driver encodes default intensities, recovery rates, funding spreads, and collateral terms.

The authors propose a two-step “Deep xVA Solver” that first applies the Deep BSDE Solver of Han et al. (2018) to approximate the clean values of all M portfolio constituents, and then — for recursive adjustments like FVA where the adjustment itself appears in the driver — applies the solver a second time to the xVA BSDE. At each time step, the control process is parametrised by a distinct feedforward neural network, and all parameters are jointly optimised via stochastic gradient descent to minimise a terminal loss. This transforms the BSDE into a model-based reinforcement learning problem over the parametrised Markovian control.

The method is validated on forward contracts, European call options, and a 100-dimensional basket option. Non-recursive adjustments (CVA, DVA) are computed via an outer Monte Carlo over the neural-network-simulated exposure paths, while recursive FVA is handled by the full two-step procedure. The authors also derive a posteriori error bounds linking the xVA approximation error to the terminal loss of the neural network solver, and demonstrate computation of VaR and Expected Shortfall on xVA without nested simulation.

Key Contributions

  • A rigorous BSDE-based computational framework for portfolio-wide xVA (CVA, DVA, FVA, ColVA) solved via recursive application of the Deep BSDE Solver
  • Distinction between non-recursive adjustments (CVA, DVA) computable by outer Monte Carlo and recursive adjustments (FVA) requiring a second BSDE solve
  • A posteriori error bounds (Eq. 3.10, 3.15) relating xVA approximation error to the neural network terminal loss and time discretisation
  • Hedge ratios and sensitivities for xVA obtained directly from the neural network control without additional differentiation
  • Demonstration on a 100-dimensional basket option with basis-point accuracy and computation of dynamic risk measures (VaR, ES) on xVA paths
  • Realistic simulation of the collateral account and its impact on FVA

Methodology

The financial framework follows Biagini et al. (2021), modelling d risky assets as Ito diffusions under the risk-neutral measure Q, with bank and counterparty defaults as exponentially distributed stopping times satisfying the immersion hypothesis. Clean values satisfy decoupled FBSDEs (Eq. 2.6), while the full portfolio value satisfies a G-BSDE (Eq. 2.8) that reduces under the reduced filtration to the F-BSDE for XVA (Eq. 2.10).

The Deep BSDE Solver discretises the BSDE on a uniform time grid and parametrises the control Z at each step by a feedforward neural network with ReLU activations and batch normalisation. The optimisation problem (Eq. 3.8) minimises the mean squared terminal error over the initial value and all network parameters via SGD. For non-recursive xVAs, Algorithm 2 - Deep xVA for non-recursive adjustments applies outer Monte Carlo to the learned clean value paths. For recursive xVAs, Algorithm 3 - Deep xVA Solver applies the Deep BSDE Solver a second time to the xVA BSDE using the learned clean values as input. Pathwise sensitivity computation via neural network differentiation yields hedge ratios and second-order Greeks.

Key Findings

  • Time-zero option prices are recovered with sub-basis-point accuracy; exposure profiles degrade with time due to accumulated hedging error in Z
  • For the 100-dimensional basket call, CVA/DVA computed by Algorithm 2 (0.8947) and Algorithm 3 (0.8952) agree to within the Monte Carlo confidence interval
  • FVA computation scales to 200 dimensions with error below 1%, and CPU time grows roughly linearly with dimension for fixed network architecture
  • Perfect collateralisation drives FVA to numerically zero, confirming theoretical expectations
  • VaR on xVA converges smoothly and matches terminal analytical values
  • Network architecture of 2 hidden layers with d+10 to d+20 nodes and ReLU activation provides consistent accuracy across dimensions

Important References

  • Longstaff, F. A. and Schwartz, E. S. (2001) - “Valuing American Options by Simulation: A Simple Least-Squares Approach” — foundational regression Monte Carlo method that the Deep BSDE approach generalises for exposure simulation (517 influential citations)
  • El Karoui, N., Peng, S. and Quenez, M.-C. (1997) - “Backward Stochastic Differential Equations in Finance” — establishes the BSDE representation of hedging portfolios central to the paper’s formulation (261 influential citations)
  • Cybenko, G. (1989) - “Approximation by superpositions of a sigmoidal function” — universal approximation theorem underpinning the expressiveness of the neural network parametrisation (431 influential citations)

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