Credit, funding and XVA risk

Credit and funding adjustments combine future exposure with counterparty, own-credit or funding assumptions. QuantSupport first generates market paths and values the portfolio, then applies a measure-specific aggregator to each path. Netting and variation margin determine the exposure supplied to that aggregator.

The formulas below describe the implemented integration and the inputs held fixed when its sensitivities are calculated.

Let \(U_{p,k}\) be exposure after the netting/collateral rules applied for the adjustment, \(D_k\) the numerical system discount factor, and \(\Delta t_k\) the ACT/365 year fraction between consecutive grid dates. The implemented integration uses right-endpoint exposure.

CVA and DVA

For counterparty survival \(Q_k\) and recovery \(R\),

\[ \mathrm{CVA}=(1-R)\frac1n \sum_p\sum_{k=1}^{m}D_k\max(U_{p,k},0)(Q_{k-1}-Q_k). \]

Flat spread \(s\) is converted to hazard \(\lambda=s/(1-R)\), then \(Q(t)=e^{-\lambda t}\). With a credit curve, survival comes from log-linear survival interpolation and the implemented terminal hazard continuation. A credit spread is not interchangeable with hazard, and recovery can affect both the flat hazard and the loss multiplier.

DVA uses own-credit survival/recovery and \(\max(-U,0)\). It therefore turns a negative exposure into a positive own-default benefit measure. DvaAggregator and its factory are lower-level facilities. XvaEngine::run automatically constructs CVA and FVA, not a fully specified bilateral CVA/DVA model.

The aggregators produce named measures; they do not prescribe a universal accounting sign for adding/subtracting those measures from a clean price. State that convention at the application boundary.

Signed funding integration

The implemented FVA is

\[ \mathrm{FVA}=\frac1n\sum_p\sum_{k=1}^{m} D_k\,U_{p,k}\,s_f(t_k)\,\Delta t_k. \]

This symmetric funding convention uses signed exposure, so positive and negative contributions can cancel. It uses one spread for both directions; separate borrowing and lending spreads are outside this model. As a result, a reduction in EPE alone does not determine the change in FVA.

Funding inputs can be flat, dated explicit spreads, or spreads derived from a configured funding curve relative to the system curve. System discount factors are optional at lower levels; high-level construction supplies them. Without them, aggregation is not automatically present-valued by another hidden discounting stage.

What the returned sensitivities mean

The engine returns one aggregated list of labeled derivatives across the run. Rate-path leaves can retain original curve calibration/IFT dependencies. Fixed model lambda and configured correlations are not automatically independent risk factors. Calibrated Gaussian sigma includes the supplied volatility Jacobian but omits curve/strike-induced recalibration.

The numerical system discount sequence \(D_k\) is held fixed during AD. A quote scenario that recalculates it can therefore disagree with the reported partial derivative for a legitimate reason.

Credit-curve CVA has another important boundary. High-level construction snapshots numerical survival pillars and their labels; CreditCurveCvaFactory creates independent survival leaves. A label such as CVA.<quote-like label> therefore describes a derivative with respect to a survival pillar, not a replayed original CDS-spread calibration derivative. Recovery is separately labeled. Quote-like text does not change a leaf’s numerical units.

Similarly, funding-index construction snapshots derived forward spreads. FundingCurveFvaFactory creates independent spread leaves, generally labeled with FVA. prefixes, rather than differentiating their derivation from all funding/system curve quotes. Explicit funding-spread curves can contribute their own dated zero-rate leaves. Distinguish these factory risks from direct deterministic credit/funding-curve sensitivities.

Value and gradient share simulated paths and sampling noise. Validate selected dependencies with same-seed runs and carefully specified bumps, but compare like with like: a full quote shock reconstructs more than a partial survival/spread/model derivative may include. Reporting the leaf meaning, units and held-fixed assumptions is as important as reporting the number.