credit cross-gamma

The credit cross-gamma ${\Gamma }_{sc}$ is the mixed second-order sensitivity of the XVA book’s value to simultaneous movements in the underlying asset $S$ and counterparty $c$‘s credit (bond price $P_c$). It measures how the asset delta of the XVA changes as credit spreads move, or equivalently, how the credit delta changes as the underlying moves.

In the context of the Hedging Valuation Adjustment, the credit cross-gamma is the dominant driver of the XVA desk’s drag HVA. This is because:

1. Cross-gammas cannot be hedged with options: unlike pure asset gamma ${\Gamma }_{ss}$ which can be offset by buying vanilla options, there are no liquid options on the joint asset-credit exposure. The XVA desk must delta-hedge these risks, incurring quadratic friction costs.

2. Credit instruments are illiquid: the friction cost ${\psi }_c$ of trading single-name credit (CDS or bonds) is typically much larger than the friction cost ${\psi }_s$ of trading the underlying asset (FX, equities, rates). This amplifies the credit cross-gamma’s contribution to HVA.

3. Portfolio-level nature of XVA: the cross-gamma is inherently a per-counterparty quantity (it depends on how that counterparty’s exposure profile correlates with credit spread movements), but it must be hedged at the portfolio level by the XVA desk.

Key Details

• In the Burnett & Williams (2021) notation: the credit drag HVA contribution is $({\psi }_c/D_c^2){\sigma }^2{\Gamma }_{sc}^2$, quadratic in ${\Gamma }_{sc}$
• For the numerical example (5Y FX forward, 100bps spread, 40% recovery): credit HVA = $-442,447$ vs asset HVA = $-3,018$
• The cross-gamma is a key challenge for XVA desks: it represents risks that “simply cannot be hedged with liquid options, because such options do not exist”
• Cross-gammas arise from the cross-asset nature of XVA — the desk simultaneously manages exposure to many asset classes and many counterparties’ credits

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