originating desk HVA vs XVA desk HVA

In the XVA-consistent Hedging Valuation Adjustment framework of Burnett & Williams (2021), the total HVA naturally separates into contributions from originating desks and the XVA desk.

Originating desk HVA ($H_j^c$ for book $j\ne X$): Each originating desk (e.g., IR trading, FX trading) bears the friction cost of hedging its own book’s $S$-dependence. The HVA on book $j$ due to counterparty $c$ satisfies:

${\sum }_c({\partial }_t+{\mathcal{A}}_t-(r_F+m_j^c{\lambda }_c))H_j^c=f_j^S,$

where $f_j^S=(\sigma {\Gamma }_j/D_j{)}^2{\psi }_s(D_j)$ is the asset friction rate. This gives a closed-form HVA proportional to ${\Gamma }_j{\gamma }_j^c$ — the product of book-level and counterparty-level gammas.

XVA desk HVA ($H_X^c$): The XVA desk bears: (1) the friction cost of hedging the XVA book’s own asset gamma (drag HVA, asset component); (2) the friction cost of hedging counterparty credit (drag HVA, credit component via credit cross-gamma); and (3) the cost of unwinding hedges at default (closeout HVA).

The key practical insight is that originating desk HVA is typically small (the linearity of many instruments, e.g. forwards, implies small originating gamma), while the XVA desk HVA is large and dominated by credit cross-gamma friction. This means the XVA desk bears the bulk of hedging costs — consistent with the practitioner observation that XVA desks face disproportionate friction relative to their P&L.

Key Details

• Originating desk HVA costs ($f_j^S$, ${\Psi }_c$) are paid by the originating desk
• Credit friction ($f_c^{\lambda }$) and closeout costs (${\Psi }_c$) are paid by the XVA desk
• For a 5Y FX forward: originating desk HVA $\approx 0$ (linear instrument), XVA desk HVA = $-451,248$
• The separation is important for transfer pricing: new trades should be charged for their marginal impact on both originating and XVA desk HVAs

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