drag HVA

The drag HVA ($H_{\text{Drag}}^c$) is the component of the XVA desk’s Hedging Valuation Adjustment that captures day-to-day friction costs from continuously rehedging asset and credit exposures. It is distinguished from the closeout HVA, which covers costs triggered by counterparty default events.

In the framework of Burnett & Williams (2021), the drag HVA for counterparty $c$ satisfies:

${\sum }_c({\partial }_t+{\mathcal{A}}_t-(r_F+m_X^c{\lambda }_c))H_{\text{Drag}}^c=f_X^S+{\sum }_c{f}_c^{\lambda },$

where $f_X^S$ is the asset friction (from hedging $S$-dependence of the XVA book) and $f_c^{\lambda }$ is the credit friction (from hedging counterparty $c$‘s credit risk). The closed-form Feynman-Kac solution is:

$H_{\text{Drag}}^c=-{\int }_t^T{\mathbb{E}}_t^{\mathbb{Q}}\left[{\mathcal{D}}_{r_F+m_X^c{\lambda }_c}(t,u)\left(\frac{{\psi }_s}{D_s^2}{\sigma }^2{\Gamma }_{ss}{\gamma }_{ss}^c+\frac{{\psi }_c}{D_c^2}{\sigma }^2{\Gamma }_{sc}^2\right)\right]\mathrm{d}u.$

The two terms correspond to:

1. Asset drag: proportional to ${\Gamma }_{ss}{\gamma }_{ss}^c$ — the XVA book-level asset gamma times the counterparty $c$ share. This is typically small since the XVA book is hedged with liquid instruments.
2. Credit drag: proportional to ${\Gamma }_{sc}^2$ — the asset-vs-credit cross-gamma. This dominates in practice because single-name credit is illiquid (${\psi }_c/D_c^2\gg {\psi }_s/D_s^2$).

Key Details

• Discounting uses $r_F+m_X^c{\lambda }_c$ rather than the standard $r_F+{\lambda }_c$, reflecting the super-contingency multiplier
• The multi-asset generalisation uses the trace form: $H_{\text{Drag}}^c=-\int \mathbb{E}[{\mathcal{D}}_{r_F+m_X^c{\lambda }_c}\mathrm{Tr}(\mathbf{F}\mathbf{\Gamma }\mathbf{\Sigma }{\mathbf{\gamma }}^c)]\mathrm{d}u$
• For the 5Y FX forward example: credit HVA = $-442,447$ vs asset HVA = $-3,018$ — a factor of $\sim$150

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