closeout HVA

The closeout HVA ($H_{\text{Closeout}}^c$) is the component of the XVA desk’s Hedging Valuation Adjustment that captures the friction cost of unwinding hedges upon counterparty default. When counterparty $c$ defaults, the XVA desk must unwind its asset ($S$) hedge for that counterparty’s contribution, incurring a bid-ask spread cost.

In Burnett & Williams (2021), the closeout HVA satisfies:

$({\partial }_t+{\mathcal{A}}_t-(r_F+m_X^c{\lambda }_c))H_{\text{Closeout}}^c={\lambda }_c{\psi }_s\left(|{\partial }_S{V}_c+{\partial }_S{U}_c+{\sum }_j{m}_j^c{\partial }_S{H}_j^c|\right).$

The quantity inside ${\psi }_s(|\cdot |)$ is the amount of asset hedge that must be unwound — the marginal contribution of counterparty $c$ to the hedge holding. Under the simplifying assumption that the HVA’s own delta contribution is negligible, this reduces to:

$H_{\text{Closeout}}^c\approx -{\int }_t^T{\mathbb{E}}_t^{\mathbb{Q}}\left[{\mathcal{D}}_{r_F+m_X^c{\lambda }_c}(t,u){\lambda }_c{\psi }_s(|{\delta }_c|)\right]\mathrm{d}u,$

where ${\delta }_c\equiv {\partial }_S{V}_c$ is the asset delta on counterparty $c$‘s MtM.

Key Details

• The closeout HVA is qualitatively different from drag HVA: it is triggered by discrete default events ($\mathrm{d}{J}_c$) rather than continuous hedging
• For the 5Y FX forward example: closeout HVA = $-5,783$, much smaller than the credit drag component but non-negligible
• The bond hedge (used for counterparty credit) is not unwound at default — it pays out directly against the credit loss
• The HVA’s “super-contingency” arises partly from the closeout: default of $c$ removes $H_j^c$ but also impacts all other counterparties’ HVAs through the book gamma ${\Gamma }_j$

---

concept