super-contingency multiplier

The super-contingency multiplier $m_j^c$ is a factor (approximately equal to 2) that appears in the Hedging Valuation Adjustment when it is made consistent with the XVA framework. It arises from the quadratic dependence of HVA on portfolio gamma.

In Burnett & Williams (2021), when counterparty $c$ defaults, the change in total portfolio value $\Delta {\hat{V}}_c$ must account not only for the standard closeout (removal of $V_c$, $U_c$), but also for the marginal impact on all book HVAs:

$\Delta {\hat{V}}_c=-(1-R_c)V_c^{+}-U_c-{\sum }_j{m}_j^c{H}_j^c.$

The multiplier $m_j^c$ captures the fact that removing counterparty $c$‘s trades affects the book gamma ${\Gamma }_j$, which in turn affects the HVA of all other counterparties through the quadratic form. The HVA on book $j$ allocated to counterparty $c$ is $H_j^c$, but the total impact of $c$‘s default on the book HVA is $m_j^c{H}_j^c$ because the gamma contribution is not simply additive.

For the vanilla case studied in Burnett (2021), $m=2$ exactly, corresponding to the “new trade HVA” result where $(\Gamma +{\gamma }_i{)}^2\approx {\Gamma }^2+2\Gamma {\gamma }_i$. In the XVA-consistent setting, $m_j^c$ can differ from 2 due to the relative size of drag vs closeout components, but generally sits between 1 and 2.

Key Details

• $m_j^c\approx 2$ when the drag effect dominates (the usual case)
• The multiplier enters the HVA discounting: ${\mathcal{D}}_{r_F+m_X^c{\lambda }_c}$ rather than ${\mathcal{D}}_{r_F+{\lambda }_c}$
• This is conceptually similar to how CVA uses $(1-R_c)V_c^{+}$ rather than $V_c$ — a “contingent” adjustment that itself depends on default
• The terminology “super-contingency” reflects that the HVA is contingent on default through two channels: the direct removal of $c$‘s HVA, and the indirect impact on all other HVAs via gamma

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concept