The Cost of Hedging XVA

Abstract

We generalise the results of Burnett (2021) to obtain a formula for the Hedging Valuation Adjustment (HVA) that is consistent with other XVAs. This HVA incorporates friction effects due to hedging both the underlying asset and counterparty credit, along with a contribution from hedge unwinds upon counterparty default. We show numerical results for a simple case, which demonstrate that in a realistic situation the HVA can be on the same scale as more classical XVAs.

Summary

This paper extends the foundational HVA framework of Burnett (2021) to make it fully consistent with the standard XVA stack (CVA, FVA). Working within the Burgard-Kjaer (2013) framework with counterparty default risk and funding costs, the authors derive HVA formulae that account for three distinct sources of friction: hedging the underlying asset, hedging counterparty credit, and unwinding hedges upon counterparty default.

The total portfolio value decomposes as $\hat{V}={\sum }_c(V_c+U_c)+{\sum }_j{H}_j$, where $V_c$ is the risk-free value, $U_c$ the standard XVAs (CVA + FVA per Burgard-Kjaer), and $H_j$ the HVA on each book $j$. A critical insight is that the HVA is not an independent sum over counterparties — instead, each counterparty’s contribution $H_j^c$ depends on the book-level gamma ${\Gamma }_j$ and a counterparty-specific gamma share ${\gamma }_j^c$, with a “super-contingency” multiplier $m_j^c\approx 2$ reflecting the quadratic dependence on gamma.

The XVA desk’s HVA separates into two components: a drag HVA covering day-to-day friction costs from hedging asset (${\Gamma }_{ss}$) and credit cross-gamma (${\Gamma }_{sc}$) risks, and a closeout HVA capturing the cost of unwinding hedges upon counterparty default. The combined XVA desk HVA is:

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

Numerical results on a 5-year FX forward with a single counterparty show the total HVA is approximately 30% of CVA (approximately $-451,000$ vs $-1,578,792$ in domestic currency). The credit HVA dominates overwhelmingly over the asset HVA, due to the illiquidity of credit instruments relative to FX. For out-of-the-money CVA, the HVA can actually exceed the CVA itself, raising important questions about optimal hedging strategies.

Key Contributions

• Extension of HVA to be fully consistent with the Burgard-Kjaer XVA framework (CVA + FVA)
• Decomposition of XVA desk HVA into drag (day-to-day friction) and closeout (default-triggered unwinding) components
• Identification that credit cross-gamma ${\Gamma }_{sc}$ dominates the XVA desk’s HVA due to credit market illiquidity
• Separation of originating desk HVA (driven by book gamma ${\Gamma }_j$ and asset friction ${\psi }_s$) from XVA desk HVA (driven by cross-gammas and credit friction ${\psi }_c$)
• Super-contingency multiplier $m_j^c\approx 2$ arising from the quadratic gamma dependence and default-induced HVA knock-on effects
• Demonstration that HVA can be comparable to CVA (~30%) and can dominate CVA for out-of-the-money portfolios

Methodology

The derivation uses the infinitesimal formulation from Burnett (2021), smoothing discrete rehedging into a continuous friction bleed $\mathrm{d}\varphi$. The hedge portfolio includes positions in the underlying asset $S$, zero-recovery counterparty bonds $P_c$, and various funding accounts. Setting the expected P&L to zero (HVA definition) and using the Burgard-Kjaer PDE for the originating desk values yields a system of coupled PDEs for the HVAs. The friction bleed decomposes as $\mathrm{d}\varphi =\mathrm{d}{\varphi }_{\mathrm{drag}}+{\sum }_c{\Psi }_c\mathrm{d}{J}_c$, separating day-to-day friction from default-triggered closeout costs. The closeout value $\Delta {\hat{V}}_c$ upon default of counterparty $c$ includes the marginal impact ${\sum }_j{m}_j^c{H}_j^c$ on all book HVAs, not just the removal of $c$‘s own XVAs.

Key Findings

• For a 5Y FX forward: CVA = $-1,578,792$; Total HVA = $-451,248$ (28.6% of CVA); Credit HVA = $-442,447$; Asset HVA = $-3,018$; Closeout HVA = $-5,783$
• Credit HVA dominates because credit instruments have much wider bid-ask spreads than FX
• The HVA delta is approximately $2\times$ the ratio of CVA gamma to CVA delta, indicating HVA sensitivities are of “higher order” than CVA sensitivities
• For out-of-the-money CVA, the HVA/CVA ratio can exceed 1, meaning hedging costs dominate the adjustment itself
• The HVA discounting uses $r_F+m_X^c{\lambda }_c$ rather than just $r_F+{\lambda }_c$, reflecting the super-contingency of the adjustment
• Cross-gammas (asset vs credit) drive the bulk of real-world HVA, and these cannot be hedged with liquid options

Important References

1. Hedging Valuation Adjustment - Fact and Friction — foundational paper introducing HVA as a friction-driven XVA
2. Burgard and Kjaer 2013 - Funding Strategies Funding Costs — XVA framework for CVA and FVA that this paper generalises to include HVA
3. Burgard and Kjaer 2017 - Derivatives funding netting and accounting — extension of the funding framework to multiple counterparties

Atomic Notes

• Hedging Valuation Adjustment
• drag HVA
• closeout HVA
• credit cross-gamma
• super-contingency multiplier
• originating desk HVA vs XVA desk HVA

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