Hedging Valuation Adjustment

The Hedging Valuation Adjustment (HVA) is a valuation adjustment that quantifies the expected cost of transaction friction (bid-ask spreads, commissions) incurred when delta-hedging a derivative portfolio. Introduced by Benedict Burnett in Burnett (2021), it treats hedging costs as an XVA-type quantity — an adjustment to the risk-free value of a book, in the same spirit as CVA, FVA, and KVA.

The HVA arises because real-world hedging is discrete: a trader rebalances their delta position whenever it breaches a threshold $D$, incurring a friction cost $\psi$ at each rebalancing event. The expected rate of friction bleeding (the “friction rate”) is $f\approx (\sigma \Gamma /D{)}^2\psi$, where $\Gamma$ is the portfolio gamma, $\sigma$ the volatility, and $D$ the delta threshold. The HVA is the discounted integral of this friction rate over the life of the book.

Two variants exist depending on whether the HVA is itself hedged:

• Hedged HVA: The HVA is included in the value being hedged. The expectation is taken under the risk-neutral measure $\mathbb{Q}$. This case gives rise to imaginary volatility when friction is large.
• Unhedged HVA: The HVA is left as a reserve (not hedged). The expectation is under the real-world measure $\mathbb{P}$.

In the multi-counterparty, multi-book setting of Burnett & Williams (2021), the HVA further decomposes into drag HVA (day-to-day friction from asset and credit hedging) and closeout HVA (cost of unwinding hedges upon counterparty default). The XVA desk’s HVA is typically dominated by credit cross-gamma friction due to the illiquidity of single-name credit instruments.

Key Details

• Generic hedged HVA: $U(t,T)=-{\int }_t^T{\mathbb{E}}_t^{\mathbb{Q}}[{\mathcal{D}}_r(t,u)(\sigma \Gamma /D{)}^2\psi (D,X_u)]\mathrm{d}u$
• Generic unhedged HVA: $U(t,T)=-{\int }_t^T{\mathbb{E}}_t^{\mathbb{P}}[{\mathcal{D}}_r(t,u)(\sigma \Gamma /D{)}^2\psi (D,X_u)]\mathrm{d}u$
• The HVA is always negative (a cost), proportional to ${\Gamma }^2$, and quadratic in risk
• New trade marginal HVA $\approx -2\int \mathbb{E}[{\mathcal{D}}_r\cdot (\psi /D^2){\sigma }^2\Gamma {\gamma }_i]\mathrm{d}u$ — the factor of 2 arises from the quadratic form $(\Gamma +{\gamma }_i{)}^2\approx {\Gamma }^2+2\Gamma {\gamma }_i$
• Numerical example: for a 5Y FX forward, HVA $\approx$ 30% of CVA

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