multi-asset HVA trace formulation

The multi-asset HVA trace formulation extends the single-asset Hedging Valuation Adjustment to portfolios with $N$ hedgeable risk factors. Derived in Burnett (2021), the $N$-asset HVA is:

$U(t,T)=-{\int }_t^T{\mathbb{E}}_t\left[{\mathcal{D}}_r(t,u)\mathrm{Tr}\left(\mathbf{F}\mathbf{\Gamma }\mathbf{\Sigma }\mathbf{\Gamma }\right)\right]\mathrm{d}u,$

where $\mathbf{F}=\mathrm{diag}({\psi }_1/D_1^2,\dots ,{\psi }_N/D_N^2)$ is a “relative cost” matrix, $\mathbf{\Gamma }$ is the $N\times N$ cross-gamma matrix of the book, and $\mathbf{\Sigma }$ is the $N\times N$ covariance matrix of the risk factors.

The trace formula reveals how the covariance structure “mixes” risks: even if asset $i$ has no direct gamma, it can contribute to HVA through cross-gamma terms ${\Gamma }_{ij}$ with correlated asset $j$. The individual asset’s threshold $D_i$ captures both asset-specific hedging costs and the projection of portfolio risk onto that asset.

The “assets” need not be traditional financial instruments — they can include rates, volatilities, or any hedgeable quantity. The thresholds $D_i$ are determined by the hedging strategy via $D_i^2={\sum }_{j,k}{\Gamma }_{ij}{\Gamma }_{ik}{\rho }_{jk}{\sigma }_j{\sigma }_k\mathbb{E}[\delta t_i]$.

Key Details

• The formula collapses to the single-asset case when $N=1$: $U=-\int \mathbb{E}[{\mathcal{D}}_r\cdot (\psi /D^2){\sigma }^2{\Gamma }^2]\mathrm{d}u$
• The $\mathbf{F}\mathbf{\Gamma }\mathbf{\Sigma }\mathbf{\Gamma }$ structure shows that HVA is a quadratic form in the cross-gamma matrix, weighted by relative costs and correlations
• In the XVA-consistent extension (The Cost of Hedging XVA), this generalises to include counterparty hazard rates as additional “assets”

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