XVA overlay on front-office hedge

The XVA overlay formulation of BUETGOLFOUSE2026 (Section 7) describes how an XVA desk manages its positions as an overlay on top of existing front-office hedges. In practice, the XVA desk inherits front-office positions $({\theta }_t^{\text{FO}},q_t^{\text{FO}})$ and manages an overlay $({\theta }_t^{\text{XVA}},q_t^{\text{XVA}})$, with total positions ${\theta }^{\text{tot}}={\theta }^{\text{FO}}+{\theta }^{\text{XVA}}$ and analogously for $q$.

The Riccati system (eq. 18-20) holds with total positions, and the optimal overlay controls are:

${\nu }_t^{\text{XVA}\ast }=-\frac{M_{\theta \theta }}{{\Lambda }_{\theta }}({\theta }_t^{\text{XVA}}-{\theta }_t^{\text{XVA,tar}}),{\theta }_t^{\text{XVA,tar}}={\theta }^{\text{tar}}(t)-{\theta }_t^{\text{FO}}$

When the front office sets ${\theta }^{\text{FO}}={\Delta }_0(t)$ (the clean delta), the XVA overlay for $\alpha =0$ becomes ${\theta }^{\text{XVA,tar}}=-b_{\theta }/(\gamma {\sigma }_S^2)-{\Delta }_0$, which is nonzero because the Q-world front-office delta misses the real-world carry $\beta$ and the repo/dividend spread.

Key Details

• The XVA desk adjusts for three effects missing from the front-office hedge: (i) real-world drift $\beta$, (ii) XVA Greeks (${\partial }_{\theta }u=M_{\theta \theta }{\theta }^{\text{XVA}}+m_{\theta }$), and (iii) funding-credit interaction
• XVA delta: ${\partial }_{\theta }u=-{\Lambda }_{\theta }{\nu }^{\ast }$, so the XVA delta equals the market-impact-weighted optimal trading rate
• XVA Greeks are linear in inventory: computable in microseconds from the Riccati solution, suitable for real-time risk management
• The XVA carry $k_0(t)$ from (eq. 20) gives the instantaneous rate of change of XVA at fixed positions

Critical Notes

Practical applicability

The overlay formulation assumes the front-office and XVA desk trade the same instruments (underlying, CDS), which is realistic. However, the constant-coefficient, zero-collateral, quadratic-friction setting is far from production reality. The linear-in-inventory Greeks may not extend to the full nonlinear HJB without the reduction hypotheses.

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