book-funding cost in XVA

The book-funding cost is the cash-flow term $-f_t{v}_{t}dt$ in the desk wealth SDE, representing the treasury charge for funding the book. A positive mark-to-market $v_t>0$ requires borrowing at the book-funding rate $f_t$, generating a running cost. When the book is fully collateralised ($v_t=C_t$), the book-funding cost cancels exactly with the collateral benefit $(f_t-c_t)C_{t}dt$, and FVA vanishes.

BUETGOLFOUSE2026 identifies this term as the mechanical source of FVA in the desk wealth dynamics (eq. 3). In the HJB equation for the XVA adjustment $u$ (eq. 8), the book-funding cost contributes $-f_{t}u$ to the discount, which combines with the linearised default term ${\lambda }^P{\zeta }_{C}u$ to give the total discount rate $r_{\text{tot}}=f_t+{\lambda }_t^P{\zeta }_C$ (eq. 11). The FVA component in the perturbative expansion (eq. 13) arises specifically from the $-(f_t-r_0)(w_0-C_t)$ term, which traces back to $-f_t{v}_t$.

Key Details

• The term $-f_t{v}_{t}dt$ is the treasury charge: the desk borrows from the treasury to fund a positive MTM position
• Collateral reduces funding cost: $(f_t-c_t)C_t$ is the net carry from holding received margin
• When $v=C$ (fully collateralised): book-funding cost and collateral benefit cancel, FVA $=0$
• The combined discount rate $f+{\lambda }^P{\zeta }_C$ emerges only after linearising the exponential default term in the perturbative expansion

Critical Notes

Novelty claim requires qualification

The paper claims this term is “absent from pre-? XVA formulations.” However, Burgard & Kjaer (2011) and Piterbarg (2010) already derive FVA from replication arguments that implicitly include funding costs in the PDE. The distinction may be pedagogical (explicit cash-flow vs. implicit PDE coefficient) rather than substantive. The claim is also unverifiable due to the unresolved ”?” reference.

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