XVA perturbative expansion

The XVA perturbative expansion of BUETGOLFOUSE2026 recovers the classical XVA stack as the leading-order term in an asymptotic expansion of the nonlinear HJB equation. Setting $\gamma =\epsilon \rho$ (small risk-aversion) and ${\phi }_{\bullet }^{\ast }=\epsilon g_{\bullet }+o(\epsilon )$ (small frictions), the XVA adjustment expands as $u^{\epsilon }=U^{(0)}+\epsilon U^{(1)}+o(\epsilon )$.

Leading order: $O(1)$ — Classical XVA stack under P

As $\epsilon \to 0$: the diffusive quadratic, $w_0$-gradient cross-terms, and friction terms all vanish. The exponential default term linearises: ${\lambda }^P[e^{\gamma {\zeta }_C{E}^{+}}-1]/\gamma \to {\lambda }^P{\zeta }_C{E}^{+}$, contributing ${\lambda }^P{\zeta }_{C}u$ to the discount (combined rate $r_{\text{tot}}=f_t+{\lambda }_t^P{\zeta }_C$, eq. 11).

The $O(1)$ BSDE (eq. 12) is linear, and Feynman-Kac with discount factor $D^P(t,s):=\exp (-{\int }_t^s{r}_{\text{tot}}(u)du)$ gives (eq. 13):

$U_t^{(0)}=-\underset{{\text{CVA}}^P}{\underbrace{{\mathbb{E}}_t^P{\int }_t^T{D}^P(t,s){\lambda }_s^P{\zeta }_C(w_0-C_s-h_C{q}_s{)}^{+}ds}}-\underset{{\text{FVA}}^P}{\underbrace{{\mathbb{E}}_t^P{\int }_t^T{D}^P(t,s)(f_s-r_0)(w_0-C_s)ds}}-\underset{{\text{ColVA}}^P}{\underbrace{{\mathbb{E}}_t^P{\int }_t^T{D}^P(t,s)(c_s-r_0)C_{s}ds}}$

plus a carry term ${\mathbb{E}}_t^P{\int }_t^T{D}^P(t,s)[{\beta }_s{\partial }_S{w}_0+({\delta }_s-r_s^{\text{repo}})S_s{\theta }_s+{\kappa }_s{q}_s]ds$ from the real-world drift.

First order: $O(\epsilon )$ — HVA and variance penalties

The first-order source term $S_t^{(1)}$ (eq. 14) contains:

• Diffusion variance: $\frac{\rho }{2}\Vert Z_t^{(0)}{\Vert }^2$ — the continuous-time mean-variance deep-hedging penalty
• Default convexity: $\frac{\rho }{2}{\lambda }_t^P{\zeta }_C^2(w_0-C_t-h_C{q}_t{)}^{+2}$
• HVA: $g_{\text{mkt}}({\partial }_{\theta }{U}_t^{(0)})+g_{cr}({\partial }_q{U}_t^{(0)})+g_C({\partial }_C{U}_t^{(0)}+1)$ — the friction cost of hedging XVA

Key Details

• The three P-world XVA components use ${\lambda }^P$ (not ${\lambda }^Q$), discount at $f+{\lambda }^P{\zeta }_C$ (not $r_0+{\lambda }^Q{\zeta }_C$), and take ${\mathbb{E}}^P$ expectations
• For ${\lambda }^Q/{\lambda }^P\approx 2$-$3$: CVA and FVA are materially smaller under P — the Q-world over-provisions by the credit-risk premium factor
• The carry term ${\beta }_s{\partial }_S{w}_0$ has no Q-world counterpart; it represents incremental alpha from the real-world drift

Critical Notes

Convergence not established

The paper does not discuss whether the perturbative series converges or is merely asymptotic. For BSDE-based expansions with nonlinear drivers (especially the exponential default term), convergence is non-trivial and should not be assumed. The result should be understood as a formal asymptotic expansion.

HVA ordering is scaling-dependent

The claim “HVA is $O(\epsilon )$, not $O(1)$” is a consequence of the specific scaling $\gamma =\epsilon \rho$, ${\phi }^{\ast }=\epsilon g+o(\epsilon )$. Under alternative parametrisations where friction costs are held fixed while risk-aversion vanishes (or vice versa), HVA could enter at leading order. The paper’s statement (Conclusion: “including HVA alongside CVA and FVA conflates orders”) should be read as “conflates orders under our scaling,” not as a universal statement.

Linearisation of the default term

The passage ${\lambda }^P[e^{\gamma {\zeta }_C{E}^{+}}-1]/\gamma \to {\lambda }^P{\zeta }_C{E}^{+}$ as $\epsilon \to 0$ is the key step that produces the combined discount $f+{\lambda }^P{\zeta }_C$. This linearisation loses the exponential tail behaviour that makes the exact BSDE driver superlinear. For portfolios with large positive exposures, the leading-order approximation may significantly underestimate the default contribution.

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