Riccati system for XVA hedging

The Riccati system for XVA hedging, derived in BUETGOLFOUSE2026, provides closed-form optimal delta and credit-hedge rates under six reduction hypotheses: pure diffusion (R1), no collateral control (R2), frozen deterministic coefficients (R3), quadratic frictions (R4), affine delta approximation (R5), and quadratic default surrogate (R6).

Under these hypotheses, the XVA adjustment takes the quadratic ansatz $u(t,\theta ,q)=\frac{1}{2}{X}^{\top }M(t)X+m(t{)}^{\top }X+k_0(t)$ with $X=(\theta ,q{)}^{\top }$. The optimal controls are mean-reverting:

${\nu }_t^{\ast }=-\frac{M_{\theta \theta }}{{\Lambda }_{\theta }}({\theta }_t-{\theta }^{\text{tar}}(t)),{\mu }_t^{\ast }=-\frac{M_{qq}}{{\Lambda }_q}(q_t-q^{\text{tar}}(t))$

Matrix Riccati ODE

$\dot{M}(t)=M(t)RM(t)-Q(t)-\alpha (t)M(t),M(T)=0$

with $R=\text{diag}(1/{\Lambda }_{\theta },1/{\Lambda }_q)$, $\alpha (t):=f(t)-{\lambda }^P(t){\zeta }_C$, and

$Q(t)=\left(\begin{array}{cc}\gamma {\sigma }_S^2& -\gamma {\sigma }_S^2{d}_q\\ -\gamma {\sigma }_S^2{d}_q& \gamma {\lambda }^P{\zeta }_C^2{h}^2\end{array}\right)$

The novel feature is the $-\alpha (t)M(t)$ coupling, absent from standard LQR theory. It arises from ${\lambda }^P{\zeta }_{C}u-f_{t}u=-\alpha u$ in the HJB: substituting the quadratic ansatz $u=\frac{1}{2}{X}^{\top }MX+\cdots$ gives $-\alpha \cdot \frac{1}{2}{X}^{\top }MX$, contributing $-\alpha M$ to the Riccati. The scalar $\alpha =f-{\lambda }^P{\zeta }_C$ is the net self-discounting rate of the XVA position.

Closed-form solution (constant coefficients, $d_q=0$)

The channels decouple into scalar Bernoulli equations with tanh solutions:

$M_{\theta \theta }(t)={\kappa }_{\theta }^{\text{eff}}{\Lambda }_{\theta }\tanh ({\kappa }_{\theta }^{\text{eff}}\tau -{\varphi }_{\theta })+\frac{\alpha {\Lambda }_{\theta }}{2},{\kappa }_{\theta }^{\text{eff}}=\sqrt{\frac{\gamma {\sigma }_S^2}{{\Lambda }_{\theta }}+\frac{{\alpha }^2}{4}}$

$M_{qq}(t)={\kappa }_q^{\text{eff}}{\Lambda }_q\tanh ({\kappa }_q^{\text{eff}}\tau -{\varphi }_q)+\frac{\alpha {\Lambda }_q}{2},{\kappa }_q^{\text{eff}}=\sqrt{\frac{\gamma {\lambda }^P{\zeta }_C^2{h}^2}{{\Lambda }_q}+\frac{{\alpha }^2}{4}}$

where $\tau =T-t$ and ${\varphi }_{\bullet }:=\text{arctanh}(\alpha /(2{\kappa }_{\bullet }^{\text{eff}}))$.

The targets decompose as:

${\theta }^{\text{tar}}=-\frac{b_{\theta }}{\gamma {\sigma }_S^2},q^{\text{tar}}=\underset{\text{carry}}{\underbrace{\frac{\kappa }{\gamma {\lambda }^P{\zeta }_C^2{h}^2}}}+\underset{\text{linear CVA}}{\underbrace{\frac{1}{\gamma {\lambda }^P{\zeta }_{C}h}}}+\underset{\text{funded book}}{\underbrace{\frac{w_0}{h}}}$

Key Details

• The gain $M_{\bullet \bullet }/{\Lambda }_{\bullet }$ is zero at maturity ($\tau =0$) and saturates at ${\kappa }_{\bullet }^{\text{eff}}+\alpha /2$ as $\tau \to \infty$
• The effective speed ${\kappa }_{\bullet }^{\text{eff}}$ is the geometric mean of the risk penalty and inverse friction, shifted by the funding-credit spread $\alpha /2$
• When $\alpha >0$ (funding cost dominates): the gain curve shifts upward, faster convergence
• When $\alpha <0$ (high default intensity): credit carry partially compensates funding cost, allowing more inventory at equilibrium
• Sign of $\varphi$: The solution uses $\tanh (\kappa \tau -\varphi )$, not $\tanh (\kappa \tau +\varphi )$. The $-\varphi$ sign ensures $M(T)=0$ and convergence to the stable fixed point $M_{\infty }=\frac{\alpha \Lambda }{2}+\sqrt{\Lambda Q^{\text{eff}}}>0$. The $+\varphi$ alternative satisfies $M(T)=0$ but converges to the unstable (negative) fixed point

Critical Notes

Strong reduction hypotheses

The six assumptions (R1-R6) are individually standard in practitioner XVA but jointly severe. In particular: $C\equiv 0$ (R2) removes the collateral dimension entirely, making the ColVA results vacuous for collateralised portfolios. The frozen-path assumption (R3) removes stochastic volatility, stochastic rates, and stochastic intensity. The affine delta approximation (R5) is ad hoc — its domain of validity is not characterised. The quadratic default surrogate (R6) assumes $|u|\ll w_0$, which fails for deep OTM options or large portfolios.

Relationship to standard LQR

While the $-\alpha M$ term is absent from the textbook LQR Riccati (e.g., Yong & Zhou 1999), discounted or weighted LQR problems commonly produce similar additional terms. The novelty is in the financial interpretation ($\alpha$ = net self-discounting rate of the XVA position), not in the mathematical structure per se.

Textbook References

Stochastic Controls - Hamiltonian Systems and HJB Equations (Yong & Zhou, 1999)

The XVA Riccati ODE $\dot{M}=MRM-Q-\alpha M$ is a special case of the general stochastic Riccati equation (6.6) after the frozen-coefficient reduction (R3). Specifically:

• The standard form is $\dot{P}+PA+A^{T}P+C^{T}PC+Q-(B^{T}P+S+D^{T}PC{)}^T(R+D^{T}PD{)}^{-1}(B^{T}P+S+D^{T}PC)=0$
• Under (R1)—(R3), $C=D=0$ and $A=-\alpha /2\cdot I$, collapsing $C^{T}PC$ and $D^{T}PD$ terms. The $-\alpha M$ coupling arises from the $PA+A^{T}P$ terms with $A=-\alpha /2\cdot I$
• Theorem 6.1 (p. 315): Solvability of the stochastic Riccati equation implies the optimal control has the feedback form $\bar{u}=-\Psi x-\Theta$, which specialises to the mean-reverting XVA hedging rates ${\nu }^{\ast }=-(M_{\theta \theta }/{\Lambda }_{\theta })(\theta -{\theta }^{\text{tar}})$
• Theorem 7.9 (p. 330): The one-dimensional constant-coefficient analysis provides the template for the tanh solutions; the XVA case corresponds to $\gamma <0$ (guaranteed finite $\Theta$) with explicit $\Theta$ formula matching the ${\kappa }^{\text{eff}}\tau -\varphi$ structure
• Theorem 7.7 (p. 328): The condition that $R$ not be “too negative” corresponds to the requirement that transaction cost parameters ${\Lambda }_{\theta },{\Lambda }_q$ be sufficiently large relative to risk penalties for the Riccati equation to have a global solution

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