The close-out convention and collateral rehypothecation rules determine the terminal/boundary conditions of the XVA BSDE at default events. They are critical ingredients in the framework of Bichuch et al. (2015a) and Part II.
Close-out Convention
The close-out value theta_tau(V_hat) is the amount the hedger’s portfolio must replicate at the first default time tau = tau_I ^ tau_C ^ T. Following ISDA market practice (risk-free closeout convention), the surviving party liquidates the position at the mark-to-market value, after netting with available collateral:
theta_tau(V_hat) = V_hat(tau, S_tau) + 1_{tau_C < tau_I} L_C (V_hat - C_tau-)^- - 1_{tau_I < tau_C} L_I (V_hat - C_tau-)^+
where L_I, L_C in [0,1] are loss-given-default rates and C_tau- is the collateral posted just before default. The two terms correspond to:
- CVA-type term: L_C (V_hat - C_tau-)^- — loss to the hedger when the counterparty defaults and V_hat - C < 0 (hedger has positive exposure)
- DVA-type term: L_I (V_hat - C_tau-)^+ — benefit to the hedger when he defaults and V_hat - C > 0 (counterparty has positive exposure)
Under the collateral specification C_t = alpha V_hat(t, S_t) with 0 ⇐ alpha ⇐ 1, the close-out simplifies to theta_I(v_hat) = v_hat - L_I((1-alpha)v_hat)^+ and theta_C(v_hat) = v_hat + L_C((1-alpha)v_hat)^-.
Collateral Specification
Cash collateral is posted as a fraction alpha of the mark-to-market value:
- alpha = 0: zero collateralization
- alpha = 1: full collateralization
- The hedger is the collateral provider when he sold the claim (C_t > 0) and the collateral taker when he purchased it (C_t < 0)
- Collateral rates rc+ (rate received by provider) and rc- (rate paid by taker) are typically Fed Funds or EONIA rates
Rehypothecation
Full rehypothecation is assumed: the collateral taker may freely use the received cash to purchase investment securities. This is consistent with standard ISDA annexes and supported by ISDA (2014) data showing approximately 90% rehypothecation of cash collateral. The rehypothecation assumption means the collateral enters the self-financing condition and affects the funding position:
psi_t^c B_t^{rc} = -C_t and xi_t^f B_t^{rf} = V_t - xi_t^I P_t^I - xi_t^C P_t^C - C_t
Impact on XVA
Higher collateralization alpha:
- Increases the close-out value that must be replicated (more risk for the hedger)
- Increases both buyer’s and seller’s XVA
- Introduces collateral funding costs proportional to alpha(rf - rc), which become the dominant XVA component at high alpha
- Reduces the DVA benefit (the uncollateralised exposure (1-alpha)V_hat shrinks)
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 5.3.6 (p. 100): In the event of counterparty default, the value of a portfolio defines the surviving party’s claim (or liability). Contractual terms (collateral, resets, termination, close-out) generally reference the base value (without xVA) rather than the actual value (with xVA), partly for simplicity and partly for historical reasons. The possible exception is close-out itself, where replacement costs — which may include xVA terms like CVA — create a recursive problem.
- Section 3.3.3 (pp. 58—59): Replacement cost defines the entry point into an equivalent transaction with another counterparty. Real costs of replacing or rehedging (e.g. bid-offer spreads for illiquid securities) can be included in the determination of the close-out amount. Documentation has generally aimed to reference replacement costs defined as objectively as possible. However, replacement costs may themselves include valuation adjustments such as CVA, creating a recursive valuation problem.
- Section 3.3.3 (p. 59): Credit exposure = max(portfolio value, 0). A positive value corresponds to a claim on the defaulted counterparty; a negative value means the surviving party is still obliged to honour its payments. The asymmetry means one cannot gain from a counterparty’s default by being released from a liability.
- Section 17.3.4 (pp. 502—503): Close-out and default correlation in bilateral CVA. Three interconnected subtleties: (i) survival adjustment (contingent vs non-contingent CVA/DVA), (ii) default dependency between party and counterparty, (iii) close-out assumptions. The 2002 ISDA documentation specifies the claim “may take into account the creditworthiness of the Determining Party,” suggesting CVA/DVA may be part of the close-out value. Risky close-out creates a recursive problem. Brigo and Morini (2010) show cancellation of the survival probability in the unilateral case with risky close-out. Gregory and German (2013) find that non-contingent bilateral formulas are the best approximation.
- Section 18.2.2 (pp. 531—534): Funding defined as “total value minus total reusable margin” (Eq. 18.1). Rehypothecable margin offsets funding; non-reusable (segregated) margin mitigates CVA but not FVA.
- Section 18.2.5 (pp. 542—544): Impact of collateralisation on FVA. Table 18.3: when margin cannot be rehypothecated, CVA is reduced but FCA persists, and total cost can exceed the uncollateralised case. One-way margin agreements remove FBA/DVA for the non-posting party. Partial collateralisation (thresholds) requires EPE/ENE modelling at the margin-set level.