X-valuation adjustment (XVA) is the collective term for post-crisis corrections applied to the theoretical fair price of OTC derivatives, reflecting counterparty credit risk, funding costs, and collateral terms. The post-crisis pricing formula is:
price = p_bar^{RN} + sum_x xVA
where p_bar^{RN} = E[DF_{r_bar}(T) xi_T] is the base risk-neutral price and the valuation adjustments include:
- CVA (Credit Valuation Adjustment): compensates for the risk that the counterparty defaults. Computed as the expected discounted loss weighted by the counterparty hazard rate h_2 and the closeout shortfall phi_hat_2.
- DVA (Debt Valuation Adjustment): the symmetric benefit from the possibility of one’s own default; enters with a negative sign. Computed via the investor hazard rate h_1 and the closeout shortfall phi_hat_1.
- FVA (Funding Valuation Adjustment): reflects the cost of funding uncollateralised derivative positions above the risk-free rate; driven by the spread (rf^0 - rD).
- ColVA (Collateral Valuation Adjustment): accounts for the difference between the collateral rate and the funding rate, applied to the posted/received collateral alpha * V_hat.
In the BSDE framework of Sekine and Tanaka (2020), the zeroth-order approximated arbitrage-free price decomposes as:
Y_bar^{0,pm}(t) = V(t) + VA_1(t) + VA_2(t) + VA_3(t) + VA_4(t) + VA_5^{pm}(t)
where V is the base price (discounted at the funding mid-rate), VA_1 ~ -DVA, VA_2 ~ CVA, VA_3 ~ FVA, VA_4 captures repo-risk-free rate mismatch, and VA_5^{pm} ~ ColVA. This decomposition is exact when rr^0 = rf^0 = rD, reducing to the standard practitioner formula. The full nonlinear BSDE price Y^{pm} includes additional terms from funding/repo spreads (epsilon_f, epsilon_r) that make the driver nonlinear; the XVA decomposition is the linear (zeroth-order) approximation with O(epsilon) error.