The buyer’s and seller’s XVA, as defined by Bichuch et al. (2015a), formalise the total valuation adjustment in a no-arbitrage framework with asymmetric rates. The key distinction from practitioner decompositions is that rate asymmetries produce two distinct XVA values rather than a single adjustment.
Definition
Following Bichuch et al. (2015a), the seller’s and buyer’s XVA are defined as:
- XVA_sell_t := V_t^+ - V_hat(t, S_t) (costs when replicating the payoff of a sold claim)
- XVA_buy_t := V_t^- - V_hat(t, S_t) (costs when replicating the payoff of a purchased claim)
where V^+ and V^- solve the nonlinear BSDEs for seller and buyer. The no-arbitrage interval is [V_0^-, V_0^+] and its width is XVA_sell_0 - XVA_buy_0.
Explicit Decomposition Under Symmetric Rates
Under Piterbarg’s setup (rf+ = rf- = rf, rr+ = rr- = rD, rc+ = rc- = rc), buyer’s and seller’s XVA coincide and admit the four-term decomposition (Proposition 5.1 and Eq. (33) of Part I):
- Funding-adjusted risk-free price: the claim price discounted at the funding rate rf instead of rD
- DVA-type term: funding-adjusted payout at trader default, scaled by (1 - (1-alpha)L_I) on positive exposure and by 1 on negative exposure
- CVA-type term: funding-adjusted payout at counterparty default, scaled by (1 - (1-alpha)L_C) on negative exposure and by 1 on positive exposure
- Collateral funding cost: integral of alpha(rf - rc) V_hat(s, S_s) Gamma_t^s ds, capturing the cost of financing the collateral at funding rate rf while only earning rc
The XVA can also be expressed as a single multiplicative factor beta_t times V_hat(t, S_t), so the adjustment is a deterministic percentage of the publicly available price.
Asymmetric Rate Case
When rates are asymmetric, buyer’s and seller’s XVA differ. The no-arbitrage band width increases with the funding spread (rf- - rf+). The numerical study in Part II reveals:
- At low collateralization, the band widens mainly via decreasing buyer’s XVA
- At high collateralization, the band widens via increasing seller’s XVA
- Counterparty default intensity can decrease both XVAs (default premium exceeds funding costs)
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 5.2.2 (p. 86): The standard practitioner decomposition is xVA = CVA + DVA + ColVA + FVA + KVA + MVA (Eq. 5.2). Gregory notes that these definitions are “relatively standard” but not unique; furthermore, there are potential overlaps (e.g. DVA and FBA share the same economics).
- Section 16.3.2 (pp. 482—483): Starting from perfect collateralisation as the base value, xVA adjustments reflect deviations from this ideal. ColVA captures margin-type and remuneration deviations; CVA/DVA captures bilateral default risk; FVA captures funding effects of being under-collateralised; KVA captures the cost of regulatory capital; MVA captures the cost of posting initial margin. Both FVA and MVA are funding costs, but FVA is the cost of under-collateralisation while MVA is the cost of over-collateralisation.
- Section 5.2.5 (pp. 90—91): The relative importance of each xVA term varies by transaction type. Uncollateralised trades have large CVA, FVA, and KVA; strongly collateralised trades have reduced CVA and negligible FVA but may have ColVA; overcollateralised trades and centrally-cleared trades have large MVA but minimal CVA and FVA.
- Section 5.2.7 (pp. 92—93): CVA is characterised as “the least real valuation adjustment” because it depends on a default event that may never occur, whereas FVA, ColVA, KVA, and MVA are ongoing production costs visible even absent default.
- Section 18.2.6 (pp. 546—548): The FVA debate. Hull and White (2012a) argue that including entity-specific funding costs in pricing breaks the “law of one price” and creates arbitrage; this corresponds to the Bichuch et al. single-price case (symmetric rates). The counterargument (Kenyon and Green 2014) is that there is no “market” for uncollateralised derivatives; Andersen et al. (2016) show that CVA + FVA maximises shareholder value. This debate maps directly onto the no-arbitrage interval: the width of [V_0^-, V_0^+] reflects the market incompleteness that the FVA proponents invoke.
- Section 19.4.1 (pp. 587—589): The CVA-KVA overlap: under Kenyon and Green (2014), the combined cost is (1-alpha) x EL + alpha x CVA + beta x KVA, where alpha represents CVA hedging intensity and beta the capital impact of hedges.
- Section 20.4 (pp. 606—608): The MVA-KVA trade-off: posting initial margin increases MVA but decreases KVA via capital relief. Joint optimisation determines the optimal IM level (typically below the regulatory requirement).