No-arbitrage pricing under funding costs extends the classical Black-Scholes framework to markets where the hedger faces asymmetric borrowing and lending rates, repo rate spreads, and counterparty default risk. Unlike the frictionless case, the presence of rate asymmetries means there is no unique arbitrage-free price; instead, a no-arbitrage interval emerges.
Key Concepts
Hedger’s arbitrage (Definition 4.1 in Bichuch et al. (2015a)): the market (S, P^I, P^C) admits hedger’s arbitrage if there exist controls and initial capital x >= 0 such that the terminal wealth dominates the risk-free return almost surely, with strict inequality on a positive-probability event.
No-arbitrage conditions (Assumption 4.2 and Proposition 4.4): the relations
- rr+ ⇐ rf+ ⇐ rr- (repo rates bracket the lending rate)
- rf+ ⇐ rf- (borrowing rate exceeds lending rate)
- rf+ v rD < r^i + h_i^P for i in {I, C} (bond returns exceed funding/discount rates)
- rc+ v rc- ⇐ rf- ⇐ (r^I + h_I^P) ^ (r^C + h_C^P)
are sufficient to preclude hedger’s arbitrage. The proof constructs an equivalent measure under which discounted asset prices are supermartingales.
No-arbitrage interval (Theorem 4.6): under the above conditions and V_0^- ⇐ V_0^+, the set of hedger’s arbitrage-free prices for the claim Phi(S_T) is exactly [pi^inf, pi^sup] = [V_0^-, V_0^+], where V^+ and V^- solve the seller’s and buyer’s BSDEs. Prices strictly above pi^sup or below pi^inf admit arbitrage.
Economic Interpretation
- The interval arises because the hedger’s funding cost depends on whether he is net long or short the funding account, creating an asymmetry between selling and buying the claim
- When all rates coincide, the interval collapses to a single price and the classical Black-Scholes formula (adjusted for funding) is recovered
- The width of the interval equals XVA_sell - XVA_buy, which increases with the funding spread rf- - rf+
Textbook References
The xVA Challenge (Gregory, 2020)
- Section 18.2.6 (pp. 546—548): The FVA debate. Hull and White (2012a) argue that FVA should not be included in pricing or valuation, invoking risk-neutral pricing and the Modigliani-Miller theorem. They identify a “DVA2” term (benefit from defaulting on general funding liabilities) that cancels FCA, so the total firm value is CVA + DVA with no separate FVA. Counterarguments: markets for uncollateralised derivatives are incomplete (Kenyon and Green 2014); the symmetric FVA framework maximises shareholder value (Andersen et al. 2016, Albanese et al. 2015). The debate reduces to whether fair value should represent total firm value (shareholders + bondholders, implying CVA + DVA) or shareholder value alone (implying CVA + FVA).
- Section 18.2.5 (pp. 539—541): The Burgard-Kjaer hedging-replication framework derives CVA, FCA, and DVA from first principles. FCA (Eq. 18.5) involves the funding spread times a risky discount factor times EPE. The framework shows DVA = FBA under the assumption that credit spread equals funding spread. Two no-double-counting frameworks: (i) CVA + FCA + FBA (symmetric funding), (ii) CVA + DVA + FCA (asymmetric funding). Table 18.5 summarises the no-FVA (Hull-White) and FVA (Burgard-Kjaer) regimes.
- Section 18.3.2 (pp. 552—554): The asymmetric FVA framework of Albanese et al. (2015) and Burgard-Kjaer (2012). Excess cash on derivatives books is an unstable funding source, so lending is at risk-free rate while borrowing is at unsecured rate. This produces FCA only (no FBA), computed at the funding-set level (Eq. 18.8).