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Arbitrage-Free Pricing of XVA - Part I Framework and Explicit Examples

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Summary

The paper introduces a rigorous no-arbitrage framework for computing the total valuation adjustment (XVA) of a European claim in the presence of funding costs, counterparty credit risk, and collateralization. The trader’s replicating portfolio consists of a default-free stock (financed via the repo market), two risky bonds (trader and counterparty), a funding account with asymmetric borrowing/lending rates, and a collateral account. Due to rate asymmetries, the replicating BSDE for a long position differs from that of a short position, yielding a no-arbitrage interval [V_0^-, V_0^+] rather than a single price. When borrowing and lending rates coincide, buyer’s and seller’s XVA agree and admit a fully explicit closed-form expression as a percentage of the Black-Scholes price.

Key Contributions

  • Derivation of nonlinear BSDEs characterizing the replicating portfolios for long and short positions under asymmetric funding, repo, and collateral rates, with counterparty credit risk
  • Definition of buyer’s and seller’s XVA as the difference between the replication cost and the Black-Scholes price, establishing the no-arbitrage interval
  • Sufficient conditions on rates (e.g., rr+ ⇐ rf+ ⇐ rr-, rf+ ⇐ rf-) that preclude hedger’s arbitrage
  • Fully explicit XVA formula under equal borrowing/lending rates: XVA is expressed as a deterministic percentage of the claim price, decomposed into funding adjustment, CVA/DVA terms, and collateral funding costs
  • Explicit replication strategy consisting of a funding-adjusted delta hedge plus a correction for the gap between funding and collateral rates
  • Extension of Piterbarg (2010)‘s model to include counterparty default risk via trader and counterparty bonds

Methodology

  • Model: continuous-time Black-Scholes stock with constant volatility, two defaultable bonds (exponential default times with constant intensities), asymmetric repo/funding/collateral rates
  • Valuation measure: equivalent measure Q chosen by a third-party valuation agent using discount rate rD
  • Replication via BSDE: the wealth process of the self-financing hedging strategy satisfies a nonlinear BSDE with jumps, with drivers f+ and f- for seller and buyer respectively
  • Close-out convention: risk-free closeout at default time, with loss-given-default parameters LI and LC for trader and counterparty
  • Collateral: cash collateral as fraction alpha of the claim’s mark-to-market value, with full rehypothecation
  • Explicit solution: under symmetric rates (Piterbarg’s setup), the BSDE linearizes and admits a closed-form solution via the Clark-Ocone formula and Malliavin calculus

Key Findings

  • XVA under symmetric rates decomposes into: (1) a funding adjustment from discounting at rf instead of rD, and (2) a collateral cost proportional to alpha(rf - rc)
  • The replication strategy adjusts the Black-Scholes delta by the factor beta_t that also scales the XVA
  • Under rate asymmetry, the no-arbitrage band width equals XVA_sell - XVA_buy = V_0^+ - V_0
  • The framework recovers Piterbarg (2010) as a special case when default risk is excluded
  • Positive homogeneity of the BSDE drivers is essential for the arbitrage pricing argument

Important References

  • Piterbarg (2010) - funding beyond discounting (citationCount: 142)
  • Crepey (2015a,b) - bilateral counterparty risk under funding constraints (citationCount: 150, 162)
  • Burgard and Kjaer (2011a,b) - PDE representations with bilateral counterparty risk (citationCount: 142, 110)
  • Bielecki and Rutkowski (2014) - valuation and hedging with funding costs (citationCount: 68)
  • El Karoui, Peng and Quenez (1997) - BSDEs in finance (citationCount: 2618)
  • Brigo and Pallavicini (2014) - funding, collateral and hedging (citationCount: 81)
  • Delong (2013) - BSDEs with jumps and applications (citationCount: 141)

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