Abstract
We study the semilinear partial differential equation (PDE) associated with the non-linear BSDE characterizing buyer’s and seller’s XVA in a framework that allows for asymmetries in funding, repo and collateral rates, as well as for early contract termination due to counterparty credit risk. We show the existence of a unique classical solution to the PDE by first proving the existence and uniqueness of a viscosity solution and then its regularity. We use the uniqueness result to conduct a thorough numerical study illustrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.
Summary
This companion paper to Part I develops the PDE theory and numerical analysis for the nonlinear BSDE characterizing XVA under asymmetric rates and counterparty default risk. The main theoretical contribution is proving that the semilinear PDE associated with the BSDE admits a unique classical solution — first establishing uniqueness of the viscosity solution (extending results of Delong (2013) by using locality and accounting for default risk), then proving regularity to obtain a classical solution. This rigorous result contrasts with earlier literature (e.g. Burgard and Kjaer) where smoothness was assumed rather than proved. The classical solution is then used to derive explicit hedging strategies and to perform a comprehensive numerical study via finite difference (Crank-Nicholson) methods.
Key Contributions
- Proof of existence and uniqueness of a viscosity solution to the semilinear PDE arising from the nonlinear XVA BSDE, extending Delong (2013) by using locality of viscosity solutions and incorporating default risk via random terminal time
- Proof that the viscosity solution is also a classical solution under mild polynomial growth assumptions on the payoff, via the transformation to bounded terminal conditions (Cannon (1984))
- Derivation of explicit replication strategies from the classical solution: stock holding xi = v_S(t,S) and bond holdings xi^j = (v(t,S) - theta_j(v_hat))/(bond price)
- Comprehensive numerical study using Crank-Nicholson finite differences on the semilinear PDE system after log-price change of variables
Methodology
- PDE derivation: the BSDE on {t < tau} is shown to be Markovian; reducing to two state variables (t, S) via Remark 3.3 (the martingale compensators $^{j,Q} are deterministic pre-default), yielding a semilinear PDE (29)-(30)
- Log-price transformation: x = log(s) converts the PDE to a Cauchy problem on R, eliminating boundary conditions at S=0
- Viscosity solution: proved by comparison with auxiliary BSDEs on small time intervals [t_0, t_0+h], using the subsolution/supersolution framework and estimates on the stopping time tau_1 at which the process exits a neighborhood
- Classical solution: the change of variables w_bar(t,x) = (1 + e^{2nx}) v_bar(t, e^x) bounds the terminal condition, enabling application of Cannon (1984) Theorem 20.2.1
- Numerical scheme: Crank-Nicholson finite differences on the coupled system (semilinear PDE for v_bar and linear Black-Scholes PDE for v_hat)
Key Findings
- Funding rate sensitivity: higher funding rate rf- widens the no-arbitrage band (XVA_sell - XVA_buy); the effect is asymmetric — at low collateralization, the widening comes from decreasing buyer’s XVA, while at high collateralization, seller’s XVA drives it
- Collateralization effect: higher alpha increases both seller’s and buyer’s XVA because the closeout value rises, requiring replication of a larger position; the trader must take more risk (more stock and counterparty bond shares)
- Counterparty default intensity: increasing h_C^Q can decrease both XVAs because the default premium compensation from holding counterparty bonds exceeds the extra funding costs of replicating the larger closeout position
- No-arbitrage band width: insensitive to counterparty default intensity at low collateralization; at high alpha, seller’s and buyer’s XVA nearly coincide
- Replication strategy: stock and bond holdings are explicit functions of the PDE solution gradient; bond positions hedge the jump risk at default
Important References
- Bichuch, Capponi, Sturm (2015a) - companion paper providing BSDE framework (citationCount: 15)
- Delong (2013) - BSDEs with jumps, viscosity solution theory (citationCount: 141)
- Cannon (1984) - one-dimensional heat equation, classical solution existence (citationCount: 357)
- Burgard and Kjaer (2011b, 2013) - PDE representations with bilateral risk (citationCount: 110, 40)
- Crepey (2015a) - bilateral counterparty risk under funding constraints (citationCount: 150)
- Jouini and Kallal (1995) - arbitrage in markets with short-sales constraints (citationCount: 169)
- Shreve and Soner (1994) - optimal investment with transaction costs (citationCount: 631)
Atomic Notes
- nonlinear BSDE with jumps
- XVA decomposition
- no-arbitrage pricing under funding costs
- viscosity solution for semilinear PDE
- close-out convention and collateral rehypothecation