When defaults cannot be hedged: an actuarial approach to xVA calculations via local risk-minimization
Francesca Biagini, Alessandro Gnoatto, Katharina Oberpriller
TL;DR
This paper develops an actuarial perspective on xVA valuation under market incompleteness due to unhedgeable default jumps. It extends local risk-minimization to a multi-curve framework and characterizes hedging strategies through a BSDE, yielding a FS decomposition that partitions the price into CVA, DVA, ColVA, FVA, and, in a later section, KVA. The approach provides a rigorous foundation for two-step CVA computations and offers a pathway to numerical treatment via deep learning or Monte Carlo, while accommodating collateral, repo-trading, and realistic funding dynamics. Practically, the framework supports pricing under default risk with physically implementable hedges, quantifying adjustments and capital costs in a coherent, mathematically solid manner.
Abstract
We consider the pricing and hedging of counterparty credit risk and funding when there is no possibility to hedge the jump to default of either the bank or the counterparty. This represents the situation which is most often encountered in practice, due to the absence of quoted corporate bonds or CDS contracts written on the counterparty and the difficulty for the bank to buy/sell protection on her own default. We apply local risk-minimization to find the optimal strategy and compute it via a BSDE.