When Capital Is a Funding Source: The Anticipated Backward Stochastic Differential Equations of X-Value Adjustments

X-value adjustments (XVAs) refer to various financial derivative pricing adjustments accounting for counterparty risk and its funding (FVA) and capital (KVA) implications for a bank. In this paper we show that the XVA equations are well-posed, including in the realistic case where capital is deemed fungible as a source of funding for variation margin. This intertwining of capital at risk and the FVA, added to the fact that the KVA is part of capital at risk, leads to a system of backward SDEs (BSDEs) of the McKean type (anticipated BSDEs) for the FVA and the KVA, with coefficients entailing a conditional risk measure of the one-year-ahead increment of the martingale part of the FVA. This is first considered in the case of a hypothetical bank without debt. In the practical case of a defaultable bank, the resulting anticipated BSDEs, which are stopped before the default of the bank, are solved likewise after reduction to a reference market filtration.