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

Stéphane Crépey (), Wissal Sabbagh () and Shiqi Song ()
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Stéphane Crépey: LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Wissal Sabbagh: LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement
Shiqi Song: LaMME - Laboratoire de Mathématiques et Modélisation d'Evry - ENSIIE - Ecole Nationale Supérieure d'Informatique pour l'Industrie et l'Entreprise - UEVE - Université d'Évry-Val-d'Essonne - Université Paris-Saclay - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement

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Abstract: 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.

Keywords: credit valuation adjustment (CVA); funding valuation adjustment (FVA); capital valuation adjustment (KVA); anticipated (or McKean) BSDE; progressive enlargement of filtration; invariance time (search for similar items in EconPapers)
Date: 2020-01
Note: View the original document on HAL open archive server: https://hal.science/hal-03910119
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Citations: View citations in EconPapers (3)

Published in SIAM Journal on Financial Mathematics, 2020, 11, pp.99 - 130. ⟨10.1137/19m1242781⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-03910119

DOI: 10.1137/19m1242781

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