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When Capital Is a Funding Source: The Anticipated Backward Stochastic Differential Equations of X-Value Adjustments
SIAM Journal on Financial Mathematics ( IF 1 ) Pub Date : 2020-02-18 , DOI: 10.1137/19m1242781 Stéphane Crépey , Wissal Sabbagh , Shiqi Song
SIAM Journal on Financial Mathematics ( IF 1 ) Pub Date : 2020-02-18 , DOI: 10.1137/19m1242781 Stéphane Crépey , Wissal Sabbagh , Shiqi Song
SIAM Journal on Financial Mathematics, Volume 11, Issue 1, Page 99-130, January 2020.
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.
中文翻译：
当资金是资金来源时：X值调整的预期后向随机微分方程
SIAM金融数学杂志，第11卷，第1期，第99-130页，2020年1月。
X值调整（XVA）是指各种金融衍生产品定价调整，其中要考虑交易对手风险及其对银行的资金（FVA）和资本（KVA）的影响。在本文中，我们证明了XVA方程是正确的，包括在现实情况下，资本被视为可替代的，作为变异保证金的资金来源。风险资本和FVA的这种相互交织，加之KVA是风险资本的一部分这一事实，导致FVA和KVA形成了McKean类型的反向SDE（BSDE）（预期BSDE），系数，需要对FVA the部的提前一年增量进行条件风险评估。在假设银行没有债务的情况下，首先要考虑这一点。在实际情况下，如果银行有违约，则预期的BSDE会产生，
更新日期：2020-02-18
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.
中文翻译：
当资金是资金来源时：X值调整的预期后向随机微分方程
SIAM金融数学杂志，第11卷，第1期，第99-130页，2020年1月。
X值调整（XVA）是指各种金融衍生产品定价调整，其中要考虑交易对手风险及其对银行的资金（FVA）和资本（KVA）的影响。在本文中，我们证明了XVA方程是正确的，包括在现实情况下，资本被视为可替代的，作为变异保证金的资金来源。风险资本和FVA的这种相互交织，加之KVA是风险资本的一部分这一事实，导致FVA和KVA形成了McKean类型的反向SDE（BSDE）（预期BSDE），系数，需要对FVA the部的提前一年增量进行条件风险评估。在假设银行没有债务的情况下，首先要考虑这一点。在实际情况下，如果银行有违约，则预期的BSDE会产生，
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