XVA analysis from the balance sheet

Abstract

XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007–2009 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key to identifying the economic meaning of the XVA terms. Our approach is first detailed in a static setup that is solved explicitly. It is then plugged into the dynamic and trade incremental context of a real derivative banking portfolio. The corresponding cost-of-capital XVA strategy ensures for bank shareholders a submartingale equity process corresponding to a target hurdle rate on their capital at risk, consistently between and throughout deals. Set on a forward/backward SDE formulation, this strategy can be solved efficiently using GPU computing combined with deep learning regression methods in a whole bank balance sheet context. A numerical case study emphasizes the workability and added value of the ensuing pathwise XVA computations.

Keywords:

• Counterparty risk
• Balance sheet of a bank
• Market incompleteness
• Wealth transfer
• X-valuation adjustment (XVA)
• Deep learning
• Quantile regression

Mathematics Subject Classification:

• 91B25
• 91B26
• 91B30
• 91G20
• 91G40
• 62G08
• 68Q32

JEL Classification:

• D52
• G13
• G24
• G28
• G33
• M41

Acknowledgments

The authors are grateful for useful discussions with Lokman Abbas-Turki, Agostino Capponi, Karl-Theodor Eisele, Chris Kenyon, Marek Rutkowski, and Michael Schmutz.

Disclosure statement

The views expressed herein by Rodney Hoskinson are his personal views and do not reflect the views of ANZ Banking Group Limited (`ANZ'). No liability shall be accepted by ANZ whatsoever for any direct or consequential loss from any use of this paper and the information, opinions and materials contained herein.

Notes

† See also Remark 2.1 regarding the meaning of the FVA in (2(2) $\mathrm{C}\mathrm{V}\mathrm{A}+\mathrm{F}\mathrm{V}\mathrm{A}+\mathrm{K}\mathrm{V}\mathrm{A},$(2) ).

† See Artzner et al. (2020, Proposition 2.1) for a proof.

† See e.g. Föllmer and Schied (2016, Section 4.4).

‡ Note that, by definition of $\mathbb{Q}$, this quantity does not depend on $L^{\bullet }$.

† i.e. remove $(-\mathrm{\Delta }\mathrm{M}\mathrm{t}\mathrm{M})$ from, if $\mathrm{\Delta }\mathrm{M}\mathrm{t}\mathrm{M}<0$.

‡ i.e. remove $(-\mathrm{\Delta }\mathrm{C}\mathrm{A})$ from, if $\mathrm{\Delta }\mathrm{C}\mathrm{A}<0$.

§ i.e. removes $(-\mathrm{\Delta }\mathrm{K}\mathrm{V}\mathrm{A})$ from, if $\mathrm{\Delta }\mathrm{K}\mathrm{V}\mathrm{A}<0$.

† See Albanese et al. (2020, Section 5) for the discussion of cheaper funding schemes for initial margin.

Additional information

Funding

The PhD thesis of Bouazza Saadeddine is funded by a CIFRE grant from CA-CIB and French ANRT.