[Paper Review] XVA Analysis From the Balance Sheet

Claudio Albanese, Stéphane Crépey+2 more|RePEc: Research Papers in Economics|Sep 1, 2020

Financial Markets and Investment Strategies4 citations

TL;DR

This paper proposes a balance sheet–driven cost-of-capital XVA framework that models counterparty risk adjustments (CVA, FVA, MVA, KVA) as a dynamic, pathwise system of forward-backward SDEs, ensuring bank equity grows as a submartingale at a target hurdle rate. It enables efficient GPU-accelerated deep learning regression for whole-bank XVA computation, validated via a numerical case study showing robust pathwise valuation and economic consistency across trades.

ABSTRACT

XVAs denote various counterparty risk related valuation adjustments that are applied to financial derivatives since the 2007--09 crisis. We root a cost-of-capital XVA strategy in a balance sheet perspective which is key in 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 in the dynamic and trade incremental context of a real derivative banking portfolio. The corresponding cost-of-capital XVA strategy ensures to 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.

Motivation & Objective

• To reframe XVA valuation from a bank’s balance sheet perspective, grounding XVA terms in economic reality rather than isolated pricing.
• To ensure a consistent, submartingale equity process for shareholders by embedding a target hurdle rate (KVA) into the XVA framework.
• To resolve market incompleteness in counterparty risk by modeling imperfect hedging of default exposure and own default risk.
• To develop a computationally efficient, scalable solution for whole-bank XVA computation using deep learning and GPU acceleration.
• To validate the framework through a numerical case study demonstrating pathwise XVA computation and economic coherence.

Proposed method

• The framework is built on a forward/backward SDE formulation that links the bank’s trading loss process (L) with FVA and KVA components through a coupled system of equations.
• It models CVA, FVA, MVA, and KVA explicitly: CVA and MVA as contingent claims on counterparty default, FVA as funding cost of variation margin, and KVA as capital remuneration for risk.
• The KVA is defined via a continuous-time risk margin formula: KVAt = Et[∫Th(s−t)max(ECs, KVAs)ds], aligning with Swiss Solvency Test methodology.
• A Picard iteration scheme disentangles the feedback between L, EC (expected shortfall), and (FVA, KVA), enabling convergence to a unique square-integrable solution.
• The system is solved using GPU-accelerated deep learning regression, allowing scalable, pathwise computation across the entire bank’s derivative portfolio.
• Collateralization schemes are modeled via CSA (no CSA, (VM/IM) CSA), with RIM and PIM computed at quantile levels (arim, apim), and netting sets distinguished by counterparty status.

Experimental results

Research questions

• RQ1 How can XVA adjustments be rooted in a bank’s balance sheet to reflect economic reality and ensure consistent capital remuneration?
• RQ2 What is the role of KVA in ensuring a submartingale equity process that meets a target hurdle rate across all trades?
• RQ3 How can the feedback between counterparty risk, own default risk, and funding costs be modeled consistently in a dynamic, incomplete market setting?
• RQ4 Can deep learning and GPU computing efficiently solve the resulting high-dimensional, forward-backward SDE system for whole-bank XVA computation?
• RQ5 How do different collateralization schemes (e.g., (VM/IM) CSA) affect the pathwise behavior of XVA components?

Key findings

• The proposed XVA framework ensures that the bank’s equity process grows as a submartingale at the target hurdle rate, consistent across all trades and time periods.
• The system of equations for CVA, FVA, MVA, and KVA is well-posed and admits a unique square-integrable solution, validated via Picard iteration convergence.
• KVA is explicitly modeled as a risk margin on the maximum of expected shortfall (EC) and prior KVA, aligning with the Swiss Solvency Test methodology.
• The framework enables efficient, scalable computation of pathwise XVA values using GPU-accelerated deep learning regression, suitable for real-time or backtesting applications.
• The numerical case study confirms the workability of the approach, showing that XVA components are consistently computed across the portfolio under different collateralization schemes.
• The model accounts for fungibility of capital at risk with variation margin, resolving feedback loops between CR, CA, and FVA in a self-consistent manner.