Arbitrage-Free XVA

Math. Finance 28:2 (2018), 582-620

39 Pages Posted: 12 Aug 2016 Last revised: 23 Feb 2020

See all articles by Maxim Bichuch

Maxim Bichuch

Johns Hopkins University

Agostino Capponi

Columbia University

Stephan Sturm

Worcester Polytechnic Institute (WPI) - Department of Mathematical Sciences

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Date Written: August 8, 2016

Abstract

We develop a framework for computing the total valuation adjustment (XVA) of a European claim accounting for funding costs, counterparty credit risk, and collateralization. Based on no-arbitrage arguments, we derive backward stochastic differential equations (BSDEs) associated with the replicating portfolios of long and short positions in the claim. This leads to the definition of buyer's and seller's XVA, which in turn identify a no-arbitrage interval.

In the case that borrowing and lending rates coincide, we provide a fully explicit expression for the unique XVA, expressed as a percentage of the price of the traded claim, and for the corresponding replication strategies. In the general case of asymmetric funding, repo and collateral rates, we study the semilinear partial differential equations (PDE) characterizing buyer's and seller's XVA and show the existence of a unique classical solution to it. To illustrate our results, we conduct a numerical study demonstrating how funding costs, repo rates, and counterparty risk contribute to determine the total valuation adjustment.

Keywords: XVA, counterparty risk, asymmetric rates, collateralization

JEL Classification: G13, C32

Suggested Citation

Bichuch, Maxim and Capponi, Agostino and Sturm, Stephan, Arbitrage-Free XVA (August 8, 2016). Math. Finance 28:2 (2018), 582-620, Available at SSRN: https://ssrn.com/abstract=2820257

Maxim Bichuch

Johns Hopkins University ( email )

Baltimore, MD 20036-1984
United States

Agostino Capponi (Contact Author)

Columbia University ( email )

S. W. Mudd Building
New York, NY 10027
United States

Stephan Sturm

Worcester Polytechnic Institute (WPI) - Department of Mathematical Sciences ( email )

United States
5088315921 (Phone)
5088315824 (Fax)

HOME PAGE: http://users.wpi.edu/~ssturm

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