Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs

C. Burgard and M. Kjaer. Partial differential equation representations of derivatives with counterparty risk and funding costs. The Journal of Credit Risk, Vol. 7, No. 3, 1-19, 2011.

19 Pages Posted: 13 May 2010 Last revised: 7 Aug 2014

Christoph Burgard

Bank of America - Bank of America Merrill Lynch

Mats Kjaer

Bloomberg L.P.

Date Written: November 23, 2010

Abstract

We derive a partial differential equation (PDE) representation for the value of financial derivatives with bilateral counterparty risk and funding costs. The model is very general in that the funding rate may be different for lending and borrowing and the mark-to-market value at default can be specified exogenously. The buying back of a party's own bonds is a key part of the delta hedging strategy; we discuss how the cash account of the replication strategy provides sufficient funds for this.

First, we assume that the mark-to-market value at default is given by the total value of the derivative, which includes counterparty risk. We find that the resulting pricing PDE becomes non-linear, except in special cases, when the non-linear terms vanish and a Feynman-Kac representation of the total value can be obtained. In these cases, the total value of the derivative can be decomposed into the default-free value plus a bilateral credit valuation and funding adjustment.

Second, we assume that the mark-to-market value at default is given by the counterparty-riskless value of the derivative. This time, the resulting PDE is linear and the corresponding Feynman-Kac representation is used to decompose the total value of the derivative into the default-free value plus bilateral credit valuation and funding cost adjustments.

A numerical example shows that the effect on the valuation adjustments of a non-zero funding spread can be significant.

The Addendum for this paper is available at the following URL: http://ssrn.com/abstract=2109723

Keywords: Counterparty risk, Credit Valuation Adjustment, Funding costs, PDE, Feynman-Kac Theorem

JEL Classification: G13, C63

Suggested Citation

Burgard, Christoph and Kjaer, Mats, Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs (November 23, 2010). C. Burgard and M. Kjaer. Partial differential equation representations of derivatives with counterparty risk and funding costs. The Journal of Credit Risk, Vol. 7, No. 3, 1-19, 2011.. Available at SSRN: https://ssrn.com/abstract=1605307 or http://dx.doi.org/10.2139/ssrn.1605307

Christoph Burgard

Bank of America - Bank of America Merrill Lynch ( email )

London
United Kingdom

Mats Kjaer (Contact Author)

Bloomberg L.P. ( email )

39-45 Finsbury Square
City Gate House
London, EC2A 1PQ
United Kingdom

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