Abstract

http://ssrn.com/abstract=1605307
 
 

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


Christoph Burgard


Barclays Capital

Mats Kjaer


Barclays Investment Bank

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.

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

Number of Pages in PDF File: 19

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

JEL Classification: G13, C63

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Date posted: May 13, 2010 ; Last revised: August 7, 2014

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: http://ssrn.com/abstract=1605307 or http://dx.doi.org/10.2139/ssrn.1605307

Contact Information

Christoph Burgard
Barclays Capital ( email )
5 The North Colonnade
Canary Wharf
London, E14 4BB
United Kingdom
Mats Kjaer (Contact Author)
Barclays Investment Bank ( email )
5 The North Colonnade
Canary Wharf
London, E14 4BB
United Kingdom
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