Abstract
We study mathematical aspects of dynamic hedging of Credit Valuation Adjustment (CVA) in a portfolio of OTC financial derivatives. Since the sub-prime crisis, the counterparty risk and the wrong way risk are crucial issues in connection with valuation and risk management of credit derivatives. In this work we first derive a general model-free equation for the dynamics of the CVA of a portfolio of OTC derivatives. We then particularize these dynamics to the counterparty risk of a portfolio of credit derivatives including, for instance, CDSs and/or CDOs, possibly netted and collateralized, considered in the so-called Markovian copula model. Wrong way risk is represented in the model by the possibility of simultaneous defaults. We establish a rigorous connection between the CVA, which represents the price of the counterparty risk, and a suitable notion of Expected Positive Exposure (EPE). Specifically, the EPE emerges as the key ingredient of the min-variance hedging ratio of the CVA by a CDS on the counterparty. Related notions of EPE have actually long been used in an ad-hoc way by practitioners for hedging their CVA. Our analysis thus justifies rigorously this market practice, making also precise the proper definition of the EPE which should be used in this regard, and the way in which the EPE should be used in the hedging strategy.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See Remark 3.2 for filtration issues.
- 2.
Consequently, any cash flows, such as dividends paid by the collateral assets, are thus channeled back to the counterparty.
- 3.
We emphesize again that the clean price process P above is the process of the ‘clean contract,’ that is the contract in which any counterparty risk is disregarded.
References
Assefa, S., Bielecki, T.R., Crepey, S., Jeanblanc, M.: CVA computation for counterparty risk assessment in credit portfolios. In: Bielecki, T.R., Brigo, D., Patras, F. (eds.) Credit Risk Frontiers: Subprime Crisis, Pricing and Hedging, CVA, MBS, Ratings and Liquidity. Wiley, New York (2011)
Bielecki, T.R., Cousin, A., Crépey, S., Herbertsson, A.: Dynamic hedging of portfolio credit risk in a Markov copula model. JOTA 157(3) (2013)
Bielecki, T.R., Crépey, S., Jeanblanc, M., Rutkowski, M.: Convertible bonds in a defaultable diffusion model. In: Kohatsu-Higa, A., Privault, N., Sheu, S.J. (eds.) Stochastic Analysis with Financial Applications. Birkhäuser, Basel (2010)
Bielecki, T., Crépey, S., Jeanblanc, M., Zargari, B.: Valuation and hedging of CDS counterparty exposure in a Markov copula model. IJTAF 15(1) (2012)
Bielecki, T.R., Vidozzi, A., Vidozzi, L.: A Markov copulae approach to pricing and hedging of credit index derivatives and ratings triggered step-up bonds. J. Credit Risk 4, 1 (2008)
Bielecki, T.R., Vidozzi, A., Vidozzi, L., Jakubowski, J.: Study of dependence for some stochastic processes. Stoch. Anal. Appl. 26(4), 903–924 (2008)
Bielecki, T.R., Rutkowski, M.: Credit Risk: Modeling, Valuation and Hedging. Springer, Berlin (2002)
Bielecki, T.R., Jeanblanc, M., Rutkowski, M.: Pricing and trading credit default swaps. Ann. Appl. Probab. 18(6), 2495–2529 (2008)
Blanchet-Scalliet, C., Jeanblanc, M.: Hazard rate for credit risk and hedging defaultable contingent claims. Finance Stoch. 8(1), 145–159 (2008)
Cesari, G., Aquilina, J., Charpillon, N.: Modelling, Pricing, and Hedging Counterparty Credit Exposure. Springer Finance (2010)
Crépey, S.: About the pricing equations in finance. Forthcoming in Paris-Princeton Lectures in Mathematical Finance. Lecture Notes in Mathematics. Springer (2010) (Preprint version available online at http://www.maths.univ-evry.fr/crepey)
Crépey, S., Jeanblanc, M., Zargari, B.: Counterparty risk on a CDS in a Markov chain copula model with joint defaults. In: Kijima, M., Hara, C., Muromachi, Y., Tanaka, K. (eds.) Recent Advances in Financial Engineering 2009. World Scientific, Singapore (2010). Available on http://grozny.maths.univ-evry.fr/pages_perso/crepey/
Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer Finance (2006)
Durand, C., Rutkowski, M.: Credit value adjustment for bilateral counterparty risk of collateralized contracts under systemic risk. Working paper
Gregory, J.: Counterparty Credit Risk: The New Challenge for Global Financial Markets. Wiley, New York (2009)
ISDA Collateral Steering Committee: Market Review of OTC Derivative Bilateral Collateralization Practices, ISDA Swaps and Derivatives Association, March 2010
Acknowledgements
The authors warmly thank Giovanni Cesari for stimulating discussions throughout the preparation of this work. We also thank an anonymous referee for a very careful reading of the manuscript and for valuable comments and remarks.
The research of T.R. Bielecki was supported by NSF Grant DMS-0603789 and NSF Grant DMS-0908099.
The research of S. Crépey benefited from the support of the ‘Chaire Risque de crédit’, Fédération Bancaire Française, and of the DGE.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bielecki, T.R., Crépey, S. (2014). Dynamic Hedging of Counterparty Exposure. In: Kabanov, Y., Rutkowski, M., Zariphopoulou, T. (eds) Inspired by Finance. Springer, Cham. https://doi.org/10.1007/978-3-319-02069-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-319-02069-3_3
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-02068-6
Online ISBN: 978-3-319-02069-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)