Locally Risk-Minimizing Hedging of Counterparty Risk for Portfolio of Credit Derivatives
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Locally Risk-Minimizing Hedging of Counterparty Risk for Portfolio of Credit Derivatives Lijun Bo1 · Claudia Ceci2
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We discuss dynamic hedging of counterparty risk for a portfolio of credit derivatives by the local risk-minimization approach. We study the problem from the perspective of an investor who, trading with credit default swaps (CDS) referencing the counterparty, wants to protect herself/himself against the loss incurred at the default of the counterparty. We propose a credit risk intensity-based model consisting of interacting default intensities by taking into account direct contagion effects. The portfolio of defaultable claims is of generic type, including CDS portfolios, risky bond portfolios and first-to-default claims with payments allowed to depend on the default state of the reference firms and counterparty. Using the martingale representation of the conditional expectation of the counterparty risk price payment stream under the minimal martingale measure, we recover a closed-form representation for the locally risk minimizing strategy in terms of classical solutions to nonlinear recursive systems of Cauchy problems. We also discuss applications of our framework to the most prominent classes of credit derivatives. Keywords Local risk-minimization · Counterparty risk · Recursive system of Cauchy problems AMS 2000 subject classifications 60J25 · 60J75 · 60H30 · 91B28
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Claudia Ceci [email protected] Lijun Bo [email protected]
1
School of Mathematics and Statistics, Xidian University, Xi’an 710071, People’s Republic of China
2
Department of Economics, University “G. d’Annunzio” of Chieti-Pescara, 65127 Pescara, Italy
123
Applied Mathematics & Optimization
1 Introduction Counterparty credit risk receives a lot of attention after the global financial crisis of 2007–2009; since then, the management of counterparty credit risk has become a key issue for financial institutions. This risk refers to the possibility that one of the contracting parties of derivatives transactions, carries out over the counter, defaults before maturity. The vast majority of literature has focused on the valuation of counterparty risk, i.e., credit valuation adjustment, abbreviated with CVA throughout this paper; see also Capponi [14] for a survey. Despite the importance of dynamic hedging of counterparty risk across policy makers and the financial industry, the literature on the subject is still not as well developed.1 A larger body of literature has investigated dynamic hedging of defaultable claims using mean-variance strategies, but without accounting for counterparty risk. Bielecki et al. [7] and [8] introduce a framework for hedging risks in incomplete markets, building on the classical Markowitz mean-variance portfolio selection framework. They analyze quadratic hedging methods and consider strategies adapted to the defaultfree market information as well as to the enlarged filtration inclusive of default events. Bielecki et al. [9]
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