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Derivatives Pricing under Bilateral Counterparty Risk

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  • Samim Ghamami

Abstract

We consider risk-neutral valuation of a contingent claim under bilateral counterparty risk in a reduced-form setting similar to that of Duffie and Huang [1996] and Duffie and Singleton [1999]. The probabilistic valuation formulas derived under this framework cannot be usually used for practical pricing due to their recursive path-dependencies. Instead, finite-difference methods are used to solve the quasi-linear partial differential equations that equivalently represent the claim value function. By imposing restrictions on the dynamics of the risk-free rate and the stochastic intensities of the counterparties' default times, we develop path-independent probabilistic valuation formulas that have closed-form solution or can lead to computationally efficient pricing schemes. Our framework incorporates the so-called wrong way risk (WWR) as the two counterparty default intensities can depend on the derivatives values. Inspired by the work of Ghamami and Goldberg [2014] on th e impact of WWR on credit value adjustment (CVA), we derive calibration-implied formulas that enable us to mathematically compare the derivatives values in the presence and absence of WWR. We illustrate that derivatives values under unilateral WWR need not be less than the derivatives values in the absence of WWR. A sufficient condition under which this inequality holds is that the price process follows a semimartingale with independent increments.

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  • Samim Ghamami, 2015. "Derivatives Pricing under Bilateral Counterparty Risk," Finance and Economics Discussion Series 2015-26, Board of Governors of the Federal Reserve System (U.S.).
  • Handle: RePEc:fip:fedgfe:2015-26
    DOI: 10.17016/FEDS.2015.026
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    References listed on IDEAS

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    1. Duffee, Gregory R, 1999. "Estimating the Price of Default Risk," The Review of Financial Studies, Society for Financial Studies, vol. 12(1), pages 197-226.
    2. Brian Huge & David Lando, 1999. "Swap Pricing with Two-Sided Default Risk in a Rating-Based Model," Review of Finance, European Finance Association, vol. 3(3), pages 239-268.
    3. Azizpour, Shahriar & Giesecke, Kay & Kim, Baeho, 2011. "Premia for correlated default risk," Journal of Economic Dynamics and Control, Elsevier, vol. 35(8), pages 1340-1357, August.
    4. Darrell Duffie & Lasse Heje Pedersen & Kenneth J. Singleton, 2003. "Modeling Sovereign Yield Spreads: A Case Study of Russian Debt," Journal of Finance, American Finance Association, vol. 58(1), pages 119-159, February.
    5. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    6. Samim Ghamami & Lisa R. Goldberg, 2014. "Stochastic Intensity Models of Wrong Way Risk: Wrong Way CVA Need Not Exceed Independent CVA," Finance and Economics Discussion Series 2014-54, Board of Governors of the Federal Reserve System (U.S.).
    7. Duffie, Darrell & Huang, Ming, 1996. "Swap Rates and Credit Quality," Journal of Finance, American Finance Association, vol. 51(3), pages 921-949, July.
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    Keywords

    Basel III; Counterparty Risk; Credit Value Adjustment; Reduced-Form Modeling; Wrong Way Risk;
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