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Cva And Fva To Derivatives Trades Collateralized By Cash

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  • LIXIN WU

    (Department of Mathematics, University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong)

Abstract

In this paper, we consider replication pricing of derivatives that are partially collateralized by cash. We let issuer replicate the derivatives payout using shares and cash, and let buyer replicate the loss given the counterparty default using credit default swaps. The costs of funding for replication and collateral posting are taken into account in the pricing process. A partial differential equation (PDE) for the derivatives price is established, and its solution is provided in a Feynman–Kac formula, which decomposes the derivatives value into the risk-free value of the derivative plus credit valuation adjustment (CVA) and funding valuation adjustment (FVA). For most derivatives, we show that CVAs can be evaluated analytically or semi-analytically, while FVAs as well as the derivatives values can be solved recursively through numerical procedures due to their interdependence. In numerical demonstrations, continuous and discrete margin revisions are considered, respectively, for an equity call option and a vanilla interest-rate swap.

Suggested Citation

  • Lixin Wu, 2015. "Cva And Fva To Derivatives Trades Collateralized By Cash," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-22.
    • Handle: RePEc:wsi:ijtafx:v:18:y:2015:i:05:n:s0219024915500351
    DOI: 10.1142/S0219024915500351
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    References listed on IDEAS

    as
    1. Yi Tang & Bin Li, 2007. "Quantitative Analysis, Derivatives Modeling, and Trading Strategies:In the Presence of Counterparty Credit Risk for the Fixed-Income Market," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4228, January.
    2. Masaaki Fujii & Akihiko Takahashi, 2010. "Choice of Collateral Currency," CARF F-Series CARF-F-239, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2011. "Funding Valuation Adjustment: a consistent framework including CVA, DVA, collateral,netting rules and re-hypothecation," Papers 1112.1521, arXiv.org, revised Dec 2011.
    4. Damiano Brigo & Agostino Capponi & Andrea Pallavicini & Vasileios Papatheodorou, 2011. "Collateral Margining in Arbitrage-Free Counterparty Valuation Adjustment including Re-Hypotecation and Netting," Papers 1101.3926, arXiv.org.
    5. Masaaki Fujii & Akihiko Takahashi, 2010. "Choice of Collateral Currency," CIRJE F-Series CIRJE-F-778, CIRJE, Faculty of Economics, University of Tokyo.
    6. Fries, Christian P., 2010. "Discounting Revisited. Valuations under Funding Costs, Counterparty Risk and Collateralization," MPRA Paper 23082, University Library of Munich, Germany, revised 30 May 2010.
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    Cited by:

    1. Lixin Wu & Dawei Zhang, 2020. "xVA: DEFINITION, EVALUATION AND RISK MANAGEMENT," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-24, February.
    2. BRIGO, Damiano & VRINS, Frédéric, 2018. "Disentangling wrong-way risk: pricing credit valuation adjustment via change of measures," European Journal of Operational Research, Elsevier, vol. 269(3), pages 1154-1164.
    3. Junbeom Lee & Chao Zhou, 2017. "Binary Funding Impacts in Derivative Valuation," Papers 1703.00259, arXiv.org, revised Aug 2020.
    4. Ballotta, Laura & Fusai, Gianluca & Marazzina, Daniele, 2019. "Integrated structural approach to Credit Value Adjustment," European Journal of Operational Research, Elsevier, vol. 272(3), pages 1143-1157.
    5. Junbeom Lee & Chao Zhou, 2021. "Binary funding impacts in derivative valuation," Mathematical Finance, Wiley Blackwell, vol. 31(1), pages 242-278, January.

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