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Cross Currency Valuation and Hedging in the Multiple Curve Framework

Author

Listed:
  • Alessandro Gnoatto
  • Nicole Seiffert
Abstract
We generalize the results of Bielecki and Rutkowski (2015) on funding and collateralization to a multi-currency framework and link their results with those of Piterbarg (2012), Moreni and Pallavicini (2017), and Fujii et al. (2010b). In doing this, we provide a complete study of absence of arbitrage in a multi-currency market where, in each single monetary area, multiple interest rates coexist. We first characterize absence of arbitrage in the case without collateral. After that we study collateralization schemes in a very general situation: the cash flows of the contingent claim and those associated to the collateral agreement can be specified in any currency. We study both segregation and rehypothecation and allow for cash and risky collateral in arbitrary currency specifications. Absence of arbitrage and pricing in the presence of collateral are discussed under all possible combinations of conventions. Our work provides a reference for the analysis of wealth dynamics, we also provide valuation formulas that are a useful foundation for cross-currency curve construction techniques. Our framework provides also a solid foundation for the construction of multi-currency simulation models for the generation of exposure profiles in the context of xVA calculations.

Suggested Citation

  • Alessandro Gnoatto & Nicole Seiffert, 2020. "Cross Currency Valuation and Hedging in the Multiple Curve Framework," Papers 2001.11012, arXiv.org, revised Mar 2021.
    • Handle: RePEc:arx:papers:2001.11012
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    References listed on IDEAS

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    1. St�phane Cr�pey & Zorana Grbac & Nathalie Ngor & David Skovmand, 2015. "A L�vy HJM multiple-curve model with application to CVA computation," Quantitative Finance, Taylor & Francis Journals, vol. 15(3), pages 401-419, March.
    2. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2016. "A general HJM framework for multiple yield curve modelling," Finance and Stochastics, Springer, vol. 20(2), pages 267-320, April.
    3. Francesca Biagini & Alessandro Gnoatto & Immacolata Oliva, 2019. "A unified approach to xVA with CSA discounting and initial margin," Papers 1905.11328, arXiv.org, revised Mar 2021.
    4. Damiano Brigo & Cristin Buescu & Andrea Pallavicini & Qing Liu, 2015. "A Note On The Self-Financing Condition For Funding, Collateral And Discounting," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-10.
    5. Francesca Biagini & Alessandro Gnoatto & Immacolata Oliva, 2019. "Pricing of counterparty risk and funding with CSA discounting, portfolio effects and initial margin," Working Papers 04/2019, University of Verona, Department of Economics.
    6. Andrea Pallavicini & Daniele Perini & Damiano Brigo, 2012. "Funding, Collateral and Hedging: uncovering the mechanics and the subtleties of funding valuation adjustments," Papers 1210.3811, arXiv.org, revised Dec 2012.
    7. N. Moreni & A. Pallavicini, 2014. "Parsimonious HJM modelling for multiple yield curve dynamics," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 199-210, February.
    8. Zorana Grbac & Antonis Papapantoleon & John Schoenmakers & David Skovmand, 2014. "Affine LIBOR models with multiple curves: theory, examples and calibration," Papers 1405.2450, arXiv.org, revised Aug 2015.
    9. Fabio Mercurio, 2010. "Modern Libor Market Models: Using Different Curves For Projecting Rates And For Discounting," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(01), pages 113-137.
    10. Masaaki Fujii & Yasufumi Shimada & Akihiko Takahashi, 2009. "A Market Model of Interest Rates with Dynamic Basis Spreads in the presence of Collateral and Multiple Currencies," CARF F-Series CARF-F-196, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2011.
    11. Henrard, Marc, 2007. "The irony in the derivatives discounting," MPRA Paper 3115, University Library of Munich, Germany.
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    13. Christa Cuchiero & Claudio Fontana & Alessandro Gnoatto, 2019. "Affine multiple yield curve models," Mathematical Finance, Wiley Blackwell, vol. 29(2), pages 568-611, April.
    14. Damiano Brigo & Andrea Pallavicini, 2014. "Nonlinear consistent valuation of CCP cleared or CSA bilateral trades with initial margins under credit, funding and wrong-way risks," Journal of Financial Engineering (JFE), World Scientific Publishing Co. Pte. Ltd., vol. 1(01), pages 1-60.
    15. Nicola Moreni & Andrea Pallavicini, 2017. "Derivative Pricing With Collateralization And Fx Market Dislocations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(06), pages 1-27, September.
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    Cited by:

    1. Alessandro Gnoatto & Athena Picarelli & Christoph Reisinger, 2020. "Deep xVA solver -- A neural network based counterparty credit risk management framework," Papers 2005.02633, arXiv.org, revised Dec 2022.
    2. Marek Rutkowski & Matthew Bickersteth, 2021. "Pricing and Hedging of SOFR Derivatives under Differential Funding Costs and Collateralization," Papers 2112.14033, arXiv.org.
    3. Joel P. Villarino & 'Alvaro Leitao & Jos'e A. Garc'ia-Rodr'iguez, 2022. "Boundary-safe PINNs extension: Application to non-linear parabolic PDEs in counterparty credit risk," Papers 2210.02175, arXiv.org.
    4. Luca Di Persio & Alessandro Gnoatto & Marco Patacca, 2021. "A change of measure formula for recursive conditional expectations," Papers 2111.08359, arXiv.org, revised Jul 2022.

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