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Using copulas to construct bivariate foreign exchange distributions with an application to the sterling exchange rate index


  • Matthew Hurd
  • Mark Salmon
  • Christoph Schleicher


We model the joint risk-neutral distribution of the euro-sterling and the dollar-sterling exchange rates using option-implied marginal distributions that are connected via a copula function that satisfies the triangular no-arbitrage condition. We then derive a univariate distribution for a simplified sterling effective exchange rate index. Our results indicate that standard parametric copula functions, such as the commonly used Normal and Frank copulas, fail to capture the degree of asymmetry observed in the data. We overcome this problem by using a non-parametric dependence function in the form of a Bernstein copula which is shown to produce a very close fit. We further give an example of how our approach can be used to price currency index options.

Suggested Citation

  • Matthew Hurd & Mark Salmon & Christoph Schleicher, 2007. "Using copulas to construct bivariate foreign exchange distributions with an application to the sterling exchange rate index," Bank of England working papers 334, Bank of England.
  • Handle: RePEc:boe:boeewp:334

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    References listed on IDEAS

    1. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    2. Joshua V. Rosenberg, 2003. "Nonparametric pricing of multivariate contingent claims," Staff Reports 162, Federal Reserve Bank of New York.
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    4. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    5. Sancetta, A., 2003. "Nonparametric Estimation of Multivariate Distributions with Given Marginals," Cambridge Working Papers in Economics 0320, Faculty of Economics, University of Cambridge.
    6. Robert R. Bliss & Nikolaos Panigirtzoglou, 2004. "Option-Implied Risk Aversion Estimates," Journal of Finance, American Finance Association, vol. 59(1), pages 407-446, February.
    7. Bliss, Robert R. & Panigirtzoglou, Nikolaos, 2002. "Testing the stability of implied probability density functions," Journal of Banking & Finance, Elsevier, vol. 26(2-3), pages 381-422, March.
    8. Sancetta, Alessio & Satchell, Stephen, 2004. "The Bernstein Copula And Its Applications To Modeling And Approximations Of Multivariate Distributions," Econometric Theory, Cambridge University Press, vol. 20(03), pages 535-562, June.
    9. Andrew J. Patton, 2006. "Modelling Asymmetric Exchange Rate Dependence," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 47(2), pages 527-556, May.
    10. Garman, Mark B. & Kohlhagen, Steven W., 1983. "Foreign currency option values," Journal of International Money and Finance, Elsevier, vol. 2(3), pages 231-237, December.
    11. Eytan, T Hanan & Harpaz, Giora, 1986. " The Pricing of Futures and Options Contracts on the Value Line Index," Journal of Finance, American Finance Association, vol. 41(4), pages 843-855, September.
    12. Vivek Bhargava & John M. Clark, 2003. "Pricing U.S. Dollar Index Futures Options: An Empirical Investigation," The Financial Review, Eastern Finance Association, vol. 38(4), pages 571-590, November.
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    Cited by:

    1. repec:eee:phsmap:v:486:y:2017:i:c:p:595-609 is not listed on IDEAS
    2. Alvise De Col & Alessandro Gnoatto & Martino Grasselli, 2012. "Smiles all around: FX joint calibration in a multi-Heston model," Papers 1201.1782,, revised Jun 2013.
    3. Mendoza-Velázquez, Alfonso & Galvanovskis, Evalds, 2009. "Introducing the GED-Copula with an application to Financial Contagion in Latin America," MPRA Paper 46669, University Library of Munich, Germany, revised 01 Feb 2010.
    4. Fabrizio Durante & Erich Klement & Carlo Sempi & Manuel Úbeda-Flores, 2010. "Measures of non-exchangeability for bivariate random vectors," Statistical Papers, Springer, vol. 51(3), pages 687-699, September.
    5. Zhichao Zhang & Li Ding & Fan Zhang & Zhuang Zhang, 2015. "Optimal Currency Composition for China's Foreign Reserves: A Copula Approach," The World Economy, Wiley Blackwell, vol. 38(12), pages 1947-1965, December.
    6. Mark Salmon & Christoph Schleicher, 2006. "Pricing Multivariate Currency Options with Copulas," Working Papers wpn06-10, Warwick Business School, Finance Group.
    7. Tavin, Bertrand, 2015. "Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 158-178.
    8. De Col, Alvise & Gnoatto, Alessandro & Grasselli, Martino, 2013. "Smiles all around: FX joint calibration in a multi-Heston model," Journal of Banking & Finance, Elsevier, vol. 37(10), pages 3799-3818.
    9. Mendoza, Alfonso. & Galvanovskis, Evalds., 2014. "La cópula GED bivariada. Una aplicación en entornos de crisis," El Trimestre Económico, Fondo de Cultura Económica, vol. 0(323), pages .721-746, julio-sep.
    10. Alessandro Gnoatto & Martino Grasselli, 2013. "An analytic multi-currency model with stochastic volatility and stochastic interest rates," Papers 1302.7246,, revised Mar 2013.

    More about this item

    JEL classification:

    • F31 - International Economics - - International Finance - - - Foreign Exchange
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates


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