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Article

A Multi-Factor Cross-Currency LIBOR Market Model

Wolfgang Benner, Lyudmil Zyapkov and Stephan Jortzik
The Journal of Derivatives Summer 2009, 16 (4) 53-71; DOI: https://doi.org/10.3905/JOD.2009.16.4.053
Wolfgang Benner
is professor emeritus at the Institute of Finance and Banking at the University of Goettingen in Goettingen, Germany.
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  • For correspondence: ifbg@uni-goettingen.de
Lyudmil Zyapkov
is a model analyst at Risk-Capital Markets at BNP Paribas in London, U.K.
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  • For correspondence: lyudmil.zyapkov@uk.bnpparibas.com
Stephan Jortzik
is Director of European Structured Finance at FitchRatings in Frankfurt am Main, Germany.
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  • For correspondence: stephan.jortzik@fitchratings.com
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Abstract

The authors develop a rigorous two-currency pricing framework that can be constructed under either a domestic or a foreign currency numeraire. While plain vanilla interest rate derivative prices are recovered by design, exotic cross-currency interest rate products can be priced by determining no-arbitrage drifts for both the domestic and the foreign LIBORs under a uniform probability measure and by specifying the dynamics of the domestic and foreign currency leg of the exotic product. In a single-currency world, no-arbitrage drifts can always be found by specifying the evolution of the terminal LIBOR as a function of bond price volatilities, first, and solving for the drifts of all remaining LIBORs by backward induction. After introducing a second currency, we show that traditional backward induction for the second currency must fail due to interdependence between the respective bond price volatilities and LIBOR dynamics. In order to resolve any such interdependence, we propose calibrating the volatility function of the spot exchange rate to the terminal maturity spectrum of FX options and specifying a functional form for all dates prior to the terminal one. By choosing a multi-factor model setup, rather than relying on terminal decorrelation within a single-factor model, we allow for model calibration to an exogenous market correlation mix. Extending the model, we outline modifications to account for volatility skews by introducing displaced-diffusion to the LIBOR and FX rate dynamics.

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A Multi-Factor Cross-Currency LIBOR Market Model
Wolfgang Benner, Lyudmil Zyapkov, Stephan Jortzik
The Journal of Derivatives May 2009, 16 (4) 53-71; DOI: 10.3905/JOD.2009.16.4.053

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A Multi-Factor Cross-Currency LIBOR Market Model
Wolfgang Benner, Lyudmil Zyapkov, Stephan Jortzik
The Journal of Derivatives May 2009, 16 (4) 53-71; DOI: 10.3905/JOD.2009.16.4.053
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  • Article
    • Abstract
    • A CROSS-CURRENCY LIBOR MARKET MODEL SPECIFICATION
    • CROSS-CURRENCY INTEREST RATE PRODUCTS UNDER THE CROSS-CURRENCY LIBOR MARKET MODEL SETUP
    • DOMESTIC AND FOREIGN LIBOR DYNAMICS USING BACKWARD INDUCTION UNDER A DOMESTIC NUMERAIRE
    • CALIBRATING THE FX RATE VOLATILITY STRUCTURE
    • CALIBRATING THE FX RATE DRIFT
    • A SIMULATION ALGORITHM
    • ACCOUNTING FOR VOLATILITY SMILES
    • CONCLUSION
    • APPENDIX A
    • APPENDIX B
    • APPENDIX C
    • APPENDIX D
    • APPENDIX E
    • APPENDIX F
    • APPENDIX G
    • ENDNOTE
    • REFERENCES
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