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On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates

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  • Lech A. Grzelak
  • Cornelis W. Oosterlee

Abstract

We construct multi-currency models with stochastic volatility (SV) and correlated stochastic interest rates with a full matrix of correlations. We first deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull--White (Hull, J. and White, A. [1990] Pricing interest-rate derivative securities, Review of Financial Studies , 3, pp. 573--592). We then extend the framework by modelling the interest rate by an SV displaced-diffusion (DD) Libor Market Model (Andersen, L. B. G. and Andreasen, J. [2000] Volatility skews and extensions of the libor market model, Applied Mathematics Finance , 1[7], pp. 1--32), which can model an interest rate smile. We provide semi-closed form approximations which lead to efficient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity--FX--interest rate hybrid pay-offs.

Suggested Citation

  • Lech A. Grzelak & Cornelis W. Oosterlee, 2012. "On Cross-Currency Models with Stochastic Volatility and Correlated Interest Rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 19(1), pages 1-35, February.
    • Handle: RePEc:taf:apmtfi:v:19:y:2012:i:1:p:1-35
    DOI: 10.1080/1350486X.2011.570492
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    Cited by:

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    2. Kenji Nagami, 2019. "Expansion method for pricing foreign exchange options under stochastic volatility and interest rates," Papers 1908.09640, arXiv.org.
    3. Rehez Ahlip & Laurence A. F. Park & Ante Prodan, 2017. "Pricing currency options in the Heston/CIR double exponential jump-diffusion model," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-30, March.
    4. Kirkby, J. Lars, 2023. "Hybrid equity swap, cap, and floor pricing under stochastic interest by Markov chain approximation," European Journal of Operational Research, Elsevier, vol. 305(2), pages 961-978.
    5. Fazlollah Soleymani & Andrey Itkin, 2019. "Pricing foreign exchange options under stochastic volatility and interest rates using an RBF--FD method," Papers 1903.00937, arXiv.org.
    6. M. Briani & L. Caramellino & A. Zanette, 2015. "A hybrid tree/finite-difference approach for Heston-Hull-White type models," Papers 1503.03705, arXiv.org, revised Dec 2017.
    7. Benjamin Tin Chun Cheng, 2017. "Pricing and Hedging of Long-Dated Commodity Derivatives," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 2-2017.
    8. repec:uts:finphd:37 is not listed on IDEAS
    9. Ascione, Giacomo & Mehrdoust, Farshid & Orlando, Giuseppe & Samimi, Oldouz, 2023. "Foreign Exchange Options on Heston-CIR Model Under Lévy Process Framework," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    10. Yijuan Liang & Xiuchuan Xu, 2019. "Variance and Dimension Reduction Monte Carlo Method for Pricing European Multi-Asset Options with Stochastic Volatilities," Sustainability, MDPI, vol. 11(3), pages 1-21, February.
    11. Lai, Van Son & Parcollet, Mathieu & Lamond, Bernard F., 2014. "The valuation of catastrophe bonds with exposure to currency exchange risk," International Review of Financial Analysis, Elsevier, vol. 33(C), pages 243-252.
    12. Simonella, Roberta & Vázquez, Carlos, 2023. "XVA in a multi-currency setting with stochastic foreign exchange rates," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 59-79.
    13. Griselda Deelstra & Gr�gory Ray�e, 2013. "Local Volatility Pricing Models for Long-Dated FX Derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 20(4), pages 380-402, September.
    14. Anatoliy Swishchuk & Maksym Tertychnyi & Robert Elliott, 2014. "Pricing Currency Derivatives with Markov-modulated Levy Dynamics," Papers 1402.1953, arXiv.org.
    15. Grzelak, Lech A., 2022. "Sparse grid method for highly efficient computation of exposures for xVA," Applied Mathematics and Computation, Elsevier, vol. 434(C).
    16. Ahmet Akca & Ethem Çanakoğlu, 2021. "Adaptive stochastic risk estimation of firm operating profit," Economia e Politica Industriale: Journal of Industrial and Business Economics, Springer;Associazione Amici di Economia e Politica Industriale, vol. 48(3), pages 463-504, September.
    17. Lech A. Grzelak, 2021. "Sparse Grid Method for Highly Efficient Computation of Exposures for xVA," Papers 2104.14319, arXiv.org, revised May 2022.
    18. Benjamin Cheng & Christina Nikitopoulos-Sklibosios & Erik Schlogl, 2015. "Pricing of Long-dated Commodity Derivatives with Stochastic Volatility and Stochastic Interest Rates," Research Paper Series 366, Quantitative Finance Research Centre, University of Technology, Sydney.
    19. P. Karlsson & K. F. Pilz & E. Schlögl, 2017. "Calibrating a market model with stochastic volatility to commodity and interest rate risk," Quantitative Finance, Taylor & Francis Journals, vol. 17(6), pages 907-925, June.
    20. Singor, Stefan N. & Grzelak, Lech A. & van Bragt, David D.B. & Oosterlee, Cornelis W., 2013. "Pricing inflation products with stochastic volatility and stochastic interest rates," Insurance: Mathematics and Economics, Elsevier, vol. 52(2), pages 286-299.
    21. Alessandro Gnoatto & Martino Grasselli, 2013. "An analytic multi-currency model with stochastic volatility and stochastic interest rates," Papers 1302.7246, arXiv.org, revised Mar 2013.
    22. S. Simaitis & C. S. L. de Graaf & N. Hari & D. Kandhai, 2016. "Smile and default: the role of stochastic volatility and interest rates in counterparty credit risk," Quantitative Finance, Taylor & Francis Journals, vol. 16(11), pages 1725-1740, November.

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    More about this item

    JEL classification:

    • G1 - Financial Economics - - General Financial Markets
    • F3 - International Economics - - International Finance
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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