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An Equity-Interest Rate Hybrid Model With Stochastic Volatility and the Interest Rate Smile

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  • Grzelak, Lech
  • Oosterlee, Kees

Abstract

We define an equity-interest rate hybrid model in which the equity part is driven by the Heston stochastic volatility [Hes93], and the interest rate (IR) is generated by the displaced-diffusion stochastic volatility Libor Market Model [AA02]. We assume a non-zero correlation between the main processes. By an appropriate change of measure the dimension of the corresponding pricing PDE can be greatly reduced. We place by a number of approximations the model in the class of affine processes [DPS00], for which we then provide the corresponding forward characteristic function. We discuss in detail the accuracy of the approximations and the efficient calibration. Finally, by experiments, we show the effect of the correlations and interest rate smile/skew on typical equity-interest rate hybrid product prices. For a whole strip of strikes this approximate hybrid model can be evaluated for equity plain vanilla options in just milliseconds.

Suggested Citation

  • Grzelak, Lech & Oosterlee, Kees, 2010. "An Equity-Interest Rate Hybrid Model With Stochastic Volatility and the Interest Rate Smile," MPRA Paper 20574, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:20574
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    References listed on IDEAS

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    1. Lixin Wu & Fan Zhang, 2008. "Fast swaption pricing under the market model with a square-root volatility process," Quantitative Finance, Taylor & Francis Journals, vol. 8(2), pages 163-180.
    2. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    3. John C. Cox & Jonathan E. Ingersoll Jr. & Stephen A. Ross, 2005. "A Theory Of The Term Structure Of Interest Rates," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 5, pages 129-164, World Scientific Publishing Co. Pte. Ltd..
    4. Darrell Duffie & Jun Pan & Kenneth Singleton, 2000. "Transform Analysis and Asset Pricing for Affine Jump-Diffusions," Econometrica, Econometric Society, vol. 68(6), pages 1343-1376, November.
    5. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    6. Erik Schlögl, 2001. "Arbitrage-Free Interpolation in Models of Market Observable Interest Rates," Research Paper Series 71, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Fang, Fang & Oosterlee, Kees, 2008. "A Novel Pricing Method For European Options Based On Fourier-Cosine Series Expansions," MPRA Paper 9319, University Library of Munich, Germany.
    8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    9. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-430, March.
    10. Grzelak, Lech & Oosterlee, Kees, 2009. "On The Heston Model with Stochastic Interest Rates," MPRA Paper 20620, University Library of Munich, Germany, revised 18 Jan 2010.
    11. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    12. Alan Brace & Dariusz G¸atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155, April.
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    Cited by:

    1. 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.
    2. 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).

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    More about this item

    Keywords

    hybrid models; Heston equity model; Libor Market Model with stochastic volatility; displaced diffusion; affine diffusion; fast calibration.;
    All these keywords.

    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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