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Approximate option pricing under a two-factor Heston–Kou stochastic volatility model

Author

Listed:
  • Youssef El-Khatib

    (United Arab Emirates University)

  • Zororo S. Makumbe

    (Universitat de Barcelona)

  • Josep Vives

    (Universitat de Barcelona)

Abstract

Under a two-factor stochastic volatility jump (2FSVJ) model we obtain an exact decomposition formula for a plain vanilla option price and a second-order approximation of this formula, using Itô calculus techniques. The 2FSVJ model is a generalization of several models described in the literature such as Heston (Rev Financ Stud 6(2):327–343, 1993); Bates (Rev Financ Stud 9(1):69–107, 1996); Kou (Manag Sci 48(8):1086–1101, 2002); Christoffersen et al. (Manag Sci 55(12):1914–1932, 2009) models. Thus, the aim of this study is to extend some approximate pricing formulas described in the literature, like formulas in Alòs (Finance Stoch 16(3):403–422, 2012); Merino et al. (Int J Theor Appl Finance 21(08):1850052, 2018); Gulisashvili et al. (J Comput Finance 24(1), 2020), to pricing under the more general 2FSVJ model. Moreover, we provide numerical illustrations of our pricing method and its accuracy and computational advantage under double exponential and log-normal jumps. Numerically, our pricing method performs very well compared to the Fourier integral method. The performance is ideal for out-of-the-money options as well as for short maturities.

Suggested Citation

  • Youssef El-Khatib & Zororo S. Makumbe & Josep Vives, 2024. "Approximate option pricing under a two-factor Heston–Kou stochastic volatility model," Computational Management Science, Springer, vol. 21(1), pages 1-28, June.
  • Handle: RePEc:spr:comgts:v:21:y:2024:i:1:d:10.1007_s10287-023-00486-8
    DOI: 10.1007/s10287-023-00486-8
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    References listed on IDEAS

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    1. S. G. Kou, 2002. "A Jump-Diffusion Model for Option Pricing," Management Science, INFORMS, vol. 48(8), pages 1086-1101, August.
    2. Elisa Alòs & Rafael De Santiago & Josep Vives, 2015. "Calibration Of Stochastic Volatility Models Via Second-Order Approximation: The Heston Case," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(06), pages 1-31.
    3. Peter Christoffersen & Steven Heston & Kris Jacobs, 2009. "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work So Well," Management Science, INFORMS, vol. 55(12), pages 1914-1932, December.
    4. Raúl Merino & Jan Pospíšil & Tomáš Sobotka & Tommi Sottinen & Josep Vives, 2021. "Decomposition Formula For Rough Volterra Stochastic Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 24(02), pages 1-47, March.
    5. Raúl Merino & Josep Vives, 2017. "Option Price Decomposition in Spot-Dependent Volatility Models and Some Applications," International Journal of Stochastic Analysis, Hindawi, vol. 2017, pages 1-16, July.
    6. 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.
    7. Raúl Merino & Josep Vives, 2015. "A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models," International Journal of Stochastic Analysis, Hindawi, vol. 2015, pages 1-11, June.
    8. R. Merino & J. Pospíšil & T. Sobotka & J. Vives, 2018. "Decomposition Formula For Jump Diffusion Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-36, December.
    9. Falko Baustian & Milan Mrázek & Jan Pospíšil & Tomáš Sobotka, 2017. "Unifying pricing formula for several stochastic volatility models with jumps," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 33(4), pages 422-442, August.
    10. Pacati, Claudio & Pompa, Gabriele & Renò, Roberto, 2018. "Smiling twice: The Heston++ model," Journal of Banking & Finance, Elsevier, vol. 96(C), pages 185-206.
    11. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    12. R. Merino & J. Pospíšil & T. Sobotka & J. Vives, 2018. "Decomposition Formula For Jump Diffusion Models," Journal of Enterprising Culture (JEC), World Scientific Publishing Co. Pte. Ltd., vol. 21(08), pages 1-36, December.
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