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On the density of log-spot in the Heston volatility model

Author

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  • del Baño Rollin, Sebastian
  • Ferreiro-Castilla, Albert
  • Utzet, Frederic
Abstract
This paper proves that the log-spot in the Heston model has a density and gives an expression of this density as an infinite convolution of Bessel type densities. Such properties are deduced from a factorization of the characteristic function, mainly obtained through an analysis of the complex moment generating function. As an application a new algorithm to simulate spot is developed.

Suggested Citation

  • del Baño Rollin, Sebastian & Ferreiro-Castilla, Albert & Utzet, Frederic, 2010. "On the density of log-spot in the Heston volatility model," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 2037-2063, September.
    • Handle: RePEc:eee:spapps:v:120:y:2010:i:10:p:2037-2063
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    References listed on IDEAS

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    1. Adrian Dragulescu & Victor Yakovenko, 2002. "Probability distribution of returns in the Heston model with stochastic volatility," Quantitative Finance, Taylor & Francis Journals, vol. 2(6), pages 443-453.
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    4. Roger Lord & Remmert Koekkoek & Dick Van Dijk, 2010. "A comparison of biased simulation schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 10(2), pages 177-194.
    5. Mark Broadie & Özgür Kaya, 2006. "Exact Simulation of Stochastic Volatility and Other Affine Jump Diffusion Processes," Operations Research, INFORMS, vol. 54(2), pages 217-231, April.
    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. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
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    Citations

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    Cited by:

    1. Arunangshu Biswas & Anindya Goswami & Ludger Overbeck, 2017. "Option Pricing in a Regime Switching Stochastic Volatility Model," Papers 1707.01237, arXiv.org, revised Jan 2018.
    2. Cui, Yiran & del Baño Rollin, Sebastian & Germano, Guido, 2017. "Full and fast calibration of the Heston stochastic volatility model," European Journal of Operational Research, Elsevier, vol. 263(2), pages 625-638.
    3. Archil Gulisashvili & Josep Vives, 2014. "Asymptotic analysis of stock price densities and implied volatilities in mixed stochastic models," Papers 1403.5302, arXiv.org.
    4. Stefano Pagliarani & Andrea Pascucci, 2017. "The exact Taylor formula of the implied volatility," Finance and Stochastics, Springer, vol. 21(3), pages 661-718, July.
    5. Eudald Romo & Luis Ortiz-Gracia, 2021. "SWIFT calibration of the Heston model," Papers 2103.01570, arXiv.org.
    6. Hervé Andres & Pierre-Edouard Arrouy & Paul Bonnefoy & Alexandre Boumezoued & Sophian Mehalla, 2020. "Fast calibration of the LIBOR Market Model with Stochastic Volatility based on analytical gradient," Working Papers hal-02875623, HAL.
    7. Carole Bernard & Zhenyu Cui & Don McLeish, 2013. "On the martingale property in stochastic volatility models based on time-homogeneous diffusions," Papers 1310.0092, arXiv.org, revised Jul 2014.
    8. Coqueret, Guillaume & Tavin, Bertrand, 2016. "An investigation of model risk in a market with jumps and stochastic volatility," European Journal of Operational Research, Elsevier, vol. 253(3), pages 648-658.
    9. Gudmundsson, Hilmar & Vyncke, David, 2019. "On the calibration of the 3/2 model," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1178-1192.
    10. Takashi Kato & Jun Sekine & Kenichi Yoshikawa, 2013. "Order Estimates for the Exact Lugannani-Rice Expansion," Papers 1310.3347, arXiv.org, revised Jun 2014.
    11. Eudald Romo & Luis Ortiz-Gracia, 2021. "SWIFT Calibration of the Heston Model," Mathematics, MDPI, vol. 9(5), pages 1-20, March.
    12. Biswas, Arunangshu & Goswami, Anindya & Overbeck, Ludger, 2018. "Option pricing in a regime switching stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 138(C), pages 116-126.
    13. Jan Baldeaux & Dale Roberts, 2012. "Quasi-Monte Carlo methods for the Heston model," Papers 1202.3217, arXiv.org, revised May 2012.
    14. Herv'e Andres & Pierre-Edouard Arrouy & Paul Bonnefoy & Alexandre Boumezoued & Sophian Mehalla, 2020. "Fast calibration of the LIBOR Market Model with Stochastic Volatility based on analytical gradient," Papers 2006.13521, arXiv.org.

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