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Unifying pricing formula for several stochastic volatility models with jumps

Author

Listed:
  • Falko Baustian
  • Milan Mrázek
  • Jan Pospíšil
  • Tomáš Sobotka

Abstract

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts. Copyright © 2017 John Wiley & Sons, Ltd.

Suggested Citation

  • 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.
    • Handle: RePEc:wly:apsmbi:v:33:y:2017:i:4:p:422-442
    DOI: 10.1002/asmb.2248
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    Cited by:

    1. Jan Pospíšil & Tomáš Sobotka & Philipp Ziegler, 2019. "Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure," Empirical Economics, Springer, vol. 57(6), pages 1935-1958, December.
    2. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Tommi Sottinen & Josep Vives, 2019. "Decomposition formula for rough Volterra stochastic volatility models," Papers 1906.07101, arXiv.org, revised Aug 2019.
    3. 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.
    4. Raul Merino & Jan Posp'iv{s}il & Tom'av{s} Sobotka & Josep Vives, 2019. "Decomposition formula for jump diffusion models," Papers 1906.06930, arXiv.org.
    5. 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.
    6. Jan Posp'iv{s}il & Vladim'ir v{S}v'igler, 2019. "Isogeometric analysis in option pricing," Papers 1910.00258, arXiv.org.
    7. Jan Posp'iv{s}il & Tom'av{s} Sobotka & Philipp Ziegler, 2019. "Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure," Papers 1912.06709, arXiv.org.

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