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Valuation equations for stochastic volatility models

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  • Erhan Bayraktar
  • Constantinos Kardaras
  • Hao Xing

Abstract

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset-price is a martingale.

Suggested Citation

  • Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2010. "Valuation equations for stochastic volatility models," Papers 1004.3299, arXiv.org, revised Dec 2011.
    • Handle: RePEc:arx:papers:1004.3299
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    Cited by:

    1. Jan Baldeaux & Alexander Badran, 2014. "Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(4), pages 299-312, September.
    2. Irina Penner & Anthony Réveillac, 2015. "Risk measures for processes and BSDEs," Finance and Stochastics, Springer, vol. 19(1), pages 23-66, January.
    3. Kexin Chen & Hoi Ying Wong, 2024. "Duality in optimal consumption–investment problems with alternative data," Finance and Stochastics, Springer, vol. 28(3), pages 709-758, July.
    4. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Aug 2018.
    5. Irina Penner & Anthony Réveillac, 2014. "Risk measures for processes and BSDEs," Post-Print hal-00814702, HAL.
    6. Dareiotis, Konstantinos & Ekström, Erik, 2019. "Density symmetries for a class of 2-D diffusions with applications to finance," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 452-472.
    7. Baldeaux, Jan & Grasselli, Martino & Platen, Eckhard, 2015. "Pricing currency derivatives under the benchmark approach," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 34-48.
    8. Paul M. N. Feehan & Ruoting Gong & Jian Song, 2015. "Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions," Papers 1509.03864, arXiv.org.
    9. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Papers 1304.4853, arXiv.org.
    10. Irina Penner & Anthony Réveillac, 2013. "Risk measures for processes and BSDEs," Working Papers hal-00814702, HAL.
    11. Xiaoshan Chen & Yu-Jui Huang & Qingshuo Song & Chao Zhu, 2013. "The Stochastic Solution to a Cauchy Problem for Degenerate Parabolic Equations," Papers 1309.0046, arXiv.org, revised Mar 2017.
    12. Kexin Chen & Hoi Ying Wong, 2022. "Duality in optimal consumption--investment problems with alternative data," Papers 2210.08422, arXiv.org, revised Jul 2023.
    13. Keller-Ressel, Martin, 2015. "Simple examples of pure-jump strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4142-4153.
    14. Chen Xiaoshan & Song Qingshuo, 2013. "American option of stochastic volatility model with negative Fichera function on degenerate boundary," Papers 1306.0345, arXiv.org.
    15. Martin Keller-Ressel, 2014. "Simple examples of pure-jump strict local martingales," Papers 1405.2669, arXiv.org, revised Jun 2015.

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    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General

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