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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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    References listed on IDEAS

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    1. Alexander Cox & David Hobson, 2005. "Local martingales, bubbles and option prices," Finance and Stochastics, Springer, vol. 9(4), pages 477-492, October.
    2. Aleksandar Mijatovi'c & Mikhail Urusov, 2011. "A note on a paper by Wong and Heyde," Papers 1105.3918, arXiv.org.
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    5. Robert A. Jarrow, 2015. "Asset Price Bubbles," Annual Review of Financial Economics, Annual Reviews, vol. 7(1), pages 201-218, December.
    6. Wiggins, James B., 1987. "Option values under stochastic volatility: Theory and empirical estimates," Journal of Financial Economics, Elsevier, vol. 19(2), pages 351-372, December.
    7. Erhan Bayraktar & Hao Xing, 2009. "On the uniqueness of classical solutions of Cauchy problems," Papers 0908.1086, arXiv.org, revised Sep 2009.
    8. David Hobson, 2010. "Comparison results for stochastic volatility models via coupling," Finance and Stochastics, Springer, vol. 14(1), pages 129-152, January.
    9. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
    10. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
    11. Erhan Bayraktar & Constantinos Kardaras & Hao Xing, 2009. "Strict Local Martingale Deflators and Pricing American Call-Type Options," Papers 0908.1082, arXiv.org, revised Dec 2009.
    12. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
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    Citations

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    Cited by:

    1. Archil Gulisashvili, 2017. "Large deviation principle for Volterra type fractional stochastic volatility models," Papers 1710.10711, arXiv.org, revised Jan 2018.
    2. 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.
    3. 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.
    4. Keller-Ressel, Martin, 2015. "Simple examples of pure-jump strict local martingales," Stochastic Processes and their Applications, Elsevier, vol. 125(11), pages 4142-4153.
    5. 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.
    6. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Working Papers hal-00814702, HAL.
    7. Irina Penner & Anthony Réveillac, 2015. "Risk measures for processes and BSDEs," Finance and Stochastics, Springer, vol. 19(1), pages 23-66, January.
    8. Irina Penner & Anthony Reveillac, 2013. "Risk measures for processes and BSDEs," Papers 1304.4853, arXiv.org.
    9. Martin Keller-Ressel, 2014. "Simple examples of pure-jump strict local martingales," Papers 1405.2669, arXiv.org, revised Jun 2015.
    10. Chen Xiaoshan & Song Qingshuo, 2013. "American option of stochastic volatility model with negative Fichera function on degenerate boundary," Papers 1306.0345, arXiv.org.

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    JEL classification:

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

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