IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1908.07417.html
   My bibliography  Save this paper

A lognormal type stochastic volatility model with quadratic drift

Author

Listed:
  • Peter Carr
  • Sander Willems

Abstract

This paper presents a novel one-factor stochastic volatility model where the instantaneous volatility of the asset log-return is a diffusion with a quadratic drift and a linear dispersion function. The instantaneous volatility mean reverts around a constant level, with a speed of mean reversion that is affine in the instantaneous volatility level. The steady-state distribution of the instantaneous volatility belongs to the class of Generalized Inverse Gaussian distributions. We show that the quadratic term in the drift is crucial to avoid moment explosions and to preserve the martingale property of the stock price process. Using a conveniently chosen change of measure, we relate the model to the class of polynomial diffusions. This remarkable relation allows us to develop a highly accurate option price approximation technique based on orthogonal polynomial expansions.

Suggested Citation

  • Peter Carr & Sander Willems, 2019. "A lognormal type stochastic volatility model with quadratic drift," Papers 1908.07417, arXiv.org.
    • Handle: RePEc:arx:papers:1908.07417
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1908.07417
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Andersen T. G & Bollerslev T. & Diebold F. X & Labys P., 2001. "The Distribution of Realized Exchange Rate Volatility," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 42-55, March.
    2. Nelson, Daniel B., 1990. "ARCH models as diffusion approximations," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 7-38.
    3. Filipović, Damir & Gourier, Elise & Mancini, Loriano, 2016. "Quadratic variance swap models," Journal of Financial Economics, Elsevier, vol. 119(1), pages 44-68.
    4. Damir Filipović & Sander Willems, 2017. "A Term Structure Model for Dividends and Interest Rates," Swiss Finance Institute Research Paper Series 17-52, Swiss Finance Institute.
    5. Robert C. Merton, 1975. "An Asymptotic Theory of Growth Under Uncertainty," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 42(3), pages 375-393.
    6. Damir Filipovic & Damien Ackerer & Sergio Pulido, 2018. "The Jacobi Stochastic Volatility Model," Post-Print hal-01338330, HAL.
    7. Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.
    8. Andersen, Torben G. & Bollerslev, Tim & Diebold, Francis X. & Ebens, Heiko, 2001. "The distribution of realized stock return volatility," Journal of Financial Economics, Elsevier, vol. 61(1), pages 43-76, July.
    9. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    10. Peter Christoffersen & Kris Jacobs & Karim Mimouni, 2010. "Volatility Dynamics for the S&P500: Evidence from Realized Volatility, Daily Returns, and Option Prices," The Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 3141-3189, August.
    11. Damien Ackerer & Damir Filipovi'c, 2016. "Linear Credit Risk Models," Papers 1605.07419, arXiv.org, revised Jul 2019.
    12. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching to nonaffine stochastic volatility: a closed-form expansion for the Inverse Gamma model," Post-Print hal-02909113, HAL.
    13. Alan L. Lewis, 2018. "Exact Solutions for a GBM-type Stochastic Volatility Model having a Stationary Distribution," Papers 1809.08635, arXiv.org, revised May 2019.
    14. 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.
    15. Damien Ackerer & Damir Filipović & Sergio Pulido, 2018. "The Jacobi stochastic volatility model," Finance and Stochastics, Springer, vol. 22(3), pages 667-700, July.
    16. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    17. Nicolas Langren'e & Geoffrey Lee & Zili Zhu, 2015. "Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model," Papers 1507.02847, arXiv.org, revised Mar 2016.
    18. Sander Willems, 2019. "Asian option pricing with orthogonal polynomials," Quantitative Finance, Taylor & Francis Journals, vol. 19(4), pages 605-618, April.
    19. Bakshi, Gurdip & Ju, Nengjiu & Ou-Yang, Hui, 2006. "Estimation of continuous-time models with an application to equity volatility dynamics," Journal of Financial Economics, Elsevier, vol. 82(1), pages 227-249, October.
    20. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
    21. Damien Ackerer & Damir Filipovi'c & Sergio Pulido, 2016. "The Jacobi Stochastic Volatility Model," Papers 1605.07099, arXiv.org, revised Mar 2018.
    22. 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.
    23. Barone-Adesi, Giovanni & Rasmussen, Henrik & Ravanelli, Claudia, 2005. "An option pricing formula for the GARCH diffusion model," Computational Statistics & Data Analysis, Elsevier, vol. 49(2), pages 287-310, April.
    24. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    25. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.
    26. Christian Kahl & Peter Jackel, 2006. "Fast strong approximation Monte Carlo schemes for stochastic volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 6(6), pages 513-536.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hyungbin Park, 2021. "Influence of risk tolerance on long-term investments: A Malliavin calculus approach," Papers 2104.00911, arXiv.org.
    2. Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.
    3. Kaustav Das & Nicolas Langren'e, 2020. "Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework," Papers 2006.01542, arXiv.org, revised Jan 2024.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Kaustav Das & Nicolas Langren'e, 2020. "Explicit approximations of option prices via Malliavin calculus in a general stochastic volatility framework," Papers 2006.01542, arXiv.org, revised Jan 2024.
    2. Kaustav Das & Nicolas Langren'e, 2018. "Closed-form approximations with respect to the mixing solution for option pricing under stochastic volatility," Papers 1812.07803, arXiv.org, revised Oct 2021.
    3. Nicolas Langrené & Geoffrey Lee & Zili Zhu, 2016. "Switching To Nonaffine Stochastic Volatility: A Closed-Form Expansion For The Inverse Gamma Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(05), pages 1-37, August.
    4. Pierre-Edouard Arrouy & Sophian Mehalla & Bernard Lapeyre & Alexandre Boumezoued, 2020. "Jacobi Stochastic Volatility factor for the Libor Market Model," Working Papers hal-02468583, HAL.
    5. Damien Ackerer & Damir Filipović, 2020. "Option pricing with orthogonal polynomial expansions," Mathematical Finance, Wiley Blackwell, vol. 30(1), pages 47-84, January.
    6. Cui, Zhenyu & Lars Kirkby, J. & Nguyen, Duy, 2019. "A general framework for time-changed Markov processes and applications," European Journal of Operational Research, Elsevier, vol. 273(2), pages 785-800.
    7. Damir Filipović & Sander Willems, 2020. "A term structure model for dividends and interest rates," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1461-1496, October.
    8. Nicolas Langren'e & Geoffrey Lee & Zili Zhu, 2015. "Switching to non-affine stochastic volatility: A closed-form expansion for the Inverse Gamma model," Papers 1507.02847, arXiv.org, revised Mar 2016.
    9. Damien Ackerer & Damir Filipovic, 2017. "Option Pricing with Orthogonal Polynomial Expansions," Papers 1711.09193, arXiv.org, revised May 2019.
    10. Wu, Xin-Yu & Ma, Chao-Qun & Wang, Shou-Yang, 2012. "Warrant pricing under GARCH diffusion model," Economic Modelling, Elsevier, vol. 29(6), pages 2237-2244.
    11. Tong, Zhigang & Liu, Allen, 2022. "Pricing variance swaps under subordinated Jacobi stochastic volatility models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    12. Christa Cuchiero & Sara Svaluto-Ferro, 2021. "Infinite-dimensional polynomial processes," Finance and Stochastics, Springer, vol. 25(2), pages 383-426, April.
    13. Kristensen, Dennis & Mele, Antonio, 2011. "Adding and subtracting Black-Scholes: A new approach to approximating derivative prices in continuous-time models," Journal of Financial Economics, Elsevier, vol. 102(2), pages 390-415.
    14. Pierre-Edouard Arrouy & Alexandre Boumezoued & Bernard Lapeyre & Sophian Mehalla, 2022. "Jacobi stochastic volatility factor for the LIBOR market model," Finance and Stochastics, Springer, vol. 26(4), pages 771-823, October.
    15. Kaeck, Andreas & Rodrigues, Paulo & Seeger, Norman J., 2017. "Equity index variance: Evidence from flexible parametric jump–diffusion models," Journal of Banking & Finance, Elsevier, vol. 83(C), pages 85-103.
    16. Damir Filipovi'c & Kathrin Glau & Yuji Nakatsukasa & Francesco Statti, 2019. "Weighted Monte Carlo with least squares and randomized extended Kaczmarz for option pricing," Papers 1910.07241, arXiv.org.
    17. Damir Filipovi'c & Martin Larsson & Sergio Pulido, 2017. "Markov cubature rules for polynomial processes," Papers 1707.06849, arXiv.org, revised Jun 2019.
    18. Nour Meddahi, 2003. "ARMA representation of integrated and realized variances," Econometrics Journal, Royal Economic Society, vol. 6(2), pages 335-356, December.
    19. Filipović, Damir & Larsson, Martin & Pulido, Sergio, 2020. "Markov cubature rules for polynomial processes," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1947-1971.
    20. Audrino, Francesco & Fengler, Matthias R., 2015. "Are classical option pricing models consistent with observed option second-order moments? Evidence from high-frequency data," Journal of Banking & Finance, Elsevier, vol. 61(C), pages 46-63.

    More about this item

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:arx:papers:1908.07417. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: arXiv administrators (email available below). General contact details of provider: http://arxiv.org/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.