IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v388y2009i17p3503-3520.html
   My bibliography  Save this article

Perturbation expansion for option pricing with stochastic volatility

Author

Listed:
  • Jizba, Petr
  • Kleinert, Hagen
  • Haener, Patrick

Abstract

We fit the volatility fluctuations of the S&P 500 index well by a Chi distribution, and the distribution of log-returns by a corresponding superposition of Gaussian distributions. The Fourier transform of this is, remarkably, of the Tsallis type. An option pricing formula is derived from the same superposition of Black–Scholes expressions. An explicit analytic formula is deduced from a perturbation expansion around a Black–Scholes formula with the mean volatility. The expansion has two parts. The first takes into account the non-Gaussian character of the stock-fluctuations and is organized by powers of the excess kurtosis, the second is contract based, and is organized by the moments of moneyness of the option. With this expansion we show that for the Dow Jones Euro Stoxx 50 option data, a Δ-hedging strategy is close to being optimal.

Suggested Citation

  • Jizba, Petr & Kleinert, Hagen & Haener, Patrick, 2009. "Perturbation expansion for option pricing with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(17), pages 3503-3520.
    • Handle: RePEc:eee:phsmap:v:388:y:2009:i:17:p:3503-3520
    DOI: 10.1016/j.physa.2009.04.027
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S037843710900332X
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2009.04.027?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    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. Rashidi Ranjbar, Hedieh & Seifi, Abbas, 2015. "A path-independent method for barrier option pricing in hidden Markov models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 440(C), pages 1-8.
    2. Marroquı´n-Martı´nez, Naroa & Moreno, Manuel, 2013. "Optimizing bounds on security prices in incomplete markets. Does stochastic volatility specification matter?," European Journal of Operational Research, Elsevier, vol. 225(3), pages 429-442.
    3. Jean-Philippe Aguilar & Cyril Coste & Jan Korbel, 2017. "Series representation of the pricing formula for the European option driven by space-time fractional diffusion," Papers 1712.04990, arXiv.org, revised Oct 2018.
    4. Wei, Yu, 2012. "Forecasting volatility of fuel oil futures in China: GARCH-type, SV or realized volatility models?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5546-5556.
    5. D’Amico, Guglielmo & Scocchera, Stefania & Storchi, Loriano, 2018. "Financial risk distribution in European Union," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 252-267.

    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. Milan Kumar Das & Anindya Goswami, 2019. "Testing of binary regime switching models using squeeze duration analysis," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 1-20, March.
    2. Marcos Escobar-Anel & Weili Fan, 2023. "The SEV-SV Model—Applications in Portfolio Optimization," Risks, MDPI, vol. 11(2), pages 1-34, January.
    3. Josselin Garnier & Knut Sølna, 2018. "Option pricing under fast-varying and rough stochastic volatility," Annals of Finance, Springer, vol. 14(4), pages 489-516, November.
    4. Darren Shannon & Grigorios Fountas, 2021. "Extending the Heston Model to Forecast Motor Vehicle Collision Rates," Papers 2104.11461, arXiv.org, revised May 2021.
    5. Eduardo Abi Jaber, 2022. "The characteristic function of Gaussian stochastic volatility models: an analytic expression," Working Papers hal-02946146, HAL.
    6. Chen, An & Hieber, Peter & Sureth, Caren, 2022. "Pay for tax certainty? Advance tax rulings for risky investment under multi-dimensional tax uncertainty," arqus Discussion Papers in Quantitative Tax Research 273, arqus - Arbeitskreis Quantitative Steuerlehre.
    7. Peter Carr & Liuren Wu, 2014. "Static Hedging of Standard Options," The Journal of Financial Econometrics, Society for Financial Econometrics, vol. 12(1), pages 3-46.
    8. Chiarella, Carl & Kang, Boda & Nikitopoulos, Christina Sklibosios & Tô, Thuy-Duong, 2013. "Humps in the volatility structure of the crude oil futures market: New evidence," Energy Economics, Elsevier, vol. 40(C), pages 989-1000.
    9. 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.
    10. Söderlind, Paul, 2009. "The C-CAPM without ex post data," Journal of Macroeconomics, Elsevier, vol. 31(4), pages 721-729, December.
    11. Damir Filipovi'c & Martin Larsson, 2017. "Polynomial Jump-Diffusion Models," Papers 1711.08043, arXiv.org, revised Jul 2019.
    12. René Garcia & Richard Luger & Eric Renault, 2000. "Asymmetric Smiles, Leverage Effects and Structural Parameters," Working Papers 2000-57, Center for Research in Economics and Statistics.
    13. Archil Gulisashvili & Peter Laurence, 2013. "The Heston Riemannian distance function," Papers 1302.2337, arXiv.org.
    14. Luca Di Persio & Luca Prezioso & Kai Wallbaum, 2019. "Closed-End Formula for options linked to Target Volatility Strategies," Papers 1902.08821, arXiv.org.
    15. Christoffersen, Peter & Heston, Steven & Jacobs, Kris, 2010. "Option Anomalies and the Pricing Kernel," Working Papers 11-17, University of Pennsylvania, Wharton School, Weiss Center.
    16. Peng He, 2012. "Option Portfolio Value At Risk Using Monte Carlo Simulation Under A Risk Neutral Stochastic Implied Volatility Model," Global Journal of Business Research, The Institute for Business and Finance Research, vol. 6(5), pages 65-72.
    17. Yeap, Claudia & Kwok, Simon S. & Choy, S. T. Boris, 2016. "A Flexible Generalised Hyperbolic Option Pricing Model and its Special Cases," Working Papers 2016-14, University of Sydney, School of Economics.
    18. Torben G. Andersen & Tim Bollerslev & Peter Christoffersen & Francis X. Diebold, 2007. "Practical Volatility and Correlation Modeling for Financial Market Risk Management," NBER Chapters, in: The Risks of Financial Institutions, pages 513-544, National Bureau of Economic Research, Inc.
    19. O. Samimi & Z. Mardani & S. Sharafpour & F. Mehrdoust, 2017. "LSM Algorithm for Pricing American Option Under Heston–Hull–White’s Stochastic Volatility Model," Computational Economics, Springer;Society for Computational Economics, vol. 50(2), pages 173-187, August.
    20. José Valentim Machado Vicente & Jaqueline Terra Moura Marins, 2019. "A Volatility Smile-Based Uncertainty Index," Working Papers Series 502, Central Bank of Brazil, Research Department.

    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:eee:phsmap:v:388:y:2009:i:17:p:3503-3520. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.