IDEAS home Printed from https://ideas.repec.org/p/arx/papers/1205.2866.html
   My bibliography  Save this paper

The fractional volatility model: No-arbitrage, leverage and completeness

Author

Listed:
  • R. Vilela Mendes
  • M. J. Oliveira
  • A. M. Rodrigues

Abstract

Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied.

Suggested Citation

  • R. Vilela Mendes & M. J. Oliveira & A. M. Rodrigues, 2012. "The fractional volatility model: No-arbitrage, leverage and completeness," Papers 1205.2866, arXiv.org.
    • Handle: RePEc:arx:papers:1205.2866
    as

    Download full text from publisher

    File URL: http://arxiv.org/pdf/1205.2866
    File Function: Latest version
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Robert J. Elliott & John Van Der Hoek, 2003. "A General Fractional White Noise Theory And Applications To Finance," Mathematical Finance, Wiley Blackwell, vol. 13(2), pages 301-330, April.
    2. Rui Vilela Mendes & M. J. Oliveira, 2006. "A data-reconstructed fractional volatility model," Papers math/0602013, arXiv.org, revised Jun 2007.
    3. Vilela Mendes, R., 2008. "The fractional volatility model: An agent-based interpretation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3987-3994.
    4. Stephen J. Taylor, 1994. "Modeling Stochastic Volatility: A Review And Comparative Study," Mathematical Finance, Wiley Blackwell, vol. 4(2), pages 183-204, April.
    5. R. F. Engle & A. J. Patton, 2001. "What good is a volatility model?," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 237-245.
    6. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
    7. Fabienne Comte & Eric Renault, 1998. "Long memory in continuous‐time stochastic volatility models," Mathematical Finance, Wiley Blackwell, vol. 8(4), pages 291-323, October.
    8. Paolo Guasoni, 2006. "No Arbitrage Under Transaction Costs, With Fractional Brownian Motion And Beyond," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 569-582, July.
    9. Salopek, D. M., 1998. "Tolerance to arbitrage," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 217-230, August.
    10. R. Vilela Mendes & R. Lima & T. Araújo, 2002. "A Process-Reconstruction Analysis Of Market Fluctuations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(08), pages 797-821.
    11. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    12. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    13. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    14. Breidt, F. Jay & Crato, Nuno & de Lima, Pedro, 1998. "The detection and estimation of long memory in stochastic volatility," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 325-348.
    15. Bjork, Tomas, 2009. "Arbitrage Theory in Continuous Time," OUP Catalogue, Oxford University Press, edition 3, number 9780199574742.
    Full references (including those not matched with items on IDEAS)

    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. Vilela Mendes, R. & Oliveira, M.J. & Rodrigues, A.M., 2015. "No-arbitrage, leverage and completeness in a fractional volatility model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 419(C), pages 470-478.
    2. Rui Vilela Mendes & M. J. Oliveira, 2006. "A data-reconstructed fractional volatility model," Papers math/0602013, arXiv.org, revised Jun 2007.
    3. R. Vilela Mendes, 2022. "The fractional volatility model and rough volatility," Papers 2206.02205, arXiv.org.
    4. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 1999. "The Distribution of Exchange Rate Volatility," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-059, New York University, Leonard N. Stern School of Business-.
    5. Hideharu Funahashi, 2017. "Pricing derivatives with fractional volatility," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(01), pages 1-28, March.
    6. Alexander Subbotin & Thierry Chauveau & Kateryna Shapovalova, 2009. "Volatility Models: from GARCH to Multi-Horizon Cascades," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00390636, HAL.
    7. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    8. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    9. Xiao, Weilin & Zhang, Weiguo & Xu, Weijun & Zhang, Xili, 2012. "The valuation of equity warrants in a fractional Brownian environment," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(4), pages 1742-1752.
    10. Giulia Di Nunno & Kk{e}stutis Kubilius & Yuliya Mishura & Anton Yurchenko-Tytarenko, 2023. "From constant to rough: A survey of continuous volatility modeling," Papers 2309.01033, arXiv.org, revised Sep 2023.
    11. Stoyan V. Stoyanov & Svetlozar T. Rachev & Stefan Mittnik & Frank J. Fabozzi, 2019. "Pricing Derivatives In Hermite Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-27, September.
    12. Tetsuya Takaishi, 2019. "Rough volatility of Bitcoin," Papers 1904.12346, arXiv.org.
    13. Xiao, Wei-Lin & Zhang, Wei-Guo & Zhang, Xili & Zhang, Xiaoli, 2012. "Pricing model for equity warrants in a mixed fractional Brownian environment and its algorithm," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6418-6431.
    14. Ho, Hwai-Chung, 2015. "Sample quantile analysis for long-memory stochastic volatility models," Journal of Econometrics, Elsevier, vol. 189(2), pages 360-370.
    15. Daniele Angelini & Matthieu Garcin, 2024. "Market information of the fractional stochastic regularity model," Papers 2409.07159, arXiv.org.
    16. Frezza, Massimiliano, 2014. "Goodness of fit assessment for a fractal model of stock markets," Chaos, Solitons & Fractals, Elsevier, vol. 66(C), pages 41-50.
    17. Elena Andreou & Eric Ghysels, 2002. "Detecting multiple breaks in financial market volatility dynamics," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 17(5), pages 579-600.
    18. Carmen Broto & Esther Ruiz, 2004. "Estimation methods for stochastic volatility models: a survey," Journal of Economic Surveys, Wiley Blackwell, vol. 18(5), pages 613-649, December.
    19. Pedro J. F. de Lima & Michelle L. Barnes, 2000. "Modeling Financial Volatility: Extreme Observations, Nonlinearities and Nonstationarities," School of Economics and Public Policy Working Papers 2000-05, University of Adelaide, School of Economics and Public Policy.
    20. Wang, Xiao-Tian, 2011. "Scaling and long-range dependence in option pricing V: Multiscaling hedging and implied volatility smiles under the fractional Black–Scholes model with transaction costs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(9), pages 1623-1634.

    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:1205.2866. 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.