IDEAS home Printed from https://ideas.repec.org/a/wsi/nmncxx/v06y2010i01ns1793005710001633.html
   My bibliography  Save this article

Options With Underlying Asset Driven By A Fractional Brownian Motion: Crossing Barriers Estimates

Author

Listed:
  • GIULIA ROTUNDO

    (Department of Business, Technological and Quantitative Studies, Faculty of Economics, University of Tuscia, via del Paradiso 47, 01100 Viterbo, Italy;
    Department of Economic and Financial Institutions, University of Macerata, Via Crescimbeni, 20-62100 - Macerata, Italy)

  • ROY CERQUETI

    (Department of Economic and Financial Institutions, University of Macerata, Via Crescimbeni, 20-62100 - Macerata, Italy)

Abstract

This paper aims at supplying a decision support system tool to investors having options written on an underlying asset driven by a fractional Brownian motion (fBm). The results presented here rely on the theory of nonlinear transformations of fBm and provide the calculus of the probability estimate that the underlying asset crosses nonlinear barriers. Recent results stating a Black and Scholes-like pricing formula for fBm monitor the expected behaviour of options on the basis of the dynamics of the underlying asset. We rely on the results drawn for plain vanilla options, leaving their extension to barrier options for future work. The theory of speculative bubbles due to endogenous causes provides a useful suggestion for the detection of periods in which these results should be used. The application of the above results is shown through the NASDAQ case study.

Suggested Citation

  • Giulia Rotundo & Roy Cerqueti, 2010. "Options With Underlying Asset Driven By A Fractional Brownian Motion: Crossing Barriers Estimates," New Mathematics and Natural Computation (NMNC), World Scientific Publishing Co. Pte. Ltd., vol. 6(01), pages 109-118.
    • Handle: RePEc:wsi:nmncxx:v:06:y:2010:i:01:n:s1793005710001633
    DOI: 10.1142/S1793005710001633
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S1793005710001633
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S1793005710001633?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. Dittmann, Ingolf & Granger, Clive W. J., 2002. "Properties of nonlinear transformations of fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 110(2), pages 113-133, October.
    2. Tommi Sottinen, 2001. "Fractional Brownian motion, random walks and binary market models," Finance and Stochastics, Springer, vol. 5(3), pages 343-355.
    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. Garzón, J. & Gorostiza, L.G. & León, J.A., 2009. "A strong uniform approximation of fractional Brownian motion by means of transport processes," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3435-3452, October.
    2. Giorgio Canarella & Luis A. Gil-Alana & Rangan Gupta & Stephen M. Miller, 2022. "Globalization, long memory, and real interest rate convergence: a historical perspective," Empirical Economics, Springer, vol. 63(5), pages 2331-2355, November.
    3. Gapeev, Pavel V., 2004. "On arbitrage and Markovian short rates in fractional bond markets," Statistics & Probability Letters, Elsevier, vol. 70(3), pages 211-222, December.
    4. Yoon, Gawon, 2005. "An introduction to I([infinity]) processes," Economic Modelling, Elsevier, vol. 22(3), pages 473-483, May.
    5. Morana, Claudio, 2024. "A new macro-financial condition index for the euro area," Econometrics and Statistics, Elsevier, vol. 29(C), pages 64-87.
    6. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Søjmark, 2024. "Functional central limit theorems for rough volatility," Finance and Stochastics, Springer, vol. 28(3), pages 615-661, July.
    7. Cordero, Fernando & Klein, Irene & Perez-Ostafe, Lavinia, 2016. "Asymptotic proportion of arbitrage points in fractional binary markets," Stochastic Processes and their Applications, Elsevier, vol. 126(2), pages 315-336.
    8. repec:hal:wpaper:hal-03284660 is not listed on IDEAS
    9. Héctor Araya & Meryem Slaoui & Soledad Torres, 2022. "Bayesian inference for fractional Oscillating Brownian motion," Computational Statistics, Springer, vol. 37(2), pages 887-907, April.
    10. Zhang, Pu & Xiao, Wei-lin & Zhang, Xi-li & Niu, Pan-qiang, 2014. "Parameter identification for fractional Ornstein–Uhlenbeck processes based on discrete observation," Economic Modelling, Elsevier, vol. 36(C), pages 198-203.
    11. Blanka Horvath & Antoine Jacquier & Aitor Muguruza & Andreas Sojmark, 2017. "Functional central limit theorems for rough volatility," Papers 1711.03078, arXiv.org, revised Nov 2023.
    12. La Spada Gabriele & Lillo Fabrizio, 2014. "The effect of round-off error on long memory processes," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(4), pages 445-482, September.
    13. Dufera, Tamirat Temesgen, 2024. "Fractional Brownian motion in option pricing and dynamic delta hedging: Experimental simulations," The North American Journal of Economics and Finance, Elsevier, vol. 69(PB).
    14. Christian Bender & Tommi Sottinen & Esko Valkeila, 2010. "Fractional processes as models in stochastic finance," Papers 1004.3106, arXiv.org.
    15. Fernando Cordero & Lavinia Perez-Ostafe, 2014. "Critical transaction costs and 1-step asymptotic arbitrage in fractional binary markets," Papers 1407.8068, arXiv.org.
    16. Bardina, X. & Nourdin, I. & Rovira, C. & Tindel, S., 2010. "Weak approximation of a fractional SDE," Stochastic Processes and their Applications, Elsevier, vol. 120(1), pages 39-65, January.
    17. Leschinski, Christian, 2017. "On the memory of products of long range dependent time series," Economics Letters, Elsevier, vol. 153(C), pages 72-76.
    18. Fernando Cordero & Lavinia Perez-Ostafe, 2015. "Critical Transaction Costs And 1-Step Asymptotic Arbitrage In Fractional Binary Markets," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(05), pages 1-26.
    19. Dorsaf Cherif & Emmanuel Lépinette, 2023. "No-arbitrage conditions and pricing from discrete-time to continuous-time strategies," Annals of Finance, Springer, vol. 19(2), pages 141-168, June.
    20. Ghysels, Eric & Santa-Clara, Pedro & Valkanov, Rossen, 2006. "Predicting volatility: getting the most out of return data sampled at different frequencies," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 59-95.
    21. Feng, Yuanhua & Zhou, Chen, 2015. "Forecasting financial market activity using a semiparametric fractionally integrated Log-ACD," International Journal of Forecasting, Elsevier, vol. 31(2), pages 349-363.

    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:wsi:nmncxx:v:06:y:2010:i:01:n:s1793005710001633. 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: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/nmnc/nmnc.shtml .

    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.