IDEAS home Printed from https://ideas.repec.org/a/spr/finsto/v6y2002i1p63-90.html
   My bibliography  Save this article

Stochastic volatility, jumps and hidden time changes

Author

Listed:
  • Marc Yor

    () (Laboratoire de probabilités et Modeles aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, 75252 Paris Cedex, France Manuscript)

  • Dilip B. Madan

    () (Robert H. Smith School of Business, Van Munching Hall, University of Maryland, College Park, MD 20742, USA)

  • Hélyette Geman

    () (University of Paris Dauphine and ESSEC, Finance Department, 95021 Cergy-Pontoise, France)

Abstract

Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.

Suggested Citation

  • Marc Yor & Dilip B. Madan & Hélyette Geman, 2002. "Stochastic volatility, jumps and hidden time changes," Finance and Stochastics, Springer, vol. 6(1), pages 63-90.
  • Handle: RePEc:spr:finsto:v:6:y:2002:i:1:p:63-90
    Note: received: May 2000; final version received: March 2001
    as

    Download full text from publisher

    File URL: http://link.springer.de/link/service/journals/00780/papers/2006001/20060063.pdf
    Download Restriction: Access to the full text of the articles in this series is restricted

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

    Citations

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


    Cited by:

    1. S. Cawston & L. Vostrikova, 2010. "$F$-divergence minimal equivalent martingale measures and optimal portfolios for exponential Levy models with a change-point," Papers 1004.3525, arXiv.org, revised Jun 2011.
    2. Sauri, Orimar & Veraart, Almut E.D., 2017. "On the class of distributions of subordinated Lévy processes and bases," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 475-496.
    3. repec:kap:compec:v:51:y:2018:i:3:d:10.1007_s10614-016-9599-7 is not listed on IDEAS
    4. Neil Shephard & Torben G. Andersen, 2008. "Stochastic Volatility: Origins and Overview," OFRC Working Papers Series 2008fe23, Oxford Financial Research Centre.
    5. Barndorff-Nielsen, Ole E. & Shephard, Neil, 2006. "Impact of jumps on returns and realised variances: econometric analysis of time-deformed Levy processes," Journal of Econometrics, Elsevier, vol. 131(1-2), pages 217-252.
    6. Michele Leonardo Bianchi & Frank J. Fabozzi & Svetlozar T. Rachev, 2014. "Calibrating the Italian smile with time-varying volatility and heavy-tailed models," Temi di discussione (Economic working papers) 944, Bank of Italy, Economic Research and International Relations Area.
    7. Evis Këllezi & Nick Webber, 2004. "Valuing Bermudan options when asset returns are Levy processes," Quantitative Finance, Taylor & Francis Journals, vol. 4(1), pages 87-100.
    8. Yanhui Mi, 2016. "A modified stochastic volatility model based on Gamma Ornstein–Uhlenbeck process and option pricing," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 1-16, June.
    9. Rubenthaler, Sylvain & Wiktorsson, Magnus, 2003. "Improved convergence rate for the simulation of stochastic differential equations driven by subordinated Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 1-26, November.

    More about this item

    Keywords

    Brownian subordination; variance gamma and normal inverse Gaussian processes; variance swap; quadratic variation; filtering;

    JEL classification:

    • C10 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - General
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

    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:spr:finsto:v:6:y:2002:i:1:p:63-90. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Sonal Shukla) or (Rebekah McClure). General contact details of provider: http://www.springer.com .

    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.

    We have no references for this item. You can help adding them by using 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.