IDEAS home Printed from
   My bibliography  Save this article

Discretely sampled variance and volatility swaps versus their continuous approximations


  • Robert Jarrow


  • Younes Kchia


  • Martin Larsson


  • Philip Protter



Discretely sampled variance and volatility swaps trade actively in OTC markets. To price these swaps, the continuously sampled approximation is often used to simplify the computations. The purpose of this paper is to study the conditions under which this approximation is valid. Our first set of theorems characterize the conditions under which the discretely sampled swap values are finite, given that the values of the continuous approximations exist. Surprisingly, for some otherwise reasonable price processes, the discretely sampled swap prices do not exist, thereby invalidating the approximation. Examples are provided. Assuming further that both swap values exist, we study sufficient conditions under which the discretely sampled values converge to their continuous counterparts. Because of its popularity in the literature, we apply our theorems to the 3/2 stochastic volatility model. Although we can show finiteness of all swap values, we can prove convergence of the approximation only for some parameter values. Copyright Springer-Verlag 2013

Suggested Citation

  • Robert Jarrow & Younes Kchia & Martin Larsson & Philip Protter, 2013. "Discretely sampled variance and volatility swaps versus their continuous approximations," Finance and Stochastics, Springer, vol. 17(2), pages 305-324, April.
  • Handle: RePEc:spr:finsto:v:17:y:2013:i:2:p:305-324
    DOI: 10.1007/s00780-012-0183-2

    Download full text from publisher

    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

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

    References listed on IDEAS

    1. Peter Carr & Roger Lee & Liuren Wu, 2012. "Variance swaps on time-changed Lévy processes," Finance and Stochastics, Springer, vol. 16(2), pages 335-355, April.
    2. Leif Andersen & Vladimir Piterbarg, 2007. "Moment explosions in stochastic volatility models," Finance and Stochastics, Springer, vol. 11(1), pages 29-50, January.
    3. Peter Carr & Jian Sun, 2007. "A new approach for option pricing under stochastic volatility," Review of Derivatives Research, Springer, vol. 10(2), pages 87-150, May.
    4. Peter Carr & Roger Lee, 2009. "Volatility Derivatives," Annual Review of Financial Economics, Annual Reviews, vol. 1(1), pages 319-339, November.
    Full references (including those not matched with items on IDEAS)


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

    Cited by:

    1. Carol Alexander & Johannes Rauch, 2017. "The Aggregation Property and its Applications to Realised Higher Moments," Papers 1709.08188,
    2. Filipović, Damir & Gourier, Elise & Mancini, Loriano, 2016. "Quadratic variance swap models," Journal of Financial Economics, Elsevier, vol. 119(1), pages 44-68.
    3. Carol Alexander & Johannes Rauch, 2014. "Model-Free Discretisation-Invariant Swaps and S&P 500 Higher-Moment Risk Premia," Papers 1404.1351,, revised Feb 2016.
    4. Wang, Xingchun & Fu, Jianping & Wang, Guanying & Wang, Yongjin, 2015. "Quadratic hedging strategies for volatility swaps," Finance Research Letters, Elsevier, vol. 15(C), pages 125-132.
    5. Carol Alexander & Johannes Rauch, 2016. "Model-Free Discretisation-Invariant Swap Contracts," Papers 1602.00235,, revised Apr 2016.
    6. David Hobson & Martin Klimmek, 2011. "Model independent hedging strategies for variance swaps," Papers 1104.4010,, revised May 2011.
    7. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
    8. Wang, Xingchun, 2016. "Catastrophe equity put options with target variance," Insurance: Mathematics and Economics, Elsevier, vol. 71(C), pages 79-86.
    9. repec:eee:ejores:v:262:y:2017:i:1:p:381-400 is not listed on IDEAS

    More about this item


    Variance swaps; Volatility swaps; NFLVR; Semimartingales; 60G35; 60G44; C65; C69; G12;

    JEL classification:

    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates


    Access and download statistics


    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:17:y:2013:i:2:p:305-324. 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: .

    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 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.

    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.