IDEAS home Printed from https://ideas.repec.org/
MyIDEAS: Login to save this paper or follow this series

Option Pricing, Historical Volatility and Tail Risks

  • Samuel E. Vazquez
Registered author(s):

    We revisit the problem of pricing options with historical volatility estimators. We do this in the context of a generalized GARCH model with multiple time scales and asymmetry. It is argued that the reason for the observed volatility risk premium is tail risk aversion. We parametrize such risk aversion in terms of three coefficients: convexity, skew and kurtosis risk premium. We propose that option prices under the real-world measure are not martingales, but that their drift is governed by such tail risk premia. We then derive a fair-pricing equation for options and show that the solutions can be written in terms of a stochastic volatility model in continuous time and under a martingale probability measure. This gives a precise connection between the pricing and real-world probability measures, which cannot be obtained using Girsanov Theorem. We find that the convexity risk premium, not only shifts the overall implied volatility level, but also changes its term structure. Moreover, the skew risk premium makes the skewness of the volatility smile steeper than a pure historical estimate. We derive analytical formulas for certain implied moments using the Bergomi-Guyon expansion. This allows for very fast calibrations of the models. We show examples of a particular model which can reproduce the observed SPX volatility surface using very few parameters.

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

    File URL: http://arxiv.org/pdf/1402.1255
    File Function: Latest version
    Download Restriction: no

    Paper provided by arXiv.org in its series Papers with number 1402.1255.

    as
    in new window

    Length:
    Date of creation: Feb 2014
    Date of revision:
    Handle: RePEc:arx:papers:1402.1255
    Contact details of provider: Web page: http://arxiv.org/

    References listed on IDEAS
    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

    as in new window
    1. Jim Gatheral & Antoine Jacquier, 2012. "Arbitrage-free SVI volatility surfaces," Papers 1204.0646, arXiv.org, revised Mar 2013.
    2. Jin-Chuan Duan, 1995. "The Garch Option Pricing Model," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 13-32.
    Full references (including those not matched with items on IDEAS)

    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

    When requesting a correction, please mention this item's handle: RePEc:arx:papers:1402.1255. 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: (arXiv administrators)

    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 references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link 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 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.

    This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.