IDEAS home Printed from
MyIDEAS: Log in (now much improved!) to save this article

Inversion of option prices for implied risk-neutral probability density functions: general theory and its applications to the natural gas market

Listed author(s):
  • Yijun Du
  • Chen Wang
  • Yibing Du
Registered author(s):

    This paper applies inverse theory to the estimation of the implied risk-neutral probability density function (PDF) from option prices. A general framework of inverting option prices for the implied risk-neutral PDF is formulated from the option pricing formula of Cox and Ross [ J. Financial Econ. , 1976, 3 , 145--166]. To overcome the non-uniqueness and instability inherent in the option inverse problem, the smoothness requirement for the shape of the PDF and a prior model are introduced by a penalty function. Positivity constraints are included as a hard bond on the PDF values. The option inverse problem then becomes a non-negative least-squares problem that can be solved by classic methods such as the non-negative least-squares program of Lawson and Hanson [ Solving Least Squares Problems , 1974 (Prentice-Hall: Englewood Cliffs, NJ)]. The best solution is not the one that gives the best fit to the observed option prices or provides the smoothest PDF, but the one that gives the optimal trade-off between the goodness-of-fit and smoothness of the estimated risk-natural PDF. The proposed inversion technique is compared with the models of Black--Scholes (BS), a mixture of two lognormals (MLN), Jarrow and Rudd's Edgeworth expansion (JR), and jump diffusion (JD) for the estimation of the PDF from the option prices associated with the September 2007 NYMEX natural gas futures. It is found that the inversion technique not only gives the best goodness-of-fit, but also a significantly better model resolution. An empirical study for the last three months of the September 2007 futures contract shows that the shapes of the estimated PDFs become more symmetric as the futures contract becomes closer to the expiration date. The dispersion of the estimated PDFs decreases with decreasing time to expiry, indicating the resolution of uncertainty with passing time.

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

    As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

    Article provided by Taylor & Francis Journals in its journal Quantitative Finance.

    Volume (Year): 12 (2012)
    Issue (Month): 12 (December)
    Pages: 1877-1891

    in new window

    Handle: RePEc:taf:quantf:v:12:y:2012:i:12:p:1877-1891
    DOI: 10.1080/14697688.2011.586355
    Contact details of provider: Web page:

    Order Information: Web:

    No references listed on IDEAS
    You can help add them by filling out this form.

    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:taf:quantf:v:12:y:2012:i:12:p:1877-1891. 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: (Chris Longhurst)

    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.