IDEAS home Printed from
   My bibliography  Save this article

Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution


  • Milevsky, Moshe Arye
  • Posner, Steven E.


Arithmetic Asian options are difficult to price and hedge as they do not have closed-form analytic solutions. The main theoretical reason for this difficulty is that the payoff depends on the finite sum of correlated lognormal variables, which is not lognormal and for which there is no recognizable probability density function. We use elementary techniques to derive the probability density function of the infinite sum of correlated lognormal random variables and show that it is reciprocal gamma distributed, under suitable parameter restrictions. A random variable is reciprocal gamma distributed if its inverse is gamma distributed. We use this result to approximate the finite sum of correlated lognormal variables and then value arithmetic Asian options using the reciprocal gamma distribution as the state-price density function. We thus obtain a closed-form analytic expression for the value of an arithmetic Asian option, where the cumulative density function of the gamma distribution, G(d) in our formula, plays the exact same role as N(d) in the Black-Scholes/Merton formula. In addition to being theoretically justified and exact in the limit, we compare our method against other algorithms in the literature and show our method is quicker, at least as accurate, and, in our opinion, more intuitive and pedagogically appealing than any previously published result, especially when applied to high yielding currency options.

Suggested Citation

  • Milevsky, Moshe Arye & Posner, Steven E., 1998. "Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 33(03), pages 409-422, September.
  • Handle: RePEc:cup:jfinqa:v:33:y:1998:i:03:p:409-422_00

    Download full text from publisher

    File URL:
    File Function: link to article abstract page
    Download Restriction: no

    References listed on IDEAS

    1. De Schepper, A. & Teunen, M. & Goovaerts, M., 1994. "An analytical inversion of a Laplace transform related to annuities certain," Insurance: Mathematics and Economics, Elsevier, vol. 14(1), pages 33-37, April.
    2. J. A. Nielsen & K. Sandmann, 1996. "The pricing of Asian options under stochastic interest rates," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(3), pages 209-236.
    3. Alziary, Benedicte & Decamps, Jean-Paul & Koehl, Pierre-Francois, 1997. "A P.D.E. approach to Asian options: analytical and numerical evidence," Journal of Banking & Finance, Elsevier, vol. 21(5), pages 613-640, May.
    Full references (including those not matched with items on IDEAS)

    More about this item


    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:cup:jfinqa:v:33:y:1998:i:03:p:409-422_00. 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: (Keith Waters). 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.