IDEAS home Printed from https://ideas.repec.org/p/sce/scecf5/243.html
   My bibliography  Save this paper

Analytical solutions to the generalized Black-Scholes PDE with the help of an adiabatic approximation to the Schrödinger PDE

Author

Listed:
  • Haven
  • Emmanuel

Abstract

For particular forms of a general volatility function, analytical solutions of the Black-Scholes PDE can be found. However, it tends to be the case that the more 'realistic' the volatility function is, for instance with a volatility smile, analytical solutions become difficult to obtain. In this paper we first convert the Black-Scholes PDE into a time dependent Schrödinger PDE. Then by using the well known separation of variables technique we obtain the time independent version of this PDE. In this format we can exploit a useful technique, introduced by Jeffreys (Proc. London Math. Soc. (1923)) and Rayleigh (Proc. Roy. Soc. (1912)), to solving this time independent Schrödinger PDE. Since we obtain thus an adiabatic approximation of an initial value problem we are particularly careful in respecting the Merton conditions when finding expressions for the coefficients of the approximation. We show that for some particular forms of the volatility function those coefficients can not be found without violating some of the Merton conditions. Fortunately enough for other volatility functions, such as the volatility smile, this violation does not occur. Finally, to find the analytical solution to the Black-Scholes PDE for allowable volatility functions we need to convert back the Schrödinger PDE into the Black-Scholes PDE. For the cases treated in the paper, the solution is not a linear combination of cumulative distribution functions

Suggested Citation

  • Haven & Emmanuel, 2005. "Analytical solutions to the generalized Black-Scholes PDE with the help of an adiabatic approximation to the Schrödinger PDE," Computing in Economics and Finance 2005 243, Society for Computational Economics.
  • Handle: RePEc:sce:scecf5:243
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    More about this item

    Keywords

    Adiabatic approximation; volatility function;

    JEL classification:

    • C69 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Other

    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:sce:scecf5:243. 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: (Christopher F. Baum). General contact details of provider: http://edirc.repec.org/data/sceeeea.html .

    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.