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Asians and cash dividends: Exploiting symmetries in pricing theory

Author

Listed:
  • Jiri Hoogland

    (CWI, Amsterdam)

  • Dimitri Neumann

    (CWI, Amsterdam)

Abstract

In this article we present new results for the pricing of arithmetic Asian options within a Black-Scholes context. To derive these results we make extensive use of the local scale invariance that exists in the theory of contingent claim pricing. This allows us to derive, in a natural way, a simple PDE for the price of arithmetic Asians options. In the case of European average strike options, a proper choice of numeraire reduces the dimension of this PDE to one, leading to a PDE similar to the one derived by Rogers and Shi. We solve this PDE, finding a Laplace-transform representation for the price of average strike options, both seasoned and unseasoned. This extends the results of Geman and Yor, who discussed the case of average price options. Next we use symmetry arguments to show that prices of average strike and average price options can be expressed in terms of each other. Finally we show, again using symmetries, that plain vanilla options on stocks paying known cash dividends are closely related to arithmetic Asians, so that all the new techniques can be directly applied to this case.

Suggested Citation

  • Jiri Hoogland & Dimitri Neumann, 2001. "Asians and cash dividends: Exploiting symmetries in pricing theory," Finance 0105002, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:0105002
    Note: Type of Document - Acrobat PDF; prepared on NT/LaTeX; to print on PostScript; pages: 18 ; figures: None.
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    References listed on IDEAS

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    1. Joseph Abate & Ward Whitt, 1995. "Numerical Inversion of Laplace Transforms of Probability Distributions," INFORMS Journal on Computing, INFORMS, vol. 7(1), pages 36-43, February.
    2. Michael Curran, 1994. "Valuing Asian and Portfolio Options by Conditioning on the Geometric Mean Price," Management Science, INFORMS, vol. 40(12), pages 1705-1711, December.
    3. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    4. Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing," Finance 9907002, University Library of Munich, Germany.
    5. Turnbull, Stuart M. & Wakeman, Lee Macdonald, 1991. "A Quick Algorithm for Pricing European Average Options," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 26(3), pages 377-389, September.
    6. Jiri Hoogland & Dimitri Neumann, 1999. "Scale invariance and contingent claim pricing II: Path-dependent contingent claims," Finance 9907003, University Library of Munich, Germany.
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    Cited by:

    1. J. K. Hoogland & C. D. D. Neumann & M. H. Vellekoop, 2003. "Symmetries In Jump-Diffusion Models With Applications In Option Pricing And Credit Risk," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 6(02), pages 135-172.
    2. Eberlein, Ernst & Papapantoleon, Antonis, 2005. "Equivalence of floating and fixed strike Asian and lookback options," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 31-40, January.
    3. Jiri Hoogland & Dimitri Neumann, 2001. "Tradable Schemes," Finance 0105003, University Library of Munich, Germany.
    4. Jan Vecer & Mingxin Xu, 2004. "Pricing Asian options in a semimartingale model," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 170-175.

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    More about this item

    Keywords

    contingent claim pricing; scale invariance; asian options; partial differential equation;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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