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Asians and cash dividends: Exploiting symmetries in pricing theory

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  • Jiri Hoogland
  • Dimitri Neumann

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, 2000. "Asians and cash dividends: Exploiting symmetries in pricing theory," Papers cond-mat/0006133, arXiv.org.
    • Handle: RePEc:arx:papers:cond-mat/0006133
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    Cited by:

    1. Jiri Hoogland & Dimitri Neumann, 2001. "Tradable Schemes," Finance 0105003, University Library of Munich, Germany.
    2. Jan Vecer, 2013. "Asian options on the harmonic average," Quantitative Finance, Taylor & Francis Journals, vol. 14(8), pages 1315-1322, September.
    3. Vicky Henderson & David Hobson & William Shaw & Rafal Wojakowski, 2007. "Bounds for in-progress floating-strike Asian options using symmetry," Annals of Operations Research, Springer, vol. 151(1), pages 81-98, April.
    4. 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.
    5. Jan Vecer & Mingxin Xu, 2004. "Pricing Asian options in a semimartingale model," Quantitative Finance, Taylor & Francis Journals, vol. 4(2), pages 170-175.
    6. 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.

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