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Another look at the integral of exponential Brownian motion and the pricing of Asian options


  • Andrew Lyasoff

    () (Boston University)


Abstract It is shown that Marc Yor’s formula (Adv. Appl. Probab. 24:509–531, 1992) for the density of the integral of exponential Brownian motion taken over a finite time interval is an extremal member of a family of previously unknown integral formulae for the same density. The derivation is independent from the one by Yor and obtained from a simple time-reversibility feature, in conjunction with a Fokker–Planck type argument. Similar arguments lead to an independent derivation of Dufresne’s result (Scand. Actuar. J. 90:39–79, 1990) for the law of the integral taken over an infinite time interval. The numerical aspects of the new formulae are developed, with concrete applications to Asian options.

Suggested Citation

  • Andrew Lyasoff, 2016. "Another look at the integral of exponential Brownian motion and the pricing of Asian options," Finance and Stochastics, Springer, vol. 20(4), pages 1061-1096, October.
  • Handle: RePEc:spr:finsto:v:20:y:2016:i:4:d:10.1007_s00780-016-0307-1
    DOI: 10.1007/s00780-016-0307-1

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    References listed on IDEAS

    1. Donati-Martin, Catherine & Matsumoto, Hiroyuki & Yor, Marc, 2000. "On positive and negative moments of the integral of geometric Brownian motions," Statistics & Probability Letters, Elsevier, vol. 49(1), pages 45-52, August.
    2. Daniel Dufresne, 2000. "Laguerre Series for Asian and Other Options," Mathematical Finance, Wiley Blackwell, vol. 10(4), pages 407-428.
    3. Alexander Novikov & Nino Kordzakhia, 2013. "On lower and upper bounds for Asian-type options: a unified approach," Papers 1309.2383,
    4. 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.
    5. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375.
    6. Hobson, David, 2007. "A short proof of an identity for a Brownian Bridge due to Donati-Martin, Matsumoto and Yor," Statistics & Probability Letters, Elsevier, vol. 77(2), pages 148-150, January.
    7. Dufresne, Daniel, 1989. "Weak convergence of random growth processes with applications to insurance," Insurance: Mathematics and Economics, Elsevier, vol. 8(3), pages 187-201, November.
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    More about this item


    Exponential Brownian motion; Random environment; Asian options;

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing


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