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

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  • 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

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    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.
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    4. Moshe Arye Milevsky & Steven E. Posner, 1999. "Asian Options, The Sum Of Lognormals, And The Reciprocal Gamma Distribution," World Scientific Book Chapters, in: Marco Avellaneda (ed.), Quantitative Analysis In Financial Markets Collected Papers of the New York University Mathematical Finance Seminar, chapter 7, pages 203-218, World Scientific Publishing Co. Pte. Ltd..
    5. Vadim Linetsky, 2004. "Spectral Expansions for Asian (Average Price) Options," Operations Research, INFORMS, vol. 52(6), pages 856-867, December.
    6. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    7. 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.
    8. 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.
    9. 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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    Cited by:

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    2. Rupak Chatterjee & Zhenyu Cui & Jiacheng Fan & Mingzhe Liu, 2018. "An efficient and stable method for short maturity Asian options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(12), pages 1470-1486, December.

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    More about this item

    Keywords

    Exponential Brownian motion; Random environment; Asian options;
    All these keywords.

    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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