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All Moments of Discrete and Continuous Arithmetic Averages on Brownian Paths: A Closed Form

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  • Allen Abrahamson

Abstract

This note derives new expressions for the moments of the average of values taken by Wiener paths at an arbitrary number, N, of discrete times. The expressions are closed summations, which entail only the N-th powers of, and the successive differences between, the moments of the lognormal finite dimensional distribution of the process' values at the time of the first averaging. By passing to the limit of the average when the averaging frequency becomes continuous, known forms for the continuous average are generalized by a single expression.

Suggested Citation

  • Allen Abrahamson, 2002. "All Moments of Discrete and Continuous Arithmetic Averages on Brownian Paths: A Closed Form," Finance 0205004, University Library of Munich, Germany.
    • Handle: RePEc:wpa:wuwpfi:0205004
    Note: Type of Document - PDF; prepared on IBM PC pdfLatex; to print on HP; pages: 16; figures: none. none
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    References listed on IDEAS

    as
    1. 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..
    2. Hélyette Geman & Marc Yor, 1993. "Bessel Processes, Asian Options, And Perpetuities," Mathematical Finance, Wiley Blackwell, vol. 3(4), pages 349-375, October.
    3. 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.
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    Cited by:

    1. Aprahamian, Hrayer & Maddah, Bacel, 2015. "Pricing Asian options via compound gamma and orthogonal polynomials," Applied Mathematics and Computation, Elsevier, vol. 264(C), pages 21-43.

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    More about this item

    Keywords

    Brownian Motion; Wiener Processes; Moments; Asian Options; Sample Path Properties; Computation;
    All these keywords.

    JEL classification:

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

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