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A mathematical proof of the existence of trends in financial time series

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Author Info

  • Michel Fliess

    ()
    (LIX - Laboratoire d'informatique de l'école polytechnique - CNRS : UMR7161 - Polytechnique - X, INRIA Saclay - Ile de France - ALIEN - INRIA - Polytechnique - X - Ecole Centrale de Lille - CNRS : UMR8146)

  • Cédric Join

    ()
    (INRIA Saclay - Ile de France - ALIEN - INRIA - Polytechnique - X - Ecole Centrale de Lille - CNRS : UMR8146, CRAN - Centre de recherche en automatique de Nancy - CNRS : UMR7039 - Université Henri Poincaré - Nancy I - Institut National Polytechnique de Lorraine (INPL))

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    Abstract

    We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195--204). Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities are depicted. We conclude by discussing the rôle of probability theory.

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    File URL: http://hal.archives-ouvertes.fr/docs/00/35/28/34/PDF/FES-Finance.pdf
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    Bibliographic Info

    Paper provided by HAL in its series Post-Print with number inria-00352834.

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    Date of creation: 2009
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    Publication status: Published - Presented, Systems Theory: Modelling, Analysis and Control, 2009, Fes, Morocco
    Handle: RePEc:hal:journl:inria-00352834

    Note: View the original document on HAL open archive server: http://hal.archives-ouvertes.fr/inria-00352834/en/
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    Web page: http://hal.archives-ouvertes.fr/

    Related research

    Keywords: Financial time series; mathematical finance; technical analysis; trends; random walks; efficient markets; forecasting; volatility; heteroscedasticity; quickly fluctuating functions; low-pass filters; nonstandard analysis; operational calculus.;

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    References

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    1. Ait-Sahalia, Yacine & Lo, Andrew W., 2000. "Nonparametric risk management and implied risk aversion," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 9-51.
    2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
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    Cited by:
    1. Michel Fliess & C\'edric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Papers 1005.0194, arXiv.org.
    2. Michel Fliess & Cédric Join, 2008. "Time Series Technical Analysis via New Fast Estimation Methods: A Preliminary Study in Mathematical Finance," Post-Print inria-00338099, HAL.
    3. Michel Fliess & C\'edric Join, 2008. "Time Series Technical Analysis via New Fast Estimation Methods: A Preliminary Study in Mathematical Finance," Papers 0811.1561, arXiv.org, revised Nov 2008.

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