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

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  • Michel Fliess

    (LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau] - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, ALIEN - Algebra for Digital Identification and Estimation - Inria Lille - Nord Europe - Inria - Institut National de Recherche en Informatique et en Automatique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en Automatique - Centrale Lille - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique)

  • Cédric Join

    (ALIEN - Algebra for Digital Identification and Estimation - Inria Lille - Nord Europe - Inria - Institut National de Recherche en Informatique et en Automatique - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en Automatique - Centrale Lille - X - École polytechnique - IP Paris - Institut Polytechnique de Paris - CNRS - Centre National de la Recherche Scientifique, CRAN - Centre de Recherche en Automatique de Nancy - UHP - Université Henri Poincaré - Nancy 1 - INPL - Institut National Polytechnique de Lorraine - CNRS - Centre National de la Recherche Scientifique)

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.

Suggested Citation

  • Michel Fliess & Cédric Join, 2009. "A mathematical proof of the existence of trends in financial time series," Post-Print inria-00352834, HAL.
    • Handle: RePEc:hal:journl:inria-00352834
    Note: View the original document on HAL open archive server: https://inria.hal.science/inria-00352834
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    References listed on IDEAS

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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 637-654, May-June.
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    Cited by:

    1. Michel Fliess & Cédric Join, 2009. "Systematic risk analysis: first steps towards a new definition of beta," Post-Print inria-00425077, HAL.
    2. Michel Fliess & C'edric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Papers 1005.0194, arXiv.org.
    3. Michel Fliess & C'edric Join & Fr'ed'eric Hatt, 2011. "Is a probabilistic modeling really useful in financial engineering? - A-t-on vraiment besoin d'un mod\`ele probabiliste en ing\'enierie financi\`ere ?," Papers 1104.2124, arXiv.org, revised May 2011.
    4. Michel Fliess & Cédric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Post-Print inria-00479824, HAL.
    5. Michel Fliess & Cédric Join & Cyril Voyant, 2018. "Prediction bands for solar energy: New short-term time series forecasting techniques," Post-Print hal-01736518, HAL.
    6. Koussaila Hamiche & Michel Fliess & Cédric Join & Hassane Abouaïssa, 2019. "Bullwhip effect attenuation in supply chain management via control-theoretic tools and short-term forecasts: A preliminary study with an application to perishable inventories," Post-Print hal-02050480, HAL.
    7. 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.
    8. Michel Fliess & Cédric Join & Frédéric Hatt, 2011. "Is a probabilistic modeling really useful in financial engineering? [A-t-on vraiment besoin d'un modèle probabiliste en ingénierie financière ?]," Post-Print hal-00585152, HAL.
    9. 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.
    10. Mogens Graf Plessen & Alberto Bemporad, 2017. "A posteriori multi-stage optimal trading under transaction costs and a diversification constraint," Papers 1709.07527, arXiv.org, revised Apr 2018.

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