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

Author

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  • Michel Fliess

    (LIX, INRIA Saclay - Ile de France)

  • C'edric Join

    (INRIA Saclay - Ile de France, CRAN)

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 r\^ole of probability theory.

Suggested Citation

  • Michel Fliess & C'edric Join, 2009. "A mathematical proof of the existence of trends in financial time series," Papers 0901.1945, arXiv.org.
    • Handle: RePEc:arx:papers:0901.1945
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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 & 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.
    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 & Cédric Join, 2009. "Systematic risk analysis: first steps towards a new definition of beta," Post-Print inria-00425077, HAL.
    4. Michel Fliess & C'edric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Papers 1005.0194, arXiv.org.
    5. 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.
    6. 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.
    7. 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.
    8. Michel Fliess & Cédric Join, 2010. "Delta Hedging in Financial Engineering: Towards a Model-Free Approach," Post-Print inria-00479824, HAL.
    9. Michel Fliess & Cédric Join & Cyril Voyant, 2018. "Prediction bands for solar energy: New short-term time series forecasting techniques," Post-Print hal-01736518, HAL.
    10. 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.

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