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Rough paths in idealized financial markets

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  • Vladimir Vovk

Abstract

This paper considers possible price paths of a financial security in an idealized market. Its main result is that the variation index of typical price paths is at most 2, in this sense, typical price paths are not rougher than typical paths of Brownian motion. We do not make any stochastic assumptions and only assume that the price path is positive and right-continuous. The qualification "typical" means that there is a trading strategy (constructed explicitly in the proof) that risks only one monetary unit but brings infinite capital when the variation index of the realized price path exceeds 2. The paper also reviews some known results for continuous price paths and lists several open problems.

Suggested Citation

  • Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers 1005.0279, arXiv.org, revised Nov 2016.
    • Handle: RePEc:arx:papers:1005.0279
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    References listed on IDEAS

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    1. Vladimir Vovk, 2009. "Continuous-time trading and the emergence of probability," Papers 0904.4364, arXiv.org, revised May 2015.
    2. Vladimir Vovk, 2007. "Continuous-time trading and emergence of volatility," Papers 0712.1483, arXiv.org, revised Dec 2007.
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    Cited by:

    1. Vladimir Vovk, 2017. "The role of measurability in game-theoretic probability," Finance and Stochastics, Springer, vol. 21(3), pages 719-739, July.
    2. Rafa{l} M. {L}ochowski, 2015. "Integration with respect to model-free price paths with jumps," Papers 1511.08194, arXiv.org, revised Sep 2016.
    3. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    4. Mathias Beiglbock & Alexander M. G. Cox & Martin Huesmann & Nicolas Perkowski & David J. Promel, 2015. "Pathwise super-replication via Vovk's outer measure," Papers 1504.03644, arXiv.org, revised Jul 2016.

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