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Continuous-time trading and the emergence of probability

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  • Vladimir Vovk

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Abstract

This paper establishes a non-stochastic analog of the celebrated result by Dubins and Schwarz about reduction of continuous martingales to Brownian motion via time change. We consider an idealized financial security with continuous price paths, without making any stochastic assumptions. It is shown that typical price paths possess quadratic variation, where “typical” is understood in the following game-theoretic sense: there exists a trading strategy that earns infinite capital without risking more than one monetary unit if the process of quadratic variation does not exist. Replacing time by the quadratic variation process, we show that the price path becomes Brownian motion. This is essentially the same conclusion as in the Dubins–Schwarz result, except that the probabilities (constituting the Wiener measure) emerge instead of being postulated. We also give an elegant statement, inspired by Peter McCullagh’s unpublished work, of this result in terms of game-theoretic probability theory. Copyright Springer-Verlag 2012

Suggested Citation

  • Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
  • Handle: RePEc:spr:finsto:v:16:y:2012:i:4:p:561-609
    DOI: 10.1007/s00780-012-0180-5
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    References listed on IDEAS

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    1. Vladimir Vovk, 2010. "Rough paths in idealized financial markets," Papers 1005.0279, arXiv.org, revised Nov 2016.
    2. Horikoshi, Yasunori & Takemura, Akimichi, 2008. "Implications of contrarian and one-sided strategies for the fair-coin game," Stochastic Processes and their Applications, Elsevier, vol. 118(11), pages 2125-2142, November.
    3. Gianluca Cassese, 2008. "Asset Pricing With No Exogenous Probability Measure," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 23-54.
    4. Masayuki Kumon & Akimichi Takemura & Kei Takeuchi, 2005. "Capital process and optimality properties of a Bayesian Skeptic in coin-tossing games," Papers math/0510662, arXiv.org, revised Sep 2008.
    5. Alexander Cox & Jan Obłój, 2011. "Robust pricing and hedging of double no-touch options," Finance and Stochastics, Springer, vol. 15(3), pages 573-605, September.
    6. V. Vovk, 1993. "Forecasting point and continuous processes: Prequential analysis," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 2(1), pages 189-217, December.
    7. Bick, Avi & Willinger, Walter, 1994. "Dynamic spanning without probabilities," Stochastic Processes and their Applications, Elsevier, vol. 50(2), pages 349-374, April.
    8. Dawid, A. Philip & de Rooij, Steven & Shafer, Glenn & Shen, Alexander & Vereshchagin, Nikolai & Vovk, Vladimir, 2011. "Insuring against loss of evidence in game-theoretic probability," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 157-162, January.
    9. David G. Hobson, 1998. "Robust hedging of the lookback option," Finance and Stochastics, Springer, vol. 2(4), pages 329-347.
    10. Mathias Beiglbock & Walter Schachermayer & Bezirgen Veliyev, 2010. "A Direct Proof of the Bichteler--Dellacherie Theorem and Connections to Arbitrage," Papers 1004.5559, arXiv.org.
    11. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    12. Kumon, Masayuki & Takemura, Akimichi & Takeuchi, Kei, 2011. "Sequential optimizing strategy in multi-dimensional bounded forecasting games," Stochastic Processes and their Applications, Elsevier, vol. 121(1), pages 155-183, January.
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    Citations

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    Cited by:

    1. repec:spr:joecth:v:64:y:2017:i:3:d:10.1007_s00199-016-0999-7 is not listed on IDEAS
    2. repec:spr:finsto:v:21:y:2017:i:3:d:10.1007_s00780-017-0336-4 is not listed on IDEAS
    3. Nicolas Perkowski & David J. Promel, 2013. "Pathwise stochastic integrals for model free finance," Papers 1311.6187, arXiv.org, revised Jun 2016.
    4. repec:spr:finsto:v:21:y:2017:i:4:d:10.1007_s00780-017-0338-2 is not listed on IDEAS
    5. Lesiba. Ch. Galane & Rafa{l} M. {L}ochowski & Farai J. Mhlanga, 2017. "On the quadratic variation of the model-free price paths with jumps," Papers 1710.07894, arXiv.org, revised Jan 2018.
    6. 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.
    7. Daniel Bartl & Michael Kupper & David J. Promel & Ludovic Tangpi, 2017. "Duality for pathwise superhedging in continuous time," Papers 1705.02933, arXiv.org, revised Sep 2017.
    8. Daniel Bartl & Michael Kupper & Ariel Neufeld, 2017. "Pathwise superhedging on prediction sets," Papers 1711.02764, arXiv.org.
    9. Rafa{l} M. {L}ochowski, 2015. "Integration with respect to model-free price paths with jumps," Papers 1511.08194, arXiv.org, revised Sep 2016.
    10. Gianluca Cassese, 2017. "Asset pricing in an imperfect world," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 64(3), pages 539-570, October.

    More about this item

    Keywords

    Game-theoretic probability; Continuous time; Emergence of probability; Continuous price paths; Incomplete markets; 91G99; 60G17; 60G05; 60G44; C58; G13; G14;

    JEL classification:

    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G14 - Financial Economics - - General Financial Markets - - - Information and Market Efficiency; Event Studies; Insider Trading

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