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A game-theoretic derivation of the $\sqrt{dt}$ effect

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  • Vladimir Vovk
  • Glenn Shafer

Abstract

We study the origins of the $\sqrt{dt}$ effect in finance and SDE. In particular, we show, in the game-theoretic framework, that market volatility is a consequence of the absence of riskless opportunities for making money and that too high volatility is also incompatible with such opportunities. More precisely, riskless opportunities for making money arise whenever a traded security has fractal dimension below or above that of the Brownian motion and its price is not almost constant and does not become extremely large. This is a simple observation known in the measure-theoretic mathematical finance. At the end of the article we also consider the case of non-zero interest rate. This version of the article was essentially written in March 2005 but remains a working paper.

Suggested Citation

  • Vladimir Vovk & Glenn Shafer, 2018. "A game-theoretic derivation of the $\sqrt{dt}$ effect," Papers 1802.01219, arXiv.org.
    • Handle: RePEc:arx:papers:1802.01219
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    References listed on IDEAS

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    1. L. C. G. Rogers, 1997. "Arbitrage with Fractional Brownian Motion," Mathematical Finance, Wiley Blackwell, vol. 7(1), pages 95-105, January.
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