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

Vladimir Vovk and Glenn Shafer

Papers from arXiv.org

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.

Date: 2018-02
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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1802.01219

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