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

TL;DR

This paper derives the  effect in finance using a game-theoretic approach, linking market volatility to the absence of riskless profit opportunities and analyzing how fractal dimensions influence arbitrage.

Contribution

It provides a novel game-theoretic derivation of the  effect, connecting market volatility with fractal dimensions and arbitrage opportunities.

Findings

Market volatility arises from the absence of riskless opportunities.

High volatility is incompatible with arbitrage opportunities.

Fractal dimension of securities influences the existence of riskless profit.

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.