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The last systemic financial crisis has reawakened the debate on the efficient nature of financial markets, traditionally described as semimartingales. The standard approaches to endow the general notion of efficiency of an empirical content turned out to be somewhat inconclusive and misleading. We propose a topological-based approach to quantify the informational efficiency of a financial time series. The idea is to measure the efficiency by means of the pointwise regularity of a (stochastic) function, given that the signature of a martingale is that its pointwise regularity equals 12. We provide estimates for real financial time series and investigate their (in)efficient behavior by comparing three main stock indexes.

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The recent global financial crisis has threatened the financial system with total collapse of many economic sectors with a particular penetration to world’s stock markets. The large swings in the prices of international stocks or indexes have reinvigorated the debate on their mathematical modeling. The traditional approaches do not seem to be very exhaustive and satisfactory, especially when extreme events occur. We propose a fractal-based approach to model the actual prices by assuming that they follow a Multifractional Process with Random Exponent. An empirical evidence is offered that this stochastic process is able to provide an appropriate modeling of actual series in terms of goodness of fit by comparing three main stock indexes.

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