More and more empirical evidence is coming that multifractality constitutes another and perhaps the most significant financial stylised fact. Our own study shows that it applies both to the price increments and to the waiting times between the consecutive transactions. A realistic model of the financial dynamics should therefore incorporate such effects. The most promising in this respect is the Multifractal Model of Asset Returns introduced by Mandelbrot and coworkers. In this model the multifractality of returns results from a deformation of time due to the fact that at the microscale the so-called business time is dictated by the density of transactions rather than by any constant time units. Of course, several variants of this model can be considered. A principal purpose of our study is to identify the optimal one by relating their performance to the real stock market dynamics. We focus mainly on the Polish Stock Market and on the issue of forecasting volatility but carry out some comparative studies for the U.S. stock market as well.

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