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Fractional Market Model and its verification on stock markets of small size 
Marzena Kozłowska , Ryszard Kutner 
Warsaw University, Faculty of Physics, Hoża 69, Warszawa 00681, Poland 
Abstract 
We analysed the rising and relaxation of the cusplike local peaks superposed with oscillations which were well defined by the Warsaw Stock Exchange index WIG in a daily time horizon. We found that the falling paths of all index peaks were described by a generalized exponential function or the MittagLeffler (ML) one superposed with various types of oscillations. However, the rising paths (except the first one of WIG which rises exponentially and the most important last one which rises again according to the ML function) can be better described by bullish antibubbles or inverted bubbles. The ML function superposed with oscillations is a solution of the nonhomogeneous fractional relaxation equation which defines here our Fractional Market Model (FMM) of index dynamics which can be also called the Rheological Model of Market. This solution is a generalized analog of an exactly solvable fractional version of the Standard or Zener Solid Model of viscoelastic materials commonly used in modern rheology. For example, we found that the falling paths of both indexes can be considered to be a system in the intermediate state lying between two complex ones, definded by short and longtime limits of the ML function; these limits are given by the KohlrauschWilliamsWatts (KWW) law and the Nutting law, respectively. Some rising parts (i.e. the bullish antibubbles) are a kind of logperiodic oscillations of the market in the bullish state initiated by a crash. The peaks of the index can be viewed as precritical or precrash ones since: (i) the financial market changes its state too early from bullish to bearish one before it reaches a critical (scaling) region (defined by the diverging powerlaw of return per unit time), and (ii) they are affected by a finite size effect. These features could be a reminiscence of a significant risk aversion of the investors and their finite number, respectively. However, this means that the critical region (where the relaxations of index peaks are described by the KWW law or stretched exponential decay) was not observed. Hence, neither was the powerlaw of the instantaneous returns per unit time observed. Nevertheless, criticality or crash is in a natural way contained in our FMM and we found its 'finger print'. 
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Presentation: Oral at 3 Ogólnopolskie Sympozjum "Fizyka w Ekonomii i Naukach Społecznych", by Marzena Kozłowska Submitted: 20070828 10:59 Revised: 20090607 00:44 