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Model of the fractional viscoelastic market 
Ryszard Kutner , Marzena Kozłowska , Andrzej Kasprzak 
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 model of the fractional viscoelastic market which can be also called the Fractional Market Model (FMM) of index dynamics. 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. This model, whose mechanical representation is given by a hierarchical arrangement of a number (in general infinite) of springs and dashpots, is already sufficient to describe observed slowingdown relaxation. In our approach the spring could represent a purely emotional or irrational behaviour of agents (an undamped activity) while the dashpot could define a purely rational one (fear). Such an interpretation could create the possibility of constructing the mechanical model of real stock market. For example, we found that the falling paths of the index can be considered to be a system in the intermediate state lying between two complex ones, defined by short and longtime limits of the MittagLeffler function; these limits are given by the KohlrauschWilliamsWatts (KWW) law for the initial times, and the powerlaw or the Nutting law for asymptotic time. Some rising paths (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 the bullish to bearish one before it reaches a 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 scaling 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 naturalway contained in our FMM and we found its "finger print". 
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Presentation: Oral at International Conference on Economic Science with Heterogeneous Interacting Agents 2008, by Ryszard Kutner Submitted: 20080403 15:30 Revised: 20090607 00:48 