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an account of this failure, by means of non-fractal modeling, ie, of a multi-scale market analysis Here we present the main articles relating to this emphasis Teichmoeller [TEI 71], Of cer [OFF 72], Fielitz and Smith [FIE 72], Praetz [PRA 72] and Barnea and Downes [BAR 73] obtain all the values of which increase on average from 16 in high frequency to 18 in low frequency This increase led Hsu et al [HSU 74], who also veri ed it, to estimate that in an economy where factors affecting price levels (technical developments, government policies, etc) can undergo movements on a great scale, it seems unreasonable (our emphasis) to want to try to represent price variations by a single probability distribution (see [HSU 74], p1) Brenner [BRE 74], Blattberg and Gonedes [BLA 74] and Hagerman [HAG 78] continued the investigations by observing the same phenomenon Hagerman concludes that the symmetric stable model cannot reasonably (our emphasis) be regarded as a suitable description of stock market returns (see [HAG 78], p 1220) We can see a similarity of arguments between Hagerman and Hsu et al [HSU 74] for whom it does not seem to be reasonable to retain a model with in nite variance This argument was used by Bienaym against Cauchy as early as 1853 Zajdenweber [ZAJ 76] veri es the adjustment in L vy s distribution but does not test the scale invariance Upton and Shannon [UPT 79] take up the question in a different way by seeking to estimate the violation degree in normality based on observation scale by using the Kolmogorov-Smirnov (KS) method, which is a calculation of curve coef cients K and skewness S The scale invariance is not retained A new study by Fielitz and Rozelle [FIE 83] con rms the scaling anomaly Other investigations are carried out on exchange markets Wasserfallen and Zimmermann [WAS 85], Boothe and Glassman [BOO 87], Akgiray and Booth [AKG 88a], Tucker and Pond [TUC 88] and Hall et al [HAL 89] tackled, for their part, the increase of according to the decrease of observation frequency At the end of the 1980s, the iid- -stable model of stock market returns appeared to be rejected by all the research in this eld In 1986, as we read in a summarized work on the analysis of stock market variations: many researchers estimated that the hypothesis of in nite variance was not acceptable Detailed studies on stock market variations rejected L vy s distributions in a conclusive way [ ] Ten years after his article in 1965, Fama himself preferred to use a normal distribution for monthly variations and thus to give up stable distributions for daily variations (see [TAY 86], p 46) However, we can observe that theoretical scale invariance of Gaussian modeling (scaling law in square root of time) is not validated by real markets in all cases and that generalization by iid- -stable model represents a good compromise between modeling
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Research of Scaling Law on Stock Market Variations
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power and statistical cost of estimation We nd such an argument, for example, in McCulloch [MCC 78], who advocates the small number of parameters required by stable laws, as compared with the ve necessary parameters for jump models such as those proposed by Merton [MER 76] In other words, the question remains open, even if it is probable that the true process of returns is more complex than iid- -stable modeling Certain works that were carried out show that the values of can change in time10 (stationarity problem of r), which leads us to raise the question of dependence between increments of the prices process and in nding other forms of scaling laws on nancial series 13323 Scaling anomalies of parameters under iid hypothesis Systematic increase of characteristic exponent ( ) of stable laws according to constitutes what is called a scaling anomaly Indeed, in iid- -stable modeling, the following relation must be veri ed: (T ) = (n ) = ( ) T = n (1323)
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The fact that this relation is not found for all the values of n shows that scale invariance is not total on all time scales, or that the iid hypothesis is not valid More generally, a way of highlighting invariance by changing the scaling probability law, and thus being able to determine fractal hypothesis, is to examine whether its characteristic parameters have a scaling behavior, ie, seek a dilation (or contraction) law of parameters according to time scale This idea is the beginning of an important trend in the theoretical research in nance Let ( ) be a statistical parameter of distribution r(t, ): ( ) is a function of and searching for scaling laws on a market between 0 and T therefore leads to the estimation of parameter values based on each value of , then to the study of scale relation, or function : ( ) All the statistical distribution parameters are also a priori usable for the research of scaling laws on distributions The most analyzed parameters in research works are either a scaling parameter or the curve coef cient, or kurtosis K In the Gaussian case, the scaling parameter is the standard deviation and in case of iid increments, we must have the relation: (T ) = (n ) = n1/2 ( ) T = n (1324)
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This scaling relation, already postulated on variance by Bachelier [BAC 00], was introduced into research during the 1980s, and is known under the name of test of
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10 See an example in [WAL 94]
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