Scaling, Fractals and Wavelets

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variance ratio 11 Relation (1324) shows that in the case of iid returns, we must have a proportionality ( ) 1/2 Some works have highlighted a slight violation of this relation, bringing to light a proportionality of type ( ) H with H > 05 For example, Mantegna [MANT 91], and Mantegna and Stanley [MANT 00] make a list of the values close to 053 or 057 In case of non-Gaussian -stable laws, the scaling parameter noted by is tested and we must have the relation12: (T ) = (n ) = n1/ ( ) T = n (1325)

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An important parameter is Pearson s coef cient, or kurtosis K, de ned by KX = E[(X E(X))4 ]/E[(X E(X))2 ]2 3, as this makes it possible to highlight a variation in the normality of the distribution observed For a normal distribution, we have KX = 0 In the case of iid-Gaussian returns, we must have: K(T ) = K(n ) = K( )/n T = n (1326)

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Yet, for example, Cont [CON 97] nds that the kurtosis coef cient K( ) does not decrease in 1/n but rather in n1/ with 05 indicating the presence of a possible non-linear dependence between variations (see section 134) Generally, the more we improve our knowledge of the scaling behaviors of various parameters, the more it becomes possible to choose between the two alternative terms, scale invariance or characteristic scales The study of scaling behaviors of parameters thus helps in the modeling of stock market uctuations The existence of a scaling anomaly on parameter during investigations carried on between 1970 and 1980, then on K parameter during the following decade, led certain authors to try to modify Mandelbrot s model by limiting scale invariance, either to certain time scales, by introducing system changes (cross-over), or to certain parts of the distributions only on the extreme values In these two fractal metamorphoses, this led to the introduction of a multiscale market analysis 1333 Unstable iid models in partial scaling invariance 13331 Partial scaling invariances by regime switching models The question of mode changes, or partial scaling invariance on a given frequency band had already been dealt with by Mandelbrot [MAND 63], who assumed the

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11 For example, see Lo and MacKinlay s work [LO 88], who gave a list of previous works on the calculation of the variance ratio 12 This relation is veri ed by Walter [WAL 91, WAL 94, WAL 99] and Belkacem et al [BEL 00]

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Research of Scaling Law on Stock Market Variations

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existence of higher and lower limits (cut-off) in the fractality of markets (see also [MAND 97a], p 51 and pp 64 66) and introduced the concept of scaling range Akgiray and Booth [AKG 88b] used this idea to reinforce McCulloch s argument [MCC 78] on the cost-advantage ratio of a model in scaling invariance Using stable distributions between two cutoffs is appropriate because it is less costly in parameter estimations than other modeling, which is perhaps ner (like the combinations of normal laws or mixed diffusion-jumps processes) but also more complex and therefore at the origin of a greater number of estimation errors Therefore, the issue to be solved is the detection of points where change in speed occurs Bouchaud and Potters [BOU 97] and Mantegna and Stanley [MANT 00] propose such a model, combining L vy s distributions and exponential law from a given value 13332 Partial scaling invariances as compared with extremes DuMouchel [DUM 83] suggests, without making a hypothesis a priori on the entire scaling invariance, letting distribution tails speak for themselves (see [DUM 83], p 1025) For this, he uses the generalized Pareto sdistribution introduced by Pickands [PIC 75], whose distribution function is: F (x) = 1 (1 kx/ )1/k 1 exp( x/ ) k=0 k=0 (1327)

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where > 0 and k are the form parameters: the bigger k is, the thicker the distribution tail In the case where distribution is stable with characteristic exponent < 2 (scaling invariance), then we have 1/k = We can observe that, while Pareto s laws had been Mandelbrot s initial step in his introduction of the concept of scaling invariance on stock market variations, Du Mouchel operated in a manner similar to his predecessors and rediscovered Pareto s law without the invariance sought by Mandelbrot Mittnik and Rachev [MIT 89] propose to replace scale invariance on summation of iid- -stable variables by another invariance structure, invariance compared with the minimum: X(1)

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