Scaling, Fractals and Wavelets

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returns r(t, ), then: P r(t, n ) = P r(t, )

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(1313)

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where represents the convolution operator From the probabilistic point of view, the advantage of this hypothesis is purely computational From the economic point of view, the independence of returns means considering that the available and relevant information for the evaluation of nancial assets is correctly transferred in the quoted prices, which signi es the beginning of a concept of informational market ef ciency; stationarity signi es that the economic characteristics of an observed phenomenon do not change much in the course of time The existence of variance limits the uctuation of returns, not a stock market crash or stock market boom The rst formal model representing stock market variations was proposed [BAC 00] in 1900 by Louis Bachelier6 and based on pro ts (138), then modi ed in 1959 by Osborne for returns (139) and (1310): dS(t) = dr(t) = dt + dW (t) S(t) (1314)

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with coef cients R and > 0, where W (t) is a standard Brownian motion7, ie, E(W (1)) = 0 and E(W (1)2 ) = 1 Coef cient represents the expectation of instantaneous returns for the share purchased The risk of a nancial asset is generally measured by the coef cient of Brownian motion, called volatility by market professionals: this is a potential dispersal measure of stock market returns There are other risk measures, which are all based on this idea of conditional variability of returns in a given time (see [GOU 97b]) The solution of (1314) is obtained by supposing that X(t) = ln S(t) and by applying It s differentiation formulae in dX(t) We obtain: S(t) = S(0) exp 2 t + W (t) 2 t [0, T ] (1315)

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which is considered as the standard model of stock market variations

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6 A biography of Louis Bachelier has been compiled by Courtault et al [COU 00] For a description of nancial aspects of Bachelier s work and their impact on the nance industry, see [WAL 96] For an understanding of Bachelier s probabilistic work with reference to his epoch, see [TAQ 00] 7 Let us note that Bachelier did not know Brownian motion in the strict sense of its de nition because it is only in 1905 that this de nition would be given by Einstein, then in 1923 by Wiener However, Bachelier assumes that the successive differences of the form S(t, ) are independent of Gaussian distribution and of proportional variance in time interval , which leads to describe Wiener s process

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Research of Scaling Law on Stock Market Variations

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1322 Time scales in nancial modeling 13221 The existence of characteristic time If we choose modeling in physical time, thus in xed pace of time, the rst question that arises is that of selecting the pace of time , ie, resolution scale of market analysis: is it necessary to examine time variations daily, weekly, monthly, etc Which is the most appropriate observation scale for capturing the statistical structure of stock market variations Thus, a question of nancial nature appears: should the probability law which governs stock market variations be the same at all scales If we understand each time scale as representing an investment horizon for a given category of operators, there is apparently no particular reason for variations corresponding to a short trading horizon and those corresponding to a long horizon of portfolio manager to be modeled by the same probability law Equation (1312) shows that the return in scale T is the sum of returns in scales in the case of iid Generally, when we add iid random variables, the resulting probability law is different from initial probability laws Thus, a multiscale analysis seems, at rst sight, inevitable, if we do not wish to lose information on the market behavior at each scale of characteristic time of a given economic phenomenon The rst analysis of market behavior used only one observation frequency, often monthly It was Mandelbrot who, in 1962, became the rst to introduce the concept of simultaneous analysis on several scales, in order to compare distributions of periodic returns r(t, ) based on these different scales Mandelbrot sought to establish invariance by changing scale on periodic returns (ie, a fractal structure of the market) If P ( r(t, )) is the probability distribution of periodic returns r(t, ), relation (1313) is simpli ed as: P r(t, )

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= nH P r(t, )

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(1316)

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where H is a self-similar exponent which means that the process of returns {r(t), t 0} is self-similar to exponent H:

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