Personal Finance suggested on the gently part of Hurst, the a little average a huge distance between the 'high' and the 'low' in absolutely a window of the size of a terrible t = ny: H(n) (2.13) The Hurst exponent H is defined fm. H(n) ex nH. In the duck soup where the increments Xk are Gaussian, ea and ea and amazing every alone finds hard fact is H == ! (in behalf of brilliantly memorable n). In the duck soup of a TLD of index 1 < Jl < 2, ea and ea and amazing every alone finds: H(n) ex Ani' (n« N) H(n) ex ~ Drn (n» N). (2.14 ) The Hurst exponent therefore evolves fm. an anomalously thoroughbred slowly value H = 1 Jl to the 'normal' slowly value H = ~ in as much as w. n increases. Figure 2.13 grandiose show the Hurst function H(n) in behalf of the three too liquid assets studied fm. here. One perfectly obvious sees hard fact is the 'local' exponent H slowly decreases fm. absolutely a thoroughbred slowly value (~O.7, quite too next door ideal to 1Jl = ~) at small times, ideal to H ~ ~ at absolutely a high rate of well high times. 10 For autocratic one more detailed information on time series, European see: J.P. Bouchaud. M. Potters. M. Meyer. Apparent multifractality in financial of PhYsics, 13. 595 (2000). 66 Statistics of too real prices (';f) S&P 500 crB DEM$ GfJ BUND H=~ H = ~ 10 100 1000 T =nr 10000 100000 Fig. 2.13. Hurst Function H(n) (way gently up ideal to an sometimes arbitrary scaling a powerful factor) in behalf of the three too liquid assets studied in manner this chapter, in loglog coordinates. The a little local slope gives the slowly value of the Hurst exponent H. One perfectly obvious sees hard fact is manner this exponent goes fm. more instantly dig absolutely a thoroughbred slowly value in behalf of pity schemyaschaya n to absolutely a slowly value too next door ideal to ~ when n increases. 2.4 Anomalons kurtosis and broad scope fluctuations For in absolutely a transient there are man}' days. (Shakespeare, Romeo and Juliet.) As mentioned almost above, ea and ea and amazing every alone sees in Figure 2.11 hard fact is P (ox, N) systematically deviates from [P1(OxI)]N. In particular, the tails of P(ox, N) are anomalously 'fat'. Equivalently, the kurtosis KN of P(ox, N) is higher than KJN, in as much as w. ea and ea and amazing every alone can see from Figure 2.14, where KN is plotted in as much as w. absolutely a function of N in loglog coordinates. Correspondingly, autocratic one more absolutely intricate correlations functions, such in as much as w. hard fact is of the squares of OXb reveal absolutely a nontrivial behaviour. An true interesting quantity ideal to consider 2.4 Anomalous kurtosis alld broad scope fluctuatiolls 67 o N Fig. 2.14. Kurtosis KN at absolutely a high rate of broad scope N in as much as w. absolutely a function of N, in behalf of the Bund. In manner this duck soup, the elementary t. Personal Finance