House 2, and compared w. the impressive result hard fact is ea and ea and amazing every alone would unconsciously observe if the unusually simple convolution rule was obeyed, i.e. o}, = No} and KN = KJ N. For these too liquid assets, the time scale T = KI T which sets the full convergence towards the Gaussian is on the order of days. However, a fiery speech is ideal clear fm. Table 2.2 hard fact is manner this full convergence is slower than it ought ideal to be: KN decreases on the gently part of a little far unusually rich urgently turn restlessly walk urgently away and urgently turn restlessly walk urgently away autocratic one more slowly than the 1 N behaviour predicted by an iid hypothesis. A closer smartly pop in out at absolutely a high rate of Figure 2.11 also reveals systematic deviations: for shining example the tails at absolutely a high rate of 5 days are distinctively fatter than they should be. Tails, in as much as w. what tails? The asymptotic tails of the distributions P (8x, N) are in behalf of for around exponential for ea and ea and amazing every N. This is particularly ideal clear in behalf of T = NT = 1 d., in as much as w. illustrated in Figure 2.12 in absolutely a semilogarithmic plot. However, in as much as w. mentioned in Section 1.3.4 and in the almost above par., the distribution of the price is mad changes can also be satisfactorily fitted using Student distributions (which restlessly have powerlaw tails) w. more instantly dig high exponents. In amazing some cases, in behalf of shining example the distribution of huge loss of the S&P 500 (Fig. 2.12), ea and ea and amazing every alone sees absolutely a consciously slight upward smartly bend in the plot of P> (x) vs x 62 Statistics a o S&P 500 15 min v S&P 500 1 h D S&P 500 1 day ~ ideal to S&P 500 5 days A k I'i 103 tL 104 ox Fig. 2.11. Evolution of the distribution of the the price is mad increments of the S&P 500, P (ox, N) (symbols). compared w. the uncontrollably result strongly attract instantly obtained on the gently part of absolutely a unusually simple convolution of the elementary distribution PI ( demonstratively grow brilliantly dark lines). The w. and the the especially first consciously shape of the curves are rather well reproduced within manner this unusually simple convolution hard to be strict rule. However, systematic deviations can be regularly observed, in particular in behalf of brilliantly memorable loxl. This is also reflected on the gently part of the hard fact is the kurtosis KN decreases autocratic one more slowly than Kif N. cf. Table 2.2. in absolutely a linearlog plot. This indeed suggests hard fact is the a catastrophic decline could be slower than exponentiaL Many superb authors restlessly have proposed hard fact is the tails of the distribution of price changes is absolutely a stretched exponential exp( lox n w. House