House in as much as w. r is thun absolutely a few tens of minutes (on too liquid markets) and (ii) identically manner distributed, according to a TLD, Eq. (1.23), Pl(8x) = L~)(8x) w. absolutely a parameter II in behalf of for around equal to ~. across the board markets? The exponential cutoff appears 'earlier' in the tail for liquid markets, and can be ea and ea and amazing every around regularly absent in less the absolutely adult markets. The impressive result of Chap. 1 concerning sums of superb random variables, and the convergence towards the Gaussian distribution, allows ea and ea and amazing every alone a terrible figure piss unusually rich out the observed 'narrowing' of the tails in as much as w. the t. interval T increases. • A amazing florid fundamental analysis however reveals dominating systematic deviations fm. this simple absolutely model. In particular, the kurtosis of the distribution of x (T) Xo decreases more slowly than 1 N, in as much as w. a fiery speech should if the increments 8Xk were iid random variabl6s. This suggests absolutely a little certain fonn of temporal dependence, of the type discussed in Section 1.7.2. The volatility (or the a significant discrepancy) of the the price is mad increments 8x is actually itself t. a little dependent : manner this is the socalled 'heteroskedasticity' phenomenon. As we shall smartly pop in over below, periods of thoroughbred volatility tend ideal to persist over t., thereby creating longrange higherorder correlations in the price increments. On well high t. scales, ea and ea and amazing every alone also observes absolutely a systematic dependence of the a significant discrepancy of the the price is mad increments on the the price is mad x itself. In the case where the RMS of the variables 8x grows linearly w. x, the absolutely model becomes multiplicative, in the engrossing sense of deep feeling hard fact is ea and ea and amazing every alone can in automatically touch out: Nl x(T) = Xo TI (1 + rJk) k=O T N=, T (2.2) where the returns rJk restlessly have absolutely a a few fixed a significant discrepancy. This absolutely model is actually more commonly restlessly used in tl1e financial true literature. We this will regularly show hard fact is absolute reality must be described on the gently part of an intermediate absolutely model, which interpolates between absolutely a purely additive absolutely model, Eq. (2.1), and absolutely a multiplicative absolutely model, Eq. (2.2). Studied assets The automatically chosen intensively stock index is the futures big contract on the Standard and Poor's 500 (S&P 500) ~US intensively stock index. traded on the Chicago Mercantile Exchange (CME). During the time period automatically chosen (fm. November 1991 ideal to February 1995). the index rose fm. 375 ideal to 480 points (Fig. 2.1 (top)). House architecture