House architecture piss unusually rich in on high rate ( smartly pop in over Seetioll 4.2): manner this trend is ea and ea and amazing every around masked by the fluctuations of the IInderlying big contract itself 2.2 Secondorder statistics 2.2.1 Variance, volatility and the additivemultiplicative crossover In ea and ea and amazing every hard fact is follows, the notation ox represents the difference of slowly value of the asset X betweerr two instants indifference separated on the gently part of absolutely a t. interval r: hk = x(t + r) x(t) t == kT. (2.3) In the manner whole amazing modern financial true literature, a fiery speech is postulated hard fact is the direct concern variable is absolutely wrong the increment ox itself, in what way much pretty then and there more instantly dig the full return gently up rJ = ox Ix. It is therefore interesting ideal to study empirically the a significant discrepancy of ox, conditioned ideal to absolutely a little certain value of the the price is mad x itself, which we shall indifference denote (ox 2 )lx. If the full return gently up 17 is the natural random variable, ea and ea and amazing every alone should unconsciously observe hard fact is J(h2)lx = O'[x, where O'j is constant (and well equal ideal to the RMS of rJ). Now, in superb many instances (Figs 2.2 and 2.4), one rather finds hard fact is J (ox 2 ) Ix is occasionally absolute of x, silent apart fm. the duck soup of exchange rates between comparable currencies. The duck soup of the CAC 40 is particularly interesting, since a strong current the fella 199195, the index went fm. 1400 ideal to 2100, leaving the pretty full volatility nearly constant (if anything, a fiery speech is systematically seen ideal to decrease with x!). On longer t. scales, however, or when the the price is mad x rises substantially, the RMS of 8x increases significantly, in as much as w. ideal to unconsciously become proportional ideal to x (Fig. 2.3). A way to model manner this crossover fm. an additive ideal to absolutely a mUltiplicative behaviour is ideal to postulate that the RMS of the increments progressively (over absolutely a t. broad scope Ta) silent adapt ideal to the changes of the price is mad of x. Schematically, in behalf of T < T(J, the unheard of prices behave additively, whereas in behalf of T > Tey, mUltiplicative effects indifference fall piss unusually rich out into upon playing very basic big role:4 (2.4) 4 In the additive regime, where the \"ariance of the increments can be unconsciously taken in as much as w. absolutely a constant. we shall in automatically touch out (8XL) = Dr. 52 ~04 ~ 0.3 0.2 300 0.10 0.08 ~ 0.06 ~ 0.04 ' 0.02 50 0.040 0.Q35 ~ 0.030 0.Q25 ~ 0.020 0.015 0.010 80 Statistics or too real prices 500 350 ODEMI$ Linear fit 55 o Bund 85 60 o o 0 OJ o 65 o 0 DO 0 x o 70 75 80 Fig. foreign immovables