Two Fractal Overlap Time Series: Earthquakes and Market Crashes

Capturing dynamical patterns of stock prices are major challenges both for epistemologists as well as for financial analysts [1]. The statistical properties of their (time) variations or fluctuations [1] are now well studied and characterized (with established fractal properties), but are not very useful for studying and anticipating their dynamics in the market. Noting that a single fractal gives essentially a time averaged picture, a minimal twofractal overlap time series model was introduced [2,3,4] to capture the time series of earthquake magnitudes. We find that the same model can be used to mimic and study the essential features of the time series of stock prices.

Let us consider first a geometric model [2,3,4,5] of the fault dynamics occurring in overlapping tectonic plates that form the earth’s lithosphere. A geological fault is created by a fracture in the earth’s rock layers followed by a displacement of one part relative to the other. The two surfaces of the fault are known to be self-similar fractals. In the model considered here [2,3,4,5], a fault is represented by a pair of overlapping identical fractals and the fault dynamics arising out of the relative motion of the associated tectonic plates is represented by sliding one of the fractals over the other; the overlap O between the two fractals represents the energy released in an earthquake whereas log O represents the magnitude of the earthquake. In the simplest form of the model each of the two identical fractals is represented by a regular Cantor set of fractal dimension log 2/ log 3 (see Fig. 1). This is the only exactly solvable model for earthquakes known so far. The exact analysis of this model [5] for a finite generation n of the Cantor sets with periodic boundary conditions showed that the probability of the overlap O, which assumes the values O = 2 n-k (k = 0, . . . , n), fol-

Since the index of the central term (i.e., the term for the most probable event) of the above distribution is n/3 + δ, -2/3 < δ < 1/3, for large values of n Eq. (1) may be written as

by replacing nk with n/3 ± r. For r ≪ n, we can write the normal approximation to the above binomial distribution as

not mentioning the factors that do not depend on O. Now

where

is the log-normal distribution of O. As the generation index n → ∞, the normal factor spreads indefinitely (since its width is proportional to √ n) and becomes a very weak function of O so that it may be considered to be almost constant; thus G(O) asymptotically assumes the form of a simple power law with an exponent that is independent of the fractal dimension of the overlapping Cantor sets [6]:

We now consider the time series O(t) of the overlap set (of two identical fractals [4,5]), as one slides over the other with uniform velocity. Let us again consider two regular cantor sets at finite generation n. As one set slides over the other, the overlap set changes. The total overlap O(t) at any instant t changes with time (see Fig. 2(a)). In Fig. 2(b) we show the behavior of the cumulative overlap [4]

This curve, for sets with generation n = 4, is approximately a straight line [4] with slope (16/5) 4 . In general, this curve approaches a strict straight line in the limit a → ∞, asymptotically, where the overlap set comes from the Cantor sets formed of a-1 blocks, taking away the central block, giving dimension of the Cantor sets equal to ln(a -1)/lna. The cumulative curve is then almost a straight line and has then a slope (a -1) 2 /a n for sets of generation n. If one defines a ‘crash’ occurring at time t i when O(t i ) -O(t i+1 ) ≥ ∆ (a preassigned large value) and one redefines the zero of the scale at each t i , then the behavior of the cumulative overlap Q o i (t) = t ti-1 O( t)d t, t ≤ t i , has got the peak value ‘quantization’ as shown in Fig. 2(c). The reason is obvious. This justifies the simple thumb rule: one can simply count the cumulative Q o i (t) of the overlaps since the last ‘crash’ or ‘shock’ at t i-1 and if the value exceeds the minimum value (q o ), one can safely extrapolate linearly and expect growth upto αq o here and face a ‘crash’ or overlap greater than ∆ (= 150 in Fig. 2). If nothing happens there, one can again wait upto a time until which the cumulative grows upto α 2 q o and feel a ‘crash’ and so on (α = 5 in the set considered in Fig. 2).

We now consider some typical stock price time-series data, available in the internet. The data analyzed here are for the New York Stock Exchange (NYSE) Indices [7]. In Fig. 3(a), we show that the daily stock price S(t) variations for about 10 years (daily closing price of the ‘industrial index’) from January 1966 to December 1979 (3505 trading days). The cumulative Q s (t) = t 0 S(t)dt has again a straight line variation with time t (Fig. 3(b)). Similar to the Cantor set analogy, we then define the major shock by identifying those variations when δS(t) of the prices in successive days exceeded a preassigned value ∆ (Fig. 3(c)). The

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