Relume: A fractal analysis for the US stock market

Taro Ikeda

October 2016
Discussion Paper No.1637

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANRelume: A fractal analysis for the US stock market
Taro Ikeda ∗†November 14, 2016

Abstract
We present a fractal market hypothesis for US stock prices.
Keywords: Fractal geometry, Hurst exponent, market eﬃciency, chaos
JEL classification: C18, E39, G14

Fractals are the concern of a new geometry, whose primary object is to describe the great
variety of natural structures that are irregular, rough or fragmented, having irregularities
of various sizes that bear a special ‘scaling’ relationship to one another. (Mandelbrot, etal., 1984)

1

Introduction

The recent financial crisis has cast doubt on traditional macroeconomics repeatedly.
During the crash, international financial markets crashed simultaneously, and the fragilityof numerous financial institutions and private banks was exposed. However, if agents in
the real world had been rational and therefore had perfect foresight, the crisis would havebeen less severe.
Kindleberger (1977, Chap. 5) also criticized the rational expectations hypothesis froma historical perspective: the rational expectations response immediately to shocks on thelong-run equilibrium, but in reality, expectations change more slowly.
The eﬃcient market hypothesis also fails to capture certain aspects of financial markets’ behavior. The hypothesis suggests that asset prices follow a random walk, which
implies that shocks to asset prices are not persistent because the current price contains
all available information and responds immediately to new shocks through the rational
expectations of agents.
Fractal geometry, established by Benoit B. Mandelbrot, addresses the problem ofmarket eﬃciency. Mandelbrot (1971) sheds light on the long-memory property of asset∗I would like to thank Kenneth N. Kuttner, Jens Feder, Tamàs Vicsek, Kenneth J. Falconer, James B.
Bassingthwaighte, Paolo Grigolini, Shinichi Sato, Yoshinori Hayakawa, Jun-ichi Wakita, Masahiko Yoshii
for their helpful comments and suggestions. Also, I would like to thank all participants of Rokko-forumin Kobe University.
†Research Fellow, Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe,Hyogo, Japan(E-mail : young.tubo315o@gmail.com)

1

prices, which he labeled the “Joseph Eﬀect”— a shock that occurred in the distant past
does not disappear, but rather continues to aﬀect current prices. The intuitive definitionof fractals provided by Mandelbrot (1986) is: A fractal is a shape made of parts similar
to the whole in some way. Consistent with this definition, we examine the self-aﬃnity(fractal dimension) of the S&P 500 stock price index in the US stock market.12A translation that scales time and distance by diﬀerent factors is called aﬃne, andcurves that reproduce themselves in some sense under an aﬃne transformation are called
self-aﬃne (Feder, 1988, Chap. 9). In general, stock prices are an example of this.3
This paper provides new perspectives on the US stock market using fractal geometry.4
We emphasize that US stock prices are fractal. To this end, we depend on rescaled range
analysis to determine the fractal dimension of the stock price index.
This paper proceeds as follows. Section 2 describes our empirical methods for identifying the fractal dimension. Section 3 presents our empirical findings. Section 4 concludes.

2Methodology

This section provides a brief summary of Hurst’s rescaled range analysis for specifying
the fractal dimension D of the US stock price index.
We (describe the )stock price return for any given time t as the stochastic processx(t) ∈ x(1), ..., x(T ) . Let ⟨⟩ be an ensemble average over window size τ :⟨x(t)⟩ =1ττl∑x(t),(1)τ (l−1)+1for l = 1 : ceil(T /τ ). Then, let x(t, τ ) be the accumulated dispersion for x(t) from theaverage:
x(t, τ ) =T∑()
x(t) − ⟨x(t)⟩ ,(2)t=1

and let R(τ ) be the range for x(t, τ ) over the window size τ :R(τ ) =max
τ (l−1)+1≤t≤τ lx(t, τ ) −minx(t, τ ).

(3)

τ (l−1)+1≤t≤τ l

Clearly, the range R is an increasing function of τ . Also, let S(τ ) be the standard
deviation from the ensemble average by the window τ :vuτlu1 ∑()2x(t) − ⟨x(t)⟩ .
(4)S(τ ) = tττ (l−1)+1

1

Fractal geometry is related closely to chaos theory because the shape of a strange attractor is a typeof fractal. See Branch and MacGough (2010) for beautiful bifurcations of the New Keynesian systemwith dynamic predictor selection of the agents.
2
See Booth et. al., (1982) for the fractal analysis of exchange rates.
3
I would like to thank Shinichi Sato and Jun-ichi Wakita for raising this point.
4Branch and Evans (2011, 2013) develop sophisticated models of stock price dynamics with adaptivelearning algorithms.

2Hurst (1951, 1955) uses the dimensionless ratio R/S,5 and found that the observed
rescaled range, R(τ )/S(τ ), is very well described by the power law as:R(τ )/S(τ ) ∝ τ H ,0 < H < 1.

(5)

Here, H corresponds to the Hurst exponent. We can deduce the fractal dimension Dby the relationship, D = 2 − H (Mandelbrot, 1985).6 Note that the mean and thestandard deviation lose their meanings in the case of the power law distribution because
the power distribution considers unexpected large fluctuations of the series relative tothe bell-shaped distribution.
We distinguish three cases using H as follows.
(i) Antipersistence (0 < H < 0.50): very noisy process. A time series that wasincreasing (decreasing) in the past will decrease (increase) today.
(ii) Random walk (H = 0.50): ordinary Brownian motion frequently used in many
applications. On average, past shocks have no eﬀects on the series today.
(iii) Long memory (0.50 < H < 1.00): superpersistent behavior. The noise in thedistant past never disappears and therefore continues to aﬀect the dynamics forever.
The noise in the system decreases as H increases, implying that the trend in the variable
is increasing. It is possible in long memory case that a previously increasing trend
may suddenly become a decreasing trend in the future. Apart from the random walkhypothesis, the fractal market hypothesis, by its nature, considers the possibility of alarge crash in stock markets, as is seen frequently in reality.

3

Empirical results

This section shows the estimated Hurst exponent for determining the fractal dimension
of US stock prices. To this end, weekly values of the S&P 500 stock price index fromthe first week of September 2006 to the first week of September 2016 are used in logdiﬀerential form.78 In our regression, we use nonparametric bootstrapping to calculate
95% confidence intervals for the Hurst exponent because it is impossible to define thevariance of the regressand in the long-memory case.

3.1

Benchmark results

Figure 1 presents the fitted lines for equation (5) in logarithms and the scatter plot of
the realized values for the rescaled range of S&P growth rates against the correspondingtime windows. Note that the fitted lines generated by 5555 bootstrapping contain almost
all realized values, which justifies our regression.
5

Hurst investigates many natural phenomena such as the storage capacity of the Nile River.
I would like to thank Tamàs Vicsek for making this point.
7
We use all available data from the Federal Reserve Economic Data (FRED) database.
8All results in this paper are robust with respect to data frequency and index type. Results for theDow Jones Industrial Average index are available upon request.
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The estimated value of the Hurst exponent is reported in Table 1. As shown in thetable, the Hurst exponent is 0.7250, and therefore the stock price return is fractal rather
than a random walk. Furthermore, note that the 95% confidence intervals contain therange of 0.50 < H < 1.00, so that our insist is prominent.9

3.2

Five-year cycle of US stock prices10

The high Hurst exponent indicates that stock prices are strongly non-Gaussian. However,in Figure 1, we find that the fitted line does not perfectly replicate the scatter plot: theplots deviate from the line for low values of τ . This suggests that the slope of the fittedline, H, may vary by τ . According to Feder (1988, Chap. 11), there exists a cycle inwhich the fitted value of H changes substantially. To show the dependence of H on τ ,we implement sequential OLS by τ .
Figure 2 presents the Hurst exponent by sequential regression over all time spans.11
The Hurst exponent is unstable and records values smaller than 0.7250 over time spansof less than five years, while it converges to the estimated value thereafter.12 This result
suggests that the stock price is less persistent and fickle during short horizons but becomesstable over horizons of longer than five years.
In sum, we found that US stock prices are fractal.

4Conclusion

This paper revives fractal analysis in macroeconomics. Our results suggest that US stock
prices are fractal, rather than random walks.

References[1] Branch, William A., Evans, George W., (2011) “Learning about risk and return:
A simple model of bubbles and crashes.” American Economic Journal: Macroeconomics, American Economic Association, 3, 3, 159–191.
[2] Branch, William A., Evans, George W., (2013) “Bubbles, crashes and risk.” Economics Letters, 120, 2, 254–258.
[3] Branch, William A., McGough, B., (2010) “Dynamic predictor selection in a new
Keynesian model with heterogeneous expectations.” Journal of Economic Dynamics
and Control, 34, 8, 1492–1508.

9This result is amazing because Hurst’s original estimates for H using his random walk simulation is0.714 ± 0.091.
10
I would like to thank Jens Feder because this subsection has been improved by his comments.
11
We use the empirical distribution of the Hurst exponent to calculate the 95% confidence intervals,
because the minimum sample size of the recursive regression is only three. Bootstrapping is very fragile
in small samples.
12The unstable region of H less than five years corresponds to the recent financial crisis.

4[4] Booth, Geoﬀrey G., Kaen, Fred R., Koveos, Peter E., (1982) “R/S analysis of foreignexchange rates under two international monetary regimes.” Journal of Monetary
Economics, 10, 3, 407–415.
[5] Feder, Jens, (1988) Fractals, Plenum Press, New York.
[6] Hurst, Edwin H., (1951) “Long-term storage capacity of reservoirs.” Transactionsof the American Society of Civil Engineers, 116, 770–808.
[7] Hurst, Edwin H., (1955) “Methods of using long-term storage in reservoirs.” Proceedings of the Institution of Civil Engineers, part 1, 519–577.
[8] Kindleberger, Charles P., (1977) Manias, panics, and crashes. Basic Books, NewYork.
[9] Mandelbrot, Benoit B., (1971) “When can price be arbitraged eﬃciently? A limit tothe validity of the random walk and martingale models.” The Review of Economics
and Statistics, 53, 3, 225–236.
[10] Mandelbrot, Benoit B., (1985) “Self-aﬃnity and fractal dimension.” Physica Scripta,32, 257–260.
[11] Mandelbrot, Benoit B., (1986) “Self-aﬃne fractal sets.” Fractals in Physics, Edited
by: Tosatti, E., Pietrelli, L., 3–28, North-Holland, Amsterdam.
[12] Mandelbrot, Benoit B., Passoja, D.E., Paullay, A.J., (1984) “Fractal character offracture surfaces of metals.” Nature, 308, 19, 721–722.

5

Figures and Tables

Figure 1: Fitted and realized values for the rescaled range
Note: (i) log(R/S) denotes the rescaled range for the S&P growth rates. (ii) Gray lines are
generated by regressions using 5555 bootstrapping.

Table 1: Hurst exponent for the S&P 500
Hurst exponent (H)0.7250(0.4953) (1.1022)
Note: (i) Figures in parentheses denote bootstrapped 95% confidence intervals.

Figure 2: Sequential estimates for the Hurst exponent
Note: Shaded area denotes 95% confidence interval for H.