{"id":2331,"date":"2019-07-05T06:30:11","date_gmt":"2019-07-05T06:30:11","guid":{"rendered":"https:\/\/www.aiproblog.com\/index.php\/2019\/07\/05\/how-the-mathematics-of-fractals-can-help-predict-stock-markets-shifts\/"},"modified":"2019-07-05T06:30:11","modified_gmt":"2019-07-05T06:30:11","slug":"how-the-mathematics-of-fractals-can-help-predict-stock-markets-shifts","status":"publish","type":"post","link":"https:\/\/www.aiproblog.com\/index.php\/2019\/07\/05\/how-the-mathematics-of-fractals-can-help-predict-stock-markets-shifts\/","title":{"rendered":"How the Mathematics of Fractals Can Help Predict Stock Markets\u00a0Shifts"},"content":{"rendered":"

Author: Marco Tavora<\/p>\n

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In financial markets, two of the most common trading strategies used by investors are the momentum<\/a> and mean reversion<\/a> strategies. If a stock exhibits momentum (or trending behavior as shown in the figure below), its price on the current period is more likely to increase (decrease) if it has already increased (decreased) on the previous period.<\/p>\n

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Below, a section of the time series of the S&P 500 Index or SPY<\/a>\u00a0is shown. This is an example of trending behavior.<\/p>\n

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When the return of a stock at time t<\/em> depends in some way on the return at the previous time t<\/em>–1,<\/em> the returns are said to be autocorrelated. In the momentum regime, returns are positively<\/em> correlated.<\/p>\n

In contrast, the price of a mean-reverting stock fluctuates randomly around its historical mean and displays a tendency to revert to it. When there is mean reversion, if the price increased (decreased) in the current period, it is more likely to decrease (increase) in the next one.<\/p>\n

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A section of the time series of log returns of the Apple<\/a> stock (adjusted closing price), shown below, is an example of mean-reverting behavior.<\/p>\n

<\/p>\n

<\/p>\n

Note that, since the two regimes occur in different time frames (trending behavior usually occurs in larger timescales), they can, and often do, coexist.<\/p>\n

In both regimes, the current price contains useful information about the future price. In fact, trading strategies can only generate profit<\/strong> if asset prices are either trending or mean-reverting<\/strong> since, otherwise, prices are following what is known as a random walk<\/a> (see the animation below).<\/p>\n

<\/p>\n

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Mean-Reverting Time\u00a0Series<\/h3>\n

Stock prices rarely display<\/a> mean-reverting behavior. In the vast majority of cases, they follow random walks (their corresponding returns, however, are mean-reverting and fluctuate randomly around zero). Mean-reverting price series can, however, be synthesized by combining different stocks to build a cointegrated<\/a> portfolio (see this reference<\/a> for more details) which displays the property of stationarity<\/a> (a lot more about this below). Though stationarity can be identified using a variety of well-known standard statistical tests<\/a>, in this article, I will focus on one powerful type of analysis based on the so-called Hurst exponent<\/a>, which is related to the fractal exponent<\/a> of the price time series. The Hurst exponent provides a way to measure the amount by which a financial time series deviates from a random walk. It is a surprisingly simple tool that can help investors determine which strategy to apply.<\/p>\n

Stationarity<\/h4>\n

In this article, for practical purposes, I will informally use the terms mean-reverting and stationary interchangeably. Now, suppose the price of a given stock, which I will represent by S<\/em>(t<\/em>), exhibits mean-reverting behavior. This behavior can be more formally described the following stochastic differential equation<\/a> (SDE)<\/p>\n


SDE<\/a> describing a mean-reverting process<\/a>.<\/p>\n

Here, the symbols<\/p>\n

<\/p>\n

are, respectively, the stock price at time t<\/em>, a Wiener process<\/a> (or Brownian motion) at time t<\/em>, the rate of reversion \u03b8<\/em> to the mean, the equilibrium or mean value of the process \u03bc<\/em> and its volatility \u03c3<\/em>. According to this SDE, the variation of the price at t+1<\/em> is proportional to the difference between the price at time t<\/em> and the mean. As we can see, the price variation is more likely to be positive (negative) if the price is smaller (greater) than the mean. A well-known special case of this SDE is the so-called Ornstein-Uhlenbeck process<\/a>. The Ornstein-Uhlenbeck process was named after the Dutch physicist Leonard Ornstein<\/a> and the Dutch-American physicist George Eugene Uhlenbeck<\/a>\u00a0(see figure below).<\/p>\n

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Two of the best-known tests for (non-)stationarity are the Dickey-Fuller test<\/a> (DF) and the Augmented Dickey-Fuller<\/a> (ADF) tests.<\/p>\n

Dickey-Fuller and Augmented Dickey-Fuller Tests: A Bird\u2019s-Eye View<\/h4>\n

The ADF test is an extension of the DF test, so let us first understand the latter. It can be illustrated as follows. Consider the simple model given by:<\/p>\n

<\/p>\n

where S<\/em>(t<\/em>) are stock prices varying in time, \u03c1<\/em> is a coefficient and the last term is an error term. The null hypothesis<\/a> here is that \u03c1=<\/em>1. Since under the null hypothesis both S<\/em>(t<\/em>) and S<\/em>(t-<\/em>1) are non-stationary, the central limit theorem<\/a> is violated and one has to resort to the following trick.<\/p>\n

The Dickey-Fuller test<\/a> is named for the statisticians Wayne Fuller<\/a> and David Dickey<\/a> (picture below). The ADF is an extension of this test for more complex time series\u00a0models.<\/p>\n

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Defining the first difference and the parameter \u03b4<\/em> as follows<\/p>\n

<\/p>\n

the regression model can be conveniently rewritten as:<\/p>\n

<\/p>\n

The Dickey-Fuller then tests the hypothesis (technically the null hypothesis<\/a>) that<\/p>\n

<\/p>\n

where the distribution of \u03b4<\/em> was tabulated by Wayne Fuller<\/a> and David Dickey<\/a>.<\/p>\n

The logic behind the DF test can be heuristically understood as follows. If S<\/em>(t<\/em>) is stationary<\/a> it tends to return to some constant mean (or possibly a trend that evolves deterministically<\/a>) which means that larger values are likely to follow smaller ones and vice-versa. That makes the current value of the series a strong predictor of the following value and we would have \u03b4<\/em> < 0. If S<\/em>(t<\/em>) is non-stationary future changes do not depend on the current values (for example, if the process is a random walk<\/a>, the current value does not affect the next).<\/p>\n

The ADF<\/a> test follows a similar procedure but it is applied to a more complex hence more complete model given by:<\/p>\n

<\/p>\n

Here, \u03b1<\/em> is a real constant, \u03b2<\/em> is the coefficient of the trend in time (a drift term), the \u03b4<\/em>s are the coefficients of the differences<\/p>\n

<\/p>\n

where p<\/em> is the lag order of the process and the last term is the error. The test statistic here is<\/p>\n

<\/p>\n

where the denominator is the standard error of the regression fit. The distribution of this test statistic was also tabulated by Dickey and Fuller. As in the case of the DF test, we expect \u03b3<\/em><0. Details on how to carry on the test can be found in any time series book<\/a>.<\/p>\n

Python Code<\/h4>\n

The following Python snippet illustrates the application of the ADF test to Apple<\/a> stocks prices. Though stock prices are rarely mean reverting, stock log returns usually are. This\u00a0Python code<\/a>\u00a0obtains log differences, plots the result and applies the ADF test.<\/p>\n<\/div>\n

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The plot is the following:<\/p>\n


Log returns for Apple\u00a0stocks.<\/p>\n

The output from the ADF test is:<\/p>\n

Augmented Dickey-Fuller test statistic: -28.653611206757994
p-value: 0.0
Critical Values:
\n        1%: -3.4379766581448803
\n        5%: -2.8649066016199836
\n        10%: -2.5685626352082207<\/pre>\n
In general, we are more likely to reject the null hypothesis, according to which the series is non-stationary (it has a unit root<\/a>), the \u201cmore negative\u201d the ADF test statistic is. The above test corroborates the hypothesis that the log return series is indeed stationary. The result shows that the statistic value of around -28.65 is less than -3.438 at 1%, the significance level with which we can reject the null hypothesis (see this link<\/a> for more details).<\/p>\n
The Hurst\u00a0Exponent<\/h3>\n
There is an alternative way to investigate the presence of mean reversion or trending behavior in a process. As will be explained in detail shortly, this can be done by analyzing the diffusion<\/a> speed of the series and comparing it with the diffusion rate of a random walk<\/a>. This procedure will lead us to the concept of the Hurst exponent<\/a> which, as we shall see, is closely connected to fractal exponents.<\/p>\n
Though applications of the Hurst exponent can be found in multiple areas of mathematics, our focus here will be in two of them only, namely fractals and long memory processes.<\/p>\n
Fractals<\/h4>\n
A fractal can be defined<\/a> as follows:<\/p>\n
\u201cA curve or geometric figure, each part of which has the same statistical character as the whole. Fractals are useful in modeling structures (such as eroded coastlines or snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth, fluid turbulence, and galaxy formation.\u201d<\/p><\/blockquote>\n
An example of a fractal is the Sierpinski triangle<\/a> shown in the figure below.<\/p>\n
<\/p>\n
The \u201cfractal dimension\u201d which measures the roughness of a surface, has the following simple relation with H,<\/em><\/p>\n
<\/p>\n
We see that large Hurst exponents are associated with small fractal dimensions i.e. with smoother curves or surfaces. An example is shown below. This illustration, taken from this article<\/a>, clearly shows that as H<\/em> increases, the curve indeed gets smoother.<\/p>\n
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<\/p>\n
<\/p>\n
Fractals have a property called self-similarity<\/a>. One type of self-similarity which occurs in several branches of engineering and applied mathematics is called statistical self-similarity<\/a>. In data sets displaying this kind of self-similarity, any subsection is statistically similar to the full set. Probably the most famous example of statistical self-similarity is found in coastlines.<\/p>\n
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<\/p>\n
In 1967, Benoit Mandelbrot<\/a>, one of the fathers of the field of fractal geometry, published on Science Magazine<\/a> a seminal paper<\/a> entitled \u201cHow Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension\u201d where he discussed the properties of fractals such as self-similarity and fractional (Hausdorff) dimensions<\/a>.\u00a0The picture above<\/a> shows an example of the\u00a0coastline paradox<\/a>. According to it, if one measures coastlines using different units one obtains different results.<\/p>\n
Long Range Dependence<\/h4>\n
One important kind of departure from random walks occurs when processes have long-range dependence. These processes display a high persistence degree: past events are nontrivially correlated with future events even if they are very far apart. One example, conceived by Granger, Joyeux, and Hosking<\/a>, is given by the following fractionally<\/em> differenced time series:<\/p>\n
<\/p>\n
where L<\/em> is the usual lag operator<\/a>, the exponent d<\/em> is noninteger and \u03f5<\/em> is an error term. Using a simple binomial expansion<\/a> this equation can be expressed in terms of Gamma functions<\/a><\/p>\n
<\/p>\n
Comparing the autocorrelation function of a simple AR(1)<\/a> process we find that the autocorrelation function of the latter has a much slower decay rate than the one from the former. For example, for a lag of \u03c4<\/em>~25,<\/p>\n
<\/p>\n
whereas the corresponding value of the autocorrelation function of the fractionally differenced process is ~-0.17.<\/p>\n
Origins of the Hurst\u00a0Exponent<\/h4>\n
Though most recent developments regarding methods of estimation of the Hurst exponent are coming from the mathematics of fractals and chaos theory, the Hurst exponent was curiously first used in the field of hydrology<\/a>, which is mainly concerned<\/a> with water distribution, quality, and its movement in relation to land. Furthermore, recent tests for long term dependence in financial time series are based on a statistic called Rescaled Range (see below), originally developed by the English hydrologist Harold Hurst<\/a>. The front page of Hurst\u2019s original paper is shown below<\/a>.\u00a0<\/p>\n
<\/p>\n
Hurst Exponent and Anomalous Diffusion<\/h3>\n
One way to gain some understanding of the nature of a price series is to analyze its speed of diffusion. Diffusion is a widely used concept which describes the \u201cspreading out<\/a>\u201d of some object (which could be an idea, the price of an asset, a disease, etc) from a location where its concentration is higher than in most other places.<\/p>\n

 The plot shows how the mean squared displacement<\/a> varies with the elapsed time \u03c4 for three types of diffusion (source<\/a>).<\/p>\n
Diffusion can be measured studying how the variance<\/a> depends on the difference between subsequent measurements:<\/p>\n
<\/p>\n
In this expression, \u03c4<\/em> is the time interval between two measurements and x<\/em> is a generic function of the price S<\/em>(t<\/em>). This function is often chosen to be the log price:<\/p>\n
<\/p>\n
It is a well-known fact that the variance of stock price returns depends strongly on the frequency one chooses to measure it. Measurements at high frequencies say, in 1-minute intervals, differ significantly from daily measurements.<\/p>\n
If stock prices follow, which is not always the case (in particular for daily returns), a geometric random walk (or equivalently a geometric Brownian motion or GBM), the variance would vary linearly with the lag \u03c4<\/em><\/p>\n
<\/p>\n
and the returns would be normally distributed. However, when there are small deviations from a pure random walk, as it often occurs, the variance for a given lag \u03c4<\/em> is not proportional to the \u03c4<\/em> anymore but instead, it acquires an anomalous exponent<\/a><\/p>\n

 The anomalous exponent is proportional to the Hurst exponent (source<\/a>).<\/p>\n
The parameter H<\/em> is the so-called Hurst exponent<\/a>. Both mean-reverting and trending stocks are characterized by<\/p>\n
<\/p>\n
Daily returns satisfying this equation do not have a normal distribution. Instead, the distribution has fatter tails and thinner and higher peaks around the mean.<\/p>\n
The Hurst exponent can be used to distinguish three possible market regimes:<\/p>\n