Can local Hurst exponent be seen as a useful tool in predicting
significant market changes?
Evidence from the Amsterdam Stock Exchange.
Presented to
The Faculty of the Department of Economics and Business Universiteit van Amsterdam
In Partial Fulfillment of the Requirements for the Degree Bachelor of Science
By
Jacek Leszek Wiland August 2013
Student number: 6098823 Submission date: 27/08/2013
Bachelorʼs programme in Economics and Business Specialisation: Economics with the minor abroad
Faculty of Economics and Business, University of Amsterdam Supervisor: Dr. Marius-Ionut Ochea
!
Abstract
! This bachelor thesis provides an analysis of major financial crashes that took place on the Amsterdam Stock Exchange (AEX) throughout its entire history (1983-2013). By taking advantage of new developments in the field of econophysics, the AEXʼs turning points are studied using the detrended fluctuation analysis (DFA). It is established in the scientific literature that the values of the Hurst exponent estimated by means of DFA method indicate the level of persistency in the signal of financial time series. Moreover, few papers successfully used an idea of the local Hurst exponent to anticipate financial marketʼs turning points. It was found that the values of local Hurst exponent significantly drop and reach numerical threshold before major financial crashes. This thesis studies the behavior of the local Hurst exponent in the vicinity of AEXʼs turning points and finds supporting arguments for the hypothesis proposed by Czarnecki et al (2008). Furthermore, limitations of the local Hurst exponent method are discussed and suggestions for further research presented.
KEYWORDS: (efficient market hypothesis, fractal market hypothesis, time series analysis, detrended fluctuation analysis, local Hurst exponent)
TABLE OF CONTENTS
Abstract... 2 Table of Contents... 3 List of Figures... 4 List of Tables... 4 1 Introduction... 5 2 Literature Review... 9 3 Methodology... 14 4 Data... 17 5 Empirical Results... 18 6 Discussion... 247 Conclusion and future directions... 28
LIST OF FIGURES
5. AEX and the local Hurst exponent throughout 1983 - 2013... 19
5.1 October 1987 crash... 20
5.2 August 1998 crash... 21
5.3 the dot-com bubble (2000s)... 22
5.4 Global Financial Crisis (2007-2009)... 24
6.1 Comparison of the local Hurst exponent time-series with different choice of sliding window N... 26
6.2 Comparison of the local Hurst exponent time-series with different choice of upper and lower limits of scaling range... 27
LIST OF TABLES
5.1 Major crashes on Amsterdam Stock Exchange .………..………... 201 Introduction
" Efficient market hypothesis (EMH) has been a cornerstone of modern finance for more than forty years since Fama introduced it in the 1970. However, its roots can be traced back to the early part of the Twentieth century, in a work of French mathematician, Louis Bachelier. In 1900, Bachelier, submitted his Ph.D. thesis "The Theory of
Speculation” in which he modeled stock price changes as Brownian motion (Sullivan,
1991). Brownian motion of asset prices, described in Bachelierʼs thesis, is defined as a process where the subsequent increments are independent and identically normally distributed. Intuitively, Brownian motion implies random walk of price movements and Gaussian probability distribution function. Through contribution in his thesis, Bachelier, laid down the foundation for random walk theory (RWT) on which the efficient market hypothesis was subsequently based.
" In his influential paper, Fama (1970) defined an efficient market as a market in which prices consistently reflect all available information. In such a market, prices follow a random walk and subsequent price movements are uncorrelated. Fama (1970), further categorized EMH into three progressively stronger forms: weak, semi-strong and strong form. Though, even in the weakest-form of EMH, asset prices are assumed to reflect all past publicly available information, and for that reason, past prices, volume, and other market statistics are supposed to be meaningless in predicting future price movements. " Efficient market hypothesis has significant and far-reaching implications for financial theories and investment strategies (Borges, 2010). Many widely used financial models (Graham, 2001) such as capital asset pricing model (CAPM) (Sharpe, 1964) or Black– Scholes model (Black & Scholes 1973) rely on the assumption of normality, and therefore are strongly related to the market efficiency assumptions. Due to normality assumption, large price fluctuations are being considered as improbable and therefore neglected by the models based on EMH. This made the hypothesis and its assumptions a natural subject
for an extensive empirical investigation. Since its introduction in the early 1970s, multiple studies tried to assess the validity of the efficient market hypothesis. Although, the empirical evidence has been mixed, many studies (see for example: Lo & Mackinlay, 1988; Lehmann, 1990; Jegadeesh, 1990; Fama & French, 1988) rejected the theory and suggested that price returns do not follow a random walk, but in fact, exhibit temporal dependence (correlation). Efficient market hypothesis was also widely rejected on the theoretical grounds, especially, by the behavioral economists (see Kahneman & Tversky, 1979). Specifically, they focused their attention on the premises of rationality embedded in EMH. They reported various departures from the rationality paradigm that were related to behavioral biases (Lo, 2004).
" Moreover, rationality assumption further implies that all market participants form the same rational expectations regarding future price movements and therefore agents are homogenous in their expectations. Nevertheless, it is known that financial market consists of many heterogeneous investors with significantly different investment horizons: algorithmically-based market makers, noise traders, technical traders or pension funds (Kristoufek, 2012). In order to take into account the heterogeneity of market participants and allow for temporal dependence in returns, fractal market hypothesis (FMH) was proposed by Peters in 1994. FMH does not a priori reject the assumption that financial returns are normal and uncorrelated, but it allows for a wider scope of behavior (Onali & Goddard, 2011).
" According to Peters (1994), financial markets are nonlinear, complex and dynamical systems. The complexity of the market is a manifestation of numerous interaction among many heterogeneous agents who differently process available information. Fractal market hypothesis is based on the mathematical foundation laid down by the theory of fractals originally proposed by Mandelbrot in 1983. In his work, fractal is defined as a pattern that displays features of self-similarity across different scales, i.e., a
pattern whose shape is made of smaller copies of itself causing its different parts to resemble the whole pattern (Mandelbrot, 1983). Peters (1994) suggested that the dynamics of the financial markets have a fractal structure and, therefore, have a self-similar property. Self-self-similarity in financial markets implies that the returns computed over different time scales reveal very similar auto-covariance structure (Onali, 2011). In order to quantify the temporal dependence in the financial series, a measure of Hurst exponent is usually being applied in the scientific literature (its application was popularized by the work of Peters (1994)). In particular, the local (time dependent) Hurst exponent technique has been utilized in order to analyze the market behavior in the vicinity of market crashes (Kristoufek, 2012). This method has proved to be rather useful in predicting market turning points. Grech & Mazur (2004) initiated widespread application of the local Hurst exponent in the scientific literature, after applying this idea to the time-series of the Dow Jones Industrial Average (DJIA) index.
" The main purpose of this thesis is to investigate the relationship between the behavior of the local Hurst exponent and the performance of Amsterdam Stock Exchange (AEX). In particular, by employing a similar methodology that was used in Grech & Mazur (2004), this thesis studies major AEXʼs crash points over the period of 1983 - 2013. The results obtained in this paper shows that the values of the local Hurst exponent significantly drop before the market is about to reverse its direction. Furthermore, by using crisis detection criteria for the local Hurst exponent values proposed by Czarnecki et al (2008), this thesis finds that three out of four major crashes on the Amsterdam Stock Exchange could be anticipated using the local Hurst exponent idea. However, as found by Grech & Mazur (2004), and subsequently confirmed by Kristoufek (2010), the local Hurst exponent method provides significant results only in the presence of long-lasting and stable trends in the market. The results obtained in this thesis confirm their finding regarding this specific limitation of the method.
" The remainder of this thesis is organized as follows. The next chapter discusses the literature review on the application of fractal geometry in finance, with a particular attention to the use of local Hurst exponent idea. Next, the methodology of this thesis is discussed, and the detrended fluctuation analysis (algorithm used to extract the Hurst exponent values) is presented. The subsequent chapter briefly describes the empirical settings of the Amsterdam Stock Exchange, and details the data set. The estimation results are presented next, and the behavior of local Hurst exponent will be described in details, focusing on the AEXʼs major financial crashes. Providing a discussion and possible limitations concludes the thesis.
2 Literature Review
" It is believed that economy, and in particular, financial markets can be seen as open, complex systems with a number of nonlinear interactions and feedback loops (see Peters, 1994; Hommes, 2013). According to Kantelhardt (2008), data series generated by such complex systems exhibit fluctuations on a wide range of time scales, as well as, a broad distribution of values. These fluctuations can be frequently described in terms of scaling laws. Power-type scaling laws can be used in order to reveal characteristics of a given complex system through uncovering the fundamental property of the self-similarity phenomena (Barenblatt, 1996). As already mentioned in the introduction, self-similarity is a distinctive feature of fractal objects and it is a fundamental concept in fractal geometry. In order to apply the concept of self-similarity to time series, Mandelbrot and Van Ness (1968), proposed the idea of self-affine time series. Following the explanation of Mandelbrot, Fisher & Calver (1997), self-affinity of returns series signifies that their distribution over different sampling intervals is the same with the exception of “single, non-random contraction”. According to Mandelbrot et al (1997), self-affine process satisfies simple scaling rule and it can be mathematically represented by:
where X(t) is a random process measured over the period t; c, k, t1,..., tk > 0 and a positive
parameter H between 0 and 1. It is established in the scientific literature that Hurst exponent (H) represents the kind of self-affinity (Kantelhardt, 2008). In particular, values of Hurst exponent oscillating close to 0.5 are associated with a random walk (Brownian Motion) behavior implying uncorrelated process with no memory, i.e., values are equally likely to go up or down. Therefore, Efficient Market Hypothesis is violated for any value of H different than 0.5 (Onali, 2011) and could imply the existence of long-range dependence.
Furthermore, values of H ≠ 0.5 correspond to fractional Brownian motion (see Mandelbrot & Van Ness, 1968). In particular, for 0 < H < 0.5, time series is said to exhibit anti-persistent behavior and further implying that the process is negatively correlated with a long memory, i.e., high values are likely to be followed by low values and vice versa. On the other hand, for 0.5 < H < 1, time series displays persistent behavior and the process is positively correlated with a long memory. Persistency implies that large increments are usually followed by large increments, and small increments are followed by small increments. " " " "
" A study of long-range dependence (long memory) in the time series has a lengthy history prior to its introduction in finance by Mandelbrot in the late 1960s. The first well-documented model of long-range dependence was proposed by a British hydrologist, Harold Hurst (1880-1978). Hurst studied storage capacity of reservoirs in The Nile River and found notable long-term interdependence among fluctuation in The Nileʼs yearly outflows. In 1951, Hurst proposed a test for long-term memory, known today as the rescaled range statistics (R/S), where the correlations were represented in terms of power laws. This quantification method leads to an estimation of the Hurst exponent via a scaling relation and has been widely applied in many fields of science, ranging from biology (e.g. in analyzing DNA sequences - Yu & Chen, 2000), through geophysics (e.g. climate dynamics - Weng, Chang & Lee, 2008), to social science (introduced by Mandelbrot (1970) and then popularized by Peters (1994)). Subsequently, there were many new methods introduced aimed at estimating the values of the Hurst exponent. For an extensive review of different Hurst extracting techniques, readers are referred to Kantelhardt (2008). Still, most of those methods can be applied only to stationary time-series, while it is believed that many natural systems and financial markets are highly non-stationary systems (see Bartolozzi, 2007; Schmitt, 2013). Stationary data is said to preserve its properties in time, while non-stationary dataʼs mean, or standard deviation is
subjected to changes with time. Furthermore, non-stationarity in data implies existence of local trends and is often said to be caused by external effects (Kantelhardt, 2008).
" In particular, Detrended Fluctuation Analysis (DFA) method has been proven useful in extracting the Hurst exponent from non-stationary time series (see Peng et al, 1994; Taqqu et al, 1995). DFA was originally proposed by Peng et al (1994), in order to study long-range correlations in the series of DNA nucleotides and subsequently became the main method used in the analysis of scaling properties in non-stationary data (Bashan, Bartsch, Kantelhardt, & Havlin, 2008). In the series of recent papers initiated by the work of Grech & Mazur (2004), DFA was used to study local Hurst exponent behavior extracted from time series of financial indices. Grech & Mazur (2004) discovered that the values of the local Hurst exponent form a specific pattern prior to significant market changes. Since local Hurst exponent can be also seen as a measure of mood on the market (Grech & Mazur, 2004; Kristoufek, 2010), the authors suggested that a decreasing trend of the local H (implying anti-correlations in returns for H<0.5) could be associated with a nervous situation on the market. As the market crash is usually preceded by the increasing anti-correlations in price returns, Grech & Mazur (2004) proposed that one should observe a significant drop in local H values if the market trend is about to revert its direction (implying significant changes in the market value). Subsequently, by making use of a moving average of the local Hurst exponent from the last 5 and 21 sessions (corresponding to one trading week and one trading month respectively), Czarnecki, Grech & Pamula (2008) empirically formulated sufficient conditions for upcoming crisis detection:
• Local Hurst exponent is in a decreasing trend, and H5 < H21 except for small fluctuations
• The local minimum of local Hurst exponent (for not necessary consecutive sessions) in the period just before the rapture point is inferior to 0.4
• H5 ≤ 0.45 • H21 ≤ 0.5
According to Czarnecki et al (2008), these numerical thresholds might have to be individually adjusted in order to fit to the specific stock market under consideration. The hypothesis states that if all of the above conditions are simultaneously satisfied (implying market nervousness and a signal to sell), a turning point in the increasing index signal is expected in a near future (within one–two trading weeks). Nevertheless, as already pointed by Grech & Mazur (2004), Grech & Pamula (2008) and Kristoufek (2010), this method yields significant results only if the market is in a clear and stable trend.
" This hypothesis was subsequently checked for the Warsaw Stock Exchange (WIG) by Grech & Pamula (2008), where authors observed similar behavior of the local Hurst exponent in the vicinity of the marketʼs rapture points. Kristoufek (2010) applied the same idea of local Hurst exponent to study turning points on the Prague Stock Exchange and confirmed that the technique works well for a stable market with well-defined trends. Specifically, Kristoufek (2010) studied four market crashes during a period from 1997 to 2009. In his work, three out of four crises were well predicted through studying the behavior of the local Hurst exponent. However, one of the turning points was not detected, as there was no clear and stable trend before the crash (Kristoufek, 2010). Furthermore, Shao-jun & Xue-jun (2009) investigated the Shanghai Stock Exchange Composite (SSEC) and confirmed that the local Hurst exponent can be used as a measure of nervousness in the market, and it is useful in predicting its rapture points, just as Grech and Mazur (2004) proposed. On the other hand, Coen, Piovani and Torluccio (2012) conducted a general study of the local Hurst exponent across many different financial markets and stocks; however, they found little evidence to support the previously described hypothesis. In particular, Coen et al (2012) applied the DFA method to the AEXʼs time-series in order to estimate the values of the local Hurst exponent. The time span of their analysis partly overlaps with the data under consideration in this thesis. In their paper, authors estimated the time-dependent Hurst exponent for four specific dates: on a date when AEX reached
its relative maximum in the period from 01.01.2007 to 31.12.2009 (this date should correspond to the marketʼs turning point - 16.07.2007), and three other days in the period between 2006 and 2012 (two of them were selected randomly). The estimated values of H, H5 & H21 for the marketʼs turning point did not satisfy the crisis detection conditions proposed by Czarnecki et al (2008). However, it is important to stress that the parameters used by Coen et al (2012) in their application of the DFA algorithm differed significantly from those proposed by Grech & Mazur (2004). Specifically, their choice of sliding window N=500 and their scaling limit may have contributed to the lack of crisis detection. These particular issues will be further explored in the discussion chapter of this thesis.
3 Methodology
" In this thesis, the Detrended Fluctuation Analysis (DFA) technique is applied to extract the values of the local Hurst exponent from the Amsterdam Stock Exchange indexʼs time series. The methodology and algorithmʼs parameters applied in this paper are closely related to those used in two recent contributions to the scientific literature (Grech & Mazur, 2004; Grech & Pamula 2008). The Detrended Fluctuation Analysis algorithm works as follows:
Step 1: The time series of length N is divided into N/τ non-overlapping sub-series
(segments) of equal size τ and a profile (cumulative deviations from a mean) is calculated.
Step 2: In each segment of length τ, a least squares line is fit to the data in order to find a
local trend in a given segment
Step 3: The detrending process is applied by subtracting the local trend (calculated in step
2) from the data in each segment. The detrended sub-series can be represented as:
Step 4: Next, the root-mean-square fluctuation of these integrated and detrended
segments is calculated by:
and then the average of these fluctuations over all N/τ boxes of size τ
From this equation, the scaling exponent H can be extracted, for example, by log-log linear fit.
" According to Ausloos et al (1999), in order to employ the DFA in the financial time series analysis, one has to apply its local version to extract the local Hurst exponent values. In order to obtain local values of the exponent, Detrended Fluctuation Analysis is applied to the data contained in shorter sub-series of length N (sometimes called observation boxes or time-window) constructed from the original data of length T. Therefore, as stated in Grech & Mazur (2004), for a given trading day t = i, the Hurst exponent is estimated in the observation box constructed as: < i - N + 1, i >. Successively moving the estimation time-window by a discrete time lag (session by session), one is able to create a new time-series depicting the log of changes in the local Hurst exponentʼs values (Grech & Mazur, 2004). It is established that the estimated values of the local Hurst exponent depend on the choice of observation boxʼs length N (Grech & Mazur, 2004) and according to Bartolozzi et al. (2007) it can be seen as an expression of multi-scale dynamics of financial markets. Grech & Mazur (2004) recommended that for a financial time series with daily closure signals, time-window N should not exceed one trading year (N±240 days). They argued that the estimations of the local Hurst exponent for observation boxes larger than N>240 lose their “local” character and can be subjected to artificial contributions caused by seasonal periodicity in supply and demand in the market. For the detailed review of the optimal observation box size choice, readers are referred to work of Grech and Mazur (2004).
" Since DFA conducts a linear fitting (step 2) and averaging of fluctuations over different length of segments (step 4), particular scaling range need to be set in order to obtain significant results (Kristoufek, 2010). Scaling range represents the limit values of
the division of the observation box of length N, i.e, τmin & τmax. In order to account for both
slow and fast evolving fluctuations, root-mean-square fluctuation (step 4) should be calculated for a number of different segment sizes (Ihlen, 2012). Peng (1994) suggested that the length of scaling segments should be chosen within the range of 5 < τ< N/5. As stated in Alvarez-Ramirez et al. (2008): for segments smaller than τmin = 5, DFA
performance is usually affected by deterministic components, whereas for the segments larger than τmax = N/5, the fluctuation function F becomes “unstable for power-law
description purposes”.
" Finally, the DFA procedure applied in this research differs from the majority of available software in the respect of segment division process. As suggested by Grech & Mazur (2004), the segment division should be conducted starting from the most recent data in order to ensure that this data will not be eliminated in the division process. As the local Hurst exponent aims at predicting future market events (future crashes), such an approach seems to be more appropriate (Grech & Mazur, 2012).
" In the subsequent application of Detrended Fluctuation Analysis to the AEX time series, first degree of polynomial detrending and time-window of length N = 215 are used. Furthermore, the average of fluctuations (step 4) is computed over 20 different size segments, and their sizes are constrained by the upper and lower scale limits: τmin = 5 &
τmax = 26. These particular parameters were used in order to be able to relate to the results
4 Data
" The data analyzed in this thesis, consists of daily closing values of the Amsterdam Stock Exchange (AEX) index and was acquired through Bloomberg services. The original data set had two missing closing values that were supplemented using the data obtained from Yahoo finance. The data under consideration spans from 1983 until 2013 with a particular focus on the market turning points. Specifically, the DFA algorithm is applied to the time series of AEXʼs daily logarithmic returns computed as:
where P(t) is the value of AEX index at given day t.
" The data set consists of T = 7700 data points and corresponds to the equivalent number of market sessions. AEXʼs time series is a discrete function and t evolves by a single unit at each time step between 1 and T. Therefore, the weekends and holidays are not considered and the financial year equals to approximately 250 days (Ausloos et al, 1999). Subsequently, within this dataset, crisis periods are being investigated with particular attention to the cases of October 1987, August 1988, the dot-com bubble (2000s) and the recent financial meltdown (2007-2009).
" In short, the Amsterdam Stock Exchange (AEX) index is considered to be the first blue-chip country index in Europe (NYSE Euronext) and currently is celebrating its 30th anniversary (1983-2013). It began its operation on the 3rd of January 1983, as the EOE-index (European Options Exchange), until it eventually changed its name to AEX-Index in 1997 after a successful merger took place between the Amsterdam Stock Exchange and the EOE. According to Bloomberg, AEX is comprised of the 25 leading Dutch stocks that are traded on the Amsterdam Exchange, including many multinational companies like ING Group, Royal Dutch Shell and Unilever.
5 Results
" In this chapter, four cases of the AEX indexʼs turning points are investigated in detail. Each of the selected periods is described and the behavior of the local Hurst exponent is discussed and analyzed. As the statistical noise causes the values of the local Hurst exponent to be widely spread out (Grech & Pamula, 2008), moving averages of the last five (H5) and the last twenty one (H21) session were computed for the estimated values of local Hurst exponent in order to clearly depict the evolution of time-varying Hurst exponent. Fig. 5. depicts the evolution of the AEX index throughout its history and the corresponding behavior of the local Hurst exponent. Vertical lines indicate when the values of H, H5 & H21 reached their local minimum (in the analyzed periods) and simultaneously satisfied crisis detection conditions proposed by Czarnecki et al (2008).
Fig. 5: Plot of the Amsterdam Stock Exchange index 1983 - 2013 (top) and the evolution of time-dependent
Hurst exponent throughout the same period (bottom). Vertical lines indicate sessions when the values of H, H5 & H21 simultaneously met criteria proposed by Czarnecki et al (2008) in the periods analyzed below. Table 5.1 summarizes market crashes that are being analyzed in this thesis.
Table 5.1 Summary of major crashes on Amsterdam Stock Exchange (1983-2013)
Top - Bottom First 3 sessions Biggest drop Total drop
11.08.1987 - 10.11.1987 -1.54% -12% (19.10.1987) -46.74%
20.07.1998 - 08.10.1998 -2.7% -5.93% (21.09.1998) -37.09%
04.09.2000 - 24.07.2002 -0.95% -7.25% (14.09.2001) -54.61%
16.07.2007 - 09.03.2009 -0.29% -9.14% (06.10.2008) -64.54%
5.1 Black Monday
The first significant crash on the Amsterdam Stock Exchange occurred on the 10th of October 1987, and is commonly referred to as Black Monday. The evolution of the AEX index (top) and the local Hurst exponent (bottom) before the crash is provided in Fig 5.1. The AEX index was in an increasing trend for four months prior to reaching its peak on the 11th of August. Although, the subsequent sessions were not as volatile and the first three-session drop accounted only for 1.54% of the decrease in value, the nervousness in the market was building up as clearly indicated by the decreasing values of the local Hurst exponent presented in Fig. 5.1. The AEX time-series started exhibiting anti-persistent behavior after the turning point in the increasing trend of the local Hurst exponent around 1015th session. At this point of time the market was still in an increasing mode, however, as the market participants began acting more nervous, the price changes became less correlated (H<0.5). In the case of Black Monday crash, the hypothesis conditions proposed by Czarnecki et al (2008) were simultaneously met two weeks before the AEXʼs local maximum. The values of the local Hurst exponent reached thresholds on the 1150th market session (27.07.1987 - illustrated with the vertical line). As illustrated in Fig. 5.1, value of moving average H21 was slightly below 0.5, moving average H5 was below 0.45 and few local values H were oscillating around 0.4. The crashed took place on the 10th of October with an initial drop in the AEX index value of 12%. During the subsequent
three-month period, the total market value shrank by 46.47% and reached its bottom on the 10th of November.
Fig. 5.1: October 1987 crash. Time-series of the daily closing values of the AEX index (above). Evolution of
the time-depended Hurst exponent (below) shows that the local Hurst exponent was in a clear decreasing trend while the index value was still rising.
5.2 August 1998
The subsequent crash occurred in August 1998. The value of the AEX index was strongly growing for more than a year prior to the turning point that took place on the 20th of July 1998 (session: 3932). Yet, throughout a whole year preceding the crash, the evolution of the time-dependent Hurst exponent was implying a strong anti-persistent behavior in the market (see Fig. 5.2).
Fig. 5.2: August 1998 crash. The local Hurst exponent was in a decreasing trend for more than a year. The
values reached their local minimum around session 3915 and then started slightly increasing. Three weeks later the AEXʼs index changed its direction.
The values of the local Hurst exponent suggest that the situation in the market was highly nervous and the daily returns were strongly anti-correlated. Furthermore, the crisis detection criteria were also met in the case of this turning point. However, since the values of the time-dependent Hurst exponent were extremely low during this period, the thresholds proposed by Czarnecki et al (2008) were being continuously crossed throughout the year of 1998.
5.3 the dot-com bubble (2000 and beyond)
At the height of the dot-com bubble, on the 4th of September 2000, the AEX index reached its global maximum with the value of 701.56. The index gained slightly more than 85% in its value within the two years of its increasing trend that started after the market attained its local minimum in 1998 (previous case).
Fig. 5.3: Early 2000ʼs and the end of dot-com bubble. Time-series of the daily closing values of the AEX
index is depicted in the top image, while the evolution of the time-depended Hurst exponent is presented in the bottom image.
Prior to the implosion of the dot-com speculative bubble at the end of 2000, one can clearly see the decreasing trend in the time-dependent Hurst exponent values depicted in the Fig. 5.3. The shift from persistent to anti-persistent behavior took place within five months (sessions: 4300 - 4400) and the values of the local Hurst exponent reached their local minimum around a month before the AEXʼs index attained its peak. While the AEX value was still rising, investorsʼ moods were worsening as suggested by the low values of Hurst exponent in the period before the turning point. The crisis detection conditions (Czarnecki et al, 2008) were strongly met by the local and moving average Hurst exponent values. In the Fig. 5.3 one can see that the hypothesisʼ thresholds were met multiple times before the AEXʼs index reversed its direction in September 2000.
" This turning point was followed by the long-lasting and decreasing trend in the AEX value (lasting more than two years) with the total drop of 54.61%. Corresponding behavior of the time-dependent Hurst exponent seems to indicate rather uncorrelated behavior with slightly persistent tendency. After the AEXʼs maximum point at the beginning of September 2000, one can observe a rapid increase in the Hurst exponent values. However, after this initial increase the values started fluctuating around H=0.5, implying Brownian motion. This can be also seen in the chart of AEXʼs value. Although, the market trend was decreasing during that time there were several significant corrections in the trend. Furthermore, the local DFA method detects another critical period within this time frame (session: 4948). However, in this case, the crisis signaling occurs too late to be able to anticipate the market drop, since both, the AEXʼs and the local Hurst exponent values, fall simultaneously.
5.4 Global Financial Crisis (2007-2009)
The recent financial crisis of 2007-2009 had a particularly significant impact on the Amsterdam Stock Exchange, as within two years, market valued plummeted by 65%. In this time period, the local DFA method detected two possible turning points. The first critical point followed after a single market correction and took place at the end of October 2007 (session 6302). After a clearly decreasing trend, the local Hurst exponent values reached their local minimum just a few days before the turning point. Both the local Hurst exponent and its moving average met the hypothesisʼ crisis detection criteria. The second crash occurred in September of 2008, however, in this case the Hurst exponent method did not provide sufficient evidence for the crash detection. The values of the time-dependent Hurst exponent started decreasing as soon as the market changed its direction, and therefore could not be used to anticipate this turning point.
Fig. 5.4: Recent financial meltdown of 2007-2009. Two vertical lines indicate the dates when the local value
of Hurst exponent and its moving average met the crisis detection criteria. In the case of the first marketʼs turning point the method could anticipate upcoming crisis, while in the second case the method did not signal early enough.
6 Discussion
" From the above analysis of the local Hurst exponent behavior, it appears that the method can be useful in predicting future turning points on the Amsterdam Stock Exchange. Therefore, the results in this thesis confirm findings of Grech & Mazur (2004) and Grech & Pamula (2008). In this paper, four crisis periods were investigated and the method worked well in anticipating three of them. With respect to the August 1998 case, since the values of time-dependent Hurst exponent were especially low during a year prior to the crash, the crisis detection criteria were continuously met for a significant number of market sessions. Therefore, the crash predictions for the AEX index became almost
impossible between 1997-1998. Low values of the local Hurst exponent could be partially explained by highly unstable situation in many international stock markets during that time. According to Dungey, Gonzalez-Hermosillo & Martin (2002), financial markets across the world experienced several episodes of distress during the mid 1990s. Investors worried that (domestic) financial crises in Mexico (1994) or Asia (1997-1998) could spillover to other parts of the world (Dungey et al, 2002). Moreover, Grech & Mazur (2004) found that, in the period of 1995 to 2003, the Hurst exponent estimation from DJIA signal was strongly influenced by the noise signal coming from the frequently changing market value within the observation box of length N=215. They subsequently concluded that using the local Hurst exponent method for the market direction predictions became quite difficult during this period.
" In the case of the two turning points, in which the signaling of the local Hurst exponent happened too late, one could argue that the market during these periods was not in a stable and well defined state. As already concluded by Grech & Mazur (2004), Grech & Pamula (2008), and Kristoufek (2010) this method works well only in a stable environment with long-lasting trends. More importantly, initial turning points of the AEX time-series were well anticipated through the DFA method.
" Finally, as the values of the local Hurst exponent were found to depend on different lengths of the time-window N (see Bartolozzi et al., 2007 or Grech & Mazur, 2004), the DFA algorithm with a sliding window of length N=500 was applied to the period of 2007-2009 in order to relate to the results obtained in Coen et al (2012). Fig. 6.1 illustrates the evolution of the local Hurst exponent in the years of 2007 to 2008 for two different choices of time-window length: N = 215 and N = 500. These values correspond to the length of observation box chosen in this thesis and in Coenʼs paper respectively.Although, the pattern of the time-varying Hurst exponent development seems to be preserved in both cases, the magnitude of fluctuation differs. According to Grech & Mazur (2004), as N
increases, the Hurst exponent might lose its local character. Since some correlations in the data might have shorter characteristic length than the window size N, the Hurst exponent estimation might fail to account for them. This can be clearly seen in Fig. 6.1 where the Hurst exponentʼs time series with N=500 is significantly smoother than the one constructed from N = 215 sub-series.
Fig. 6.1: Comparison of the local Hurst exponent time-series with different choice of observation box size
(with the same scaling range: 5 < τ < 26). In the upper image, the DFA was calculated using sub-series of length N=215 (parameter applied in this thesis) and in the lower image, the DFA was calculated using sub-series of length N=500. The vertical line indicates when the crisis detection criteria were met in the case of N=215 DFA procedure.
Therefore, in the case of Global Financial Crisis, the DFA method applied using a sub-series of length N=500 could have failed to incorporate the signals coming from the fast evolving correlations in the AEX time-series and consequently was unable to detect upcoming crisis.
" Furthermore, there is another crucial difference in the application of the DFA
the fluctuation function (step 4 in DFA algorithm) was calculated for twenty different box sizes constrained by the particular scaling range of 5 < τ < 26. In the case of the analysis
conducted in the referenced paper (Coen et al, 2012), scaling range was chosen to be contained in the range of 10 < τ < 50. These differences in the choice of the scaling range parameter could further contribute to the dissimilarities in obtained results. In the Fig 6.2, it is clearly indicated that the estimated values of the local Hurst exponent are considerably sensitive to changes in the scaling range.
Fig. 6.2: Comparison of the local Hurst exponent time-series with different choice of upper and lower limits
of scaling range (for sub-series N=500). In the upper image, the fluctuation function F was calculated for the scaling range of 5 < τ < 26, while in the bottom image scale of 10 < τ < 50 was used. The graphs clearly indicate that the values of the local Hurst exponent are very sensitive to the choice of the scaling range parameter.
The interested reader is referred to the work of Grech and Mazur (2012) for the detailed examination of the scaling regime for the detrended fluctuation analysis.
7 Conclusion
" This thesis investigated the relationship between the behavior of the local Hurst exponent and the performance of the Amsterdam Stock Exchange. In particular, by applying the detrended fluctuation analysis to the daily sub-series of the AEX index, consisting of a signal from the past 215 trading days, the time-dependent Hurst exponent values were extracted. The results confirmed the findings of Grech & Mazur (2004) and the method proved to be a useful tool in predicting market signal evolution. In particular, the results obtained in this thesis showed that the values of the local Hurst exponent form a specific decreasing pattern just before a crash takes place. Furthermore, three out of four investigated crashes were well anticipated by the DFA analysis. This thesis concluded by testing the DFA algorithm for two different choices of observation box N and discussing the difference in obtained results.
" Since the application of the DFA method in this thesis turned out to provide advantageous insight into the AEXʼs signal evolution, it might be interesting to apply this method to the price returns of particular stocks that are being traded on the Amsterdam Exchange. Through this application, one could hopefully infer some prediction regarding specific stockʼs behavior. Furthermore, it would be interesting to research how the local Hurst exponent could contribute to the development of an early warning system (EWS) for the financial markets. More specifically, one would like to investigate how such early signaling would affect the behavior of market participants.
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