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Quantifying volatility clustering in financial time series
International Review of Financial Analysis 23 (2012) 11–19
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International Review of Financial Analysis
Quantifying volatility clustering in financial time series Jie-Jun Tseng ⁎, Sai-Ping Li Institute of Physics, Academia Sinica, Nankang, Taipei 115, Taiwan
a r t i c l e
i n f o
Available online 13 July 2011 Keywords: Econophysics Volatility clustering Financial stylized facts
a b s t r a c t A quantitative method is introduced in this work to quantify and compare the volatility clustering behavior among various financial time series. In addition to financial markets, our approach can also be applied to other complex systems and we take the earthquake as an example to demonstrate the applicability of our approach. We further propose a toy model which can mimic the stylized facts in financial markets. This model could be interpreted as the accumulation effect of the news impact on the price fluctuation in a financial market and can be viewed as a first step towards understanding the complex market behavior. © 2011 Elsevier Inc. All rights reserved.
1. Introduction With the tremendous amount of information obtained over the past decade, researchers have now come to agree on several stylized facts about financial markets, i.e., heavy tails (or fat tails in the terminology of finance) in asset return distributions, absence of autocorrelations of asset returns, volatility clustering, aggregational normality and asymmetry between rises and falls (Bouchaud & Potters, 2003; Chakraborti, Tokea, Patriarca, & Abergel, 2009; Cont, 2001; Cristelli, Pietronero, & Zaccaria, 2011; Engle & Patton, 2001; Mantegna & Stanley, 2000). Fig. 1 (a) plots the historical daily returns of NASDAQ Composite index from February 8, 1971 through June 30, 2009.1 By definition, the price return Rτ(t) at time t is the difference between the price p(t) of a financial asset (here it is the index value of NASDAQ) at time t and its price a time τ before, p(t − τ), divided by p(t− τ), pðt Þ−pðt−τ Þ Rτ ðt Þ = : pðt−τÞ
ð1Þ
Therefore, one can obtain the daily returns R1(t) by setting τ = 1 trading day and these returns reflect the price fluctuations in this time series. We will use daily returns to define fluctuations in a financial price series throughout this work. In Fig. 1 (b), we depict the probability density function of normalized daily returns of NASDAQ index. The normalized daily return is defined as (R1(t) − μR)/σR, where μR and σR denote the average and the standard deviation of R1(t). One can clearly see that there are heavy tails at the two ends of the distribution. For comparison, a Gaussian fit with μ = 0 and σ = 1 is also
⁎ Corresponding author. E-mail address: [email protected] (J.-J. Tseng). 1 In this work, we adopt the business time scale, therefore only the trading-day data are considered. 1057-5219/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2011.06.017
included. This is one of the stylized facts that were discovered back in 1960s (Fama, 1965; Mandelbrot, 1963). In addition to those heavy tails in return distributions, large fluctuations in prices seem to lump together as well (Ding, Granger, & Engle, 1993; Engle, 1982). If one examines the empirical time series shown in Fig. 1 (a), it is easy to observe that large fluctuations in prices are more often followed by large ones while small fluctuations are more likely followed by small ones. This stylized fact is known as volatility clustering (Cont, 2007). In financial time series, it is not just that there are more large fluctuations than pure random processes but also these large fluctuations tend to cluster together. It is often suggested that a more quantitative way to view this property is to look at the autocorrelations of the return series (Ding et al., 1993). The autocorrelation function (ACF) C(xt, xt + τ) is defined as ðxt −hxt iÞ xt + τ − xt + τ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q C xt ; xt + τ ≡ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ; xt + τ − xt + τ 2 xt −hxt i2
ð2Þ
where 〈x〉 denotes the expectation value of the variable x. While the returns themselves do not show the evidence of temporal correlations, the absolute returns or their squares do display a positive, pronounced slowly decaying autocorrelation which indeed exhibit power-law decay behavior. The autocorrelations of the absolute value or the square, etc. of the asset returns are often known as the nonlinear autocorrelations. We will only consider the autocorrelation of the absolute returns as an example of the nonlinear autocorrelation in this paper. Fig. 1 (c) is the plot of the ACFs of NASDAQ returns and its absolute value. One can clearly see that there is no correlation among the returns since the ACF drops to the noise level (indicated by the solid line) within a couple of days. On the other hand, the ACF of the absolute returns, i.e., the nonlinear autocorrelation does exhibit a much slower decay behavior. Researchers have fitted this with a power law decay, and it is not clear at this moment whether the slow
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
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decay should imply long-term memory of the financial time series (Cont, 2007). We should remark here that if a financial time series is long enough, there usually appear clusters of large fluctuations. In this case, bumps will appear in e.g., the nonlinear autocorrelation of the time series. Fig. 2 gives an illustration of this behavior. Fig. 2 (a) is a comparison between NASDAQ returns (solid) and the simulated returns (dash) with the cluster of large fluctuations around the 9600th day replaced by Gaussian noise fluctuations. Fig. 2 (b) shows the event counts for the largest p% fluctuations in the corresponding simulated NASDAQ returns within a moving window of size of n days while (c) is the nonlinear autocorrelation for NASDAQ (line with dots) and the simulated returns (line with open circles) with large time lags. By replacing the cluster around the 9600th day by Gaussian noise fluctuations, we observe that the bump around the 2000th day in the nonlinear autocorrelation disappears. One can further verify the bumps that appear at various time lag in the nonlinear autocorrelations are actually results of the correlations between large fluctuation clusters by replacing them by Gaussian noise fluctuations. To smooth out this behavior in nonlinear autocorrelation, one can perform a moving origin method to obtain a smooth slow decay behavior of the nonlinear autocorrelation as was done in Liu et al. (1999). We note that the slow decay of nonlinear autocorrelation is mainly due to the correlation between large fluctuation clusters. The knowledge about the origin of the volatility clustering behavior in financial markets is essential for all model builders who attempt to accurately describe the market dynamics. Therefore, in this work, we attempt to propose a new method to deal with the volatility clustering so as to gain a deeper insight into this behavior. This paper is organized
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Time lag (day) Fig. 1. (a) The historical daily returns for NASDAQ Composite index from February 8, 1971 through June 30, 2009. (b) The probability density function of the normalized daily returns. (c) The autocorrelation functions of the returns and its absolute value (the noise level is indicated by the solid line here).
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Fig. 2. (a) The comparison between NASDAQ returns (solid) and the simulated returns (dash) with the cluster of large fluctuations around the 9600th day replaced by Gaussian noise fluctuations. (b) The event counts for the largest p% fluctuations in the corresponding simulated NASDAQ returns within a moving window of size of n days. (c) The nonlinear autocorrelation for NASDAQ (line with dots) and the simulated returns (line with open circles) with large time lags.
as follows. In Section 2, an index as a quantitative measure of volatility clustering in financial time series. This would allow us to directly compare the degree of volatility clustering across different financial time series. The asymmetry between rises (gains) and falls (losses) in the time series will also be discussed. To demonstrate that our approach can indeed be applied to other complex systems, we further apply the clustering index to investigate the earthquake time series. In Section 3, a toy model which mimics the accumulation effect of the news impact on the price fluctuation in a financial market will be introduced. Section 4 will be the summary and discussion. In this work, we have carried out the analysis on seven different representative financial time series. They include (i) NASDAQ Composite Index (NASDAQ), (ii) Standard & Poor's 500 index (S&P500), (iii) Hang Seng Index (HSI), (iv) Microsoft stock price (MSFT), (v) US Dollar/New Taiwan Dollar (USD/NTD), (vi) Australian Dollar/New Taiwan Dollar (AUD/NTD) and (vii) West Texas Intermediate (WTI). While we use NASDAQ as an example throughout the paper, we will include the results of other financial time series in the Appendix. 2. Clustering index In order to discuss the volatility clustering behavior in a more quantitative way, it is better to introduce some parameters to quantitatively measure the volatility clustering of different financial time series that we can make comparison with. We here construct an index for this purpose.
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To begin with, we introduce a moving window with a certain window size to scan through the given time series. As an example, one can pick a window with size of n (where n is fixed throughout the scanning process) trading days. Similar to what we have done in previous section, we then count the total number of trading days that are among the largest p% fluctuations in returns within this window as we scan through the time series. As we will see, one can interpret this as the degree of volatility clustering of the largest p% fluctuations with respect to this particular window with size n. Fig. 3 is an illustration of the clustering of the largest 20% fluctuations in Fig. 1 (a) with a window size of 10 trading days, a span of 2 weeks in real daily life. The statistics here is obtained by using the so called moving window method. This means that we begin by putting the window on the first day of the whole series and count the number of days among largest 20% fluctuations within this 10-day window. We then move on to the second day of the whole series and again count the number of days among largest 20% fluctuations within this next 10-day window, the second step. We repeat the same procedure until we finish scanning through the whole time series. The curve with dots in Fig. 3 is a plot of the frequency distribution of the number of days among the largest 20% fluctuations within a 10-day period by using this moving window method. To make it into a quantitative measure of the degree of clustering, we need to compare it with a randomly generated time series for example, a Gaussian random noise. The curve with open circles in Fig. 3 is the frequency distribution of the number of days of the largest 20% fluctuations within a 10-day period from a simulated Gaussian random noise. From Fig. 3, one can already visually tell the difference between these two curves. To be more concise, we take the ratio of the standard deviation of the number of days of the largest p% fluctuations within the n-day window between the empirical data and the noise. Mathematically, it is defined as Rn ≡ σe/σG, where σe and σG are the standard deviation of the number of days of the largest p% fluctuations within an n-day period for the empirical data sets and simulated Gaussian noise, respectively. The larger the ratio is, the larger the degree of clustering will be. The average number of days of largest p% fluctuations within a window size of n is equal to p × n/100. This is true irrespective of whether it is the empirical data set or the noise. One can indeed see this for the simulated noise which has a peak near this value. However, if the time series displays the phenomenon of clustering of large fluctuations, there will be a higher frequency of occurrence that the number of days of the largest p% fluctuations within this window is much larger than the average value p × n/100. Similarly, there will also be a higher frequency of occurrence that the number of days of the largest p% fluctuations within this window is much smaller than the average value p × n/100. This scenario will indeed be reflected in the value of the standard deviation of the frequency distribution in Fig. 3. Thus, one can simply take the ratio of the standard deviation of the empirical data and the simulated Gaussian noise to have a quantitative
n! m n−m P ð1−P Þ ; m!ðn−mÞ!
n 1 = 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 m n−m = nP ð1−P Þ; σG ∑ ðm−PnÞ P ð1−P Þ
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which is the familiar result in statistics for the standard deviation of a sequence of n random events with occurrence probability P. The lower bound corresponds to the extreme case when the largest p% fluctuations are evenly distributed and n is a multiple of 1 = P, in which case Rn = 0. The upper limit of Rn has been derived in Tseng and Li (2011) as lim
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pffiffiffi σ lim = n: σG
ð5Þ
When the fluctuations are completely random, Rn = 1. Fig. 4 (a) gives the value of the clustering index of NASDAQ time series shown in Fig. 1 (a) as a function of window size n. We have included here the results for p = 5, 10, 15 and 20. Also included is the curve of the theoretical limit of the index. The index values all start from unity when the window size n corresponds to 1 trading day, and gradually increase as the window size increases. 2.2. Asymmetry in clustering With the clustering index in hand, one can practically study the clustering behavior of any sort of fluctuations in a financial time series, 5
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ð3Þ
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measure of the degree of clustering for the largest p% fluctuations in a financial time series. The ratio or index Rn that we introduce here has both theoretical upper and lower bounds which can be calculated analytically. Recall from above that the mean value of the average number of days of the largest p% fluctuations within a n-day window is equal to p × n/100. For a total of n days, the probability that there are m days with fluctuations among the largest p% fluctuations can be written as
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Fig. 4. (a) The clustering index, Rn, for NASDAQ return series with p = 5 (solid), 10 (dash), 15 (dot) and 20 (dash dot). The theoretical limit of the index is drawn as a thick line for comparison. (b) The asymmetry Als and A+ − with p = 15 and 20 for NASDAQ return series.
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such as the asymmetry of clustering. There exist discussions in the literature (Chen, Hong, & Stein, 2001; Henry, 1999; Peiró, 2004; Pelagatti, 2009) about the asymmetry of asset returns such as the skewness of the return distribution in Fig. 1 (b). In the study of financial time series, one can for example, ask whether there are more days that the returns are gains (rises) rather than losses (falls) as some kind of asymmetry. One can further ask whether the returns in gains would like to cluster more or the other way round, how large the difference is, and whether large fluctuations tend to cluster more, etc. These can all be viewed as asymmetries in a financial time series. To study the asymmetries in financial time series, let us first give the definitions here. In the case of the asymmetry between the largest/smallest returns, we adopt the notation that the largest and smallest fluctuations refer to the absolute returns as before. We first obtain the clustering index for the largest and smallest p% returns. The asymmetry of largest/smallest returns A1s is then defined as ð6Þ
where R1 and Rs are the indices for the largest and smallest p% fluctuations respectively. This asymmetry will give us an idea whether the large fluctuations or the small fluctuations would like to cluster more as we increase the size of the moving window. From this definition, it is clear that Als is equal to zero when the window size is equal to 1, since there are an equal number of largest and smallest fluctuations. In a similar fashion, one can define the asymmetry between the largest positive and negative returns, which we call A+ − as follows A +− =
Rþ −R− ; Rþ + R−
3. A toy model In previous sections, we introduced the stylized facts and proposed the clustering index to quantify the clustering of fluctuations in financial markets. Researchers have proposed models such as agent-based models to simulate the various stylized facts in financial markets. These agent-based models start with a certain number of artificial agents who make tradings according to the strategic rules given to them by the designer (Chakraborti et al., 2009; Cristelli et al., 2011). Many of these agent-based models impose complex trading rules in order to
ð7Þ
where R+ and R− are the indices for the largest positive and negative returns respectively. We should remind our reader here that in the case of A+ −, we first pick up the largest p% fluctuations from the absolute returns and then separate the fluctuations (returns) into positive and negative categories. In this way, we can see the asymmetry between the large positive and negative returns as well as their degree of clustering. Notice that the asymmetries as defined above are bounded by 1 and −1. Fig. 4 (b) is the results of asymmetries Als and A+ − for p = 15 and 20. From the figure, it is easy to observe that the two curves for Als are always positive, which means that the degree of clustering is more obvious for large fluctuations than for small fluctuations in NASDAQ times series. On the other hand, the two curves for A+ − are always below zero. This reflects the fact that negative returns, or big losses are likely to cluster together than big gains in the case of NASDAQ. This is in agreement with some observations (Chen et al., 2001; Engle & Patton, 2001) indicating that there are more big losses rather than big gains in financial markets since we have more big losses and these big losses are more likely to lump together. We should mention here that the window size equals to 1 corresponds to the asymmetry of distribution of returns in Fig. 1 (b). In the case of NASDAQ, the asymmetry is negative. There are however, examples of financial time series that the asymmetry for the probability density function is positive and they are included in the appendix below. By increasing the size of the moving window, one can also study the asymmetry of returns with respect to the clustering of large and small fluctuations. Therefore, the use of the index to study asymmetries in financial time series allows one to extract more information comparing to other methods which only focus on the skewness of returns.
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accompanied with its fore- and aftershocks, namely, the quakes occurring during this period might be clustered. Therefore, we would like to know that if it is also possible to study this clustering phenomenon with our index. Fig. 5 (a) shows the historical hourly data of the recorded earthquakes in Taiwan during the year 1999 (dot) and 2000 (open circle) while the magnitude of each earthquake is recorded in the Richter magnitude scale. The nonlinear ACFs and the clustering index for the above two earthquake time series are plotted in Fig. 5 (b) and (c), respectively. From Fig. 5 (b) and (c), it is interesting to note that there is a big difference between the 1999 and the 2000 data both for the nonlinear ACF and the clustering index. The main reason is because there was a big earthquake on September 21 in 1999 which is followed by many aftershocks through the subsequent weeks, as indicated in Fig. 5 (a). There is therefore a big clustering of large values in magnitude. By this example, we demonstrate that this index could also be used as an indicator to reflect the earthquake activities within a certain period of time.
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Window size,n 2.3. Application to other complex systems In addition to the financial market, the index introduced in this work could be used to study other complex systems. We here take earthquake as an example. It is observed that a large earthquake is usually
Fig. 5. (a) The historical hourly data of the recorded earthquakes in Taiwan during the year 1999 (dot) and 2000 (open circle). (b) The nonlinear autocorrelation function for the Taiwan earthquake series in 1999 (line with dots) and 2000 (line with open circles). (c) The clustering index for the Taiwan earthquake series in 1999 (solid) and 2000 (dash) with p = 2. The theoretical limit of the index is drawn as a thick line for comparison.
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imitate the trading behavior of human and many adjustable parameters are introduced. These models do have various successes in reproducing some stylized facts in financial markets. However, there are up to now no quantitative measures on how good these models are as compared to empirical data from financial markets other than autocorrelation functions. The clustering index studied here can thus be used to give a more quantitative measure of the stylized facts obtained from various agent-based models as compared to empirical data from financial markets. If one is only interested in simulating dataset that mimics financial times series with stylized facts, one can use very simple toy models. Instead of studying stylized facts using complex agent-based models as proposed by others, we here suggest a very simple way to generate time sequence that can show similar behavior as that of financial markets. The basic philosophy of this model is to mimic the effect of the news impact on the price fluctuation in a financial market. We would like to see if a model with randomly generated incoming news can also display similar behavior as observed in real financial markets. The news impact on price fluctuations of individual stocks and market indices in financial markets has been studied by researchers in the field. People have also analyzed the effect of news impact when there is an accumulation of financial news on financial markets over periods of time. We here assume that these are random incoming news and want to observe how that can affect the price fluctuations. To begin with, we generate a time sequence with random fluctuations from a Gaussian distribution function. This will be a series of Gaussian random noise. For comparison, we generate a time series of say, 5000 data points or so. In the next step, we randomly pick a segment of N (between 50 and 500 in our simulation) data points within this sequence and multiply this segment by a random factor between 1 and 2. This selected N-point segment will therefore be magnified by this factor. We then randomly pick another segment of data and again multiply this segment by another randomly chosen factor (between 1 and 2). One should keep in mind that some of the data points in this newly chosen segment might overlap with the previous one. The above procedures continue to iterate until the maximum interval between two adjacent clusters, formed by the largest p% fluctuations (p = 20 in this simulation), exceeds the maximum interval obtained from the empirical data. One will then arrive at a time series which looks like the solid line in Fig. 6, while the envelope function, which mimics the response to some exogenous news impact over this time horizon, is plotted as the dash line in the same figure. In a similar fashion, we can find the return distribution, the nonlinear autocorrelation function and the clustering index of this simulated return series. Fig. 7 shows the comparison between the results from our simulation and from NASDAQ return series. One can observe that the simulated series can generate stylized facts very close to that of the empirical dataset. The simulated series is indeed equivalent to the product of a Gaussian random noise sequence and an envelope function as shown in Fig. 6. This function is the result of the procedure that we used in the above by repeatedly magnifying random segments within the time series. One can now see that the simulated stylized facts are actually coming from the large values
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around day 600, 2500, 4000, etc. When the values of this function are large at some time points, clustering effect will occur. As we suggested above, one might interpret this function as a response to some exogenous news impact such as economic news, government policy, etc. This simple model indeed has no input parameters other than from the empirical market data for compared. The maximum segment size (between 50 and 500 in our simulation) is suggested by the maximum interval between two adjacent clusters from empirical data. In the case of the NASDAQ data, this maximum interval size is 200 days and so we pick the maximum segment size to be of this order. One can also pick the upper bound of the magnification factor to be some other values as long as it is not too large. A natural question to ask is how realistic this toy model would be. There is some indication that this simple way of generating a time series could indeed imitate what is happening in financial markets. Fig. 8 is a plot of the historical daily closing value for CBOE (Chicago Board Options Exchange) Volatility Index (VIX) which, is a key measurement of market expectations of near-term volatility conveyed by S&P500 option prices and is believed to be a major barometer of investor sentiment and market volatility. High VIX values represent investors see significant risk that the market will move sharply, either downward or upward. The fluctuation in this index could be a reflection of human expectation or response due to the accumulation of some exogenous news, government policy, etc. If one now multiplies the VIX time series by a Gaussian random noise, the resulting series will be very similar to Fig. 6. Since VIX is a measurement of the near-term volatility related to S&P500 option prices as suggested by market practitioners, one can do a
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reverse procedure to obtain a new time series by dividing the S&P500 return series by VIX series for the same period. Fig. 9 is a plot of the return distribution, the nonlinear autocorrelation function and the clustering index of this S&P500/VIX series. In Fig. 9 (a), the return distribution of this new series is now very close to a Gaussian distribution. Furthermore, the nonlinear autocorrelation and the clustering index shown in Fig. 9 (a) and (b) also indicate that this S&P500/VIX series behaves very much like a Gaussian random noise. This result suggests that S&P500 might, to a good approximation, be represented by a volatility function multiplied by a Gaussian random noise. Naively, this Gaussian noise can be thought of as an intrinsic noise function of the market itself while the volatility function (which can be viewed as the envelope function in our toy model) might be identified as the response of the market (and its traders) to some exogenous shocks 1
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such as economic news, government policy, etc. Whether this volatility function is an instantaneous response to such shocks or from the response of traders' experience (some kind of long-term memory) is not clear to us. Even though the toy model could simulate most of the empirical stylized facts in real markets, there are still some differences between the simulated time series and the empirical time series. One such difference is the clustering of big losses and big gains. Since the simulated time series is the product of an envelope function and a Gaussian random noise, the probability that a big gain or a big loss after a big gain (or loss) should be equally likely. This can be checked easily when one looks at the asymmetry of the dataset. On the other hand, the asymmetry still persists even after we divide S&P500 by VIX. The result is shown in Fig. 10. A possible interpretation of such a difference is due to the psychological effect of traders. Other effects are also possible and should be investigated. 4. Summary and discussion In this paper, we introduced an index to quantitatively measure the clustering behavior of fluctuations in financial time series and have given examples to demonstrate its advantages over the conventional methods. This index has both theoretical lower and upper bounds. It is equal to unity if the fluctuations are independent, identically distributed within the financial time series. On the other hand, its upper bound can also be analytically calculated and in the limit when the time scale of the given series is much longer than the window size n, the index Rn is pffiffiffi simply equal to n. With this index in hand, one can study not only the asymmetry of the asset returns but also the effect of clustering on the asymmetry properties in financial time series. One can see that the larger fluctuations tend to cluster more than the smaller ones. Similarly, big losses tend to lump together more severely than big gains. These findings should be helpful to people who make investments in financial markets. As a matter of fact, one can use this index to look at the financial time series at various periods of time to see how the clustering behaves over time. The clustering index studied here can also provide a good quantitative measure of the stylized facts obtained from various agentbased models as compared to empirical data from financial markets. Our approach can further be applied to other complex systems and we have taken the earthquake as an example to demonstrate the applicability of our approach. A toy model is also proposed here which can mimic the stylized facts in financial markets. This model could be interpreted as the effect of the news impact on the price fluctuation in a financial market and can be viewed as a first step towards understanding the complex market behavior. Other effects should be investigated to further improve this simple toy model.
Window size, n Acknowledgments Fig. 9. (a) The return distribution of S&P500/VIX series (dot) and S&P500 series (open circle). (b) The nonlinear autocorrelation function for S&P500/VIX series (line with dots), S&P500 series (line with open circle) and a Gaussian random noise (line with triangles). (c) The clustering index for S&P500/VIX series (solid) and S&P500 series (dash) with p = 20. The theoretical limit of the index is drawn as a thick line for comparison.
This work was supported in part by the National Science Council of Taiwan under grants NSC#98-2120-M-001-002 and NSC#97-2112M-001-008-MY3.
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
Appendix For comparison, we here show the results of other financial time series as mentioned in Section 1 with the same procedures introduced in this work. They include (i) Standard & Poor's 500 index (S&P500, from January 4, 1950 through June 30, 2009), (ii) Hang Seng Index
17
(HSI, from January 2, 1987 through June 30, 2009), (iii) Microsoft stock price (MSFT, from March 4, 1986 through June 30, 2009), (iv) US Dollar/New Taiwan Dollar (USD/NTD, from July 2, 2001 through June 30, 2009), (v) Australian Dollar/New Taiwan Dollar (AUD/NTD, from July 2, 2001 through June 30, 2009) and (vi) West Texas Intermediate (WTI, from January 6, 1986 through June 30, 2009).
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References Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1, 223–236. Engle, R. F., & Patton, A. J. (2001). What good is a volatility model? Quantitative Finance, 1, 237–245.
Chakraborti, A., Tokea, I. M., Patriarca, M., & Abergel, F. (2009). Econophysics: Empirical facts and agent-based models. arXiv:0909.1974[hep-ph]. Cristelli, M., Pietronero, L., & Zaccaria, A. (2011). Critical overview of agent-based models for economics. arXiv:1101.1847[hep-ph]. Mantegna, R. N., & Stanley, H. E. (2000). Introduction to econophysics: Correlations and complexity in finance. Cambridge: Cambridge University Press.
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19 Bouchaud, J. P., & Potters, M. (2003). Theory of financial risk and derivative pricing: From statistical physics to risk management. Cambridge: Cambridge University Press. Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36, 394. Fama, E. F. (1965). The behavior of stock market prices. Journal of Business, 38, 34–105. Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1009. Ding, Z., Granger, C., & Engle, R. F. (1993). A long memory property of stock market returns and a new model. Empirical Finance, 1, 83–106. Cont, R. (2007). Long memory in economics. Ch.. Volatility clustering in financial markets: Empirical facts and agent-based models (pp. 289–309). Heidelberg: Springer Berlin Heidelberg.
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Liu, Y., Gopikrishnan, P., Cizeau1, P., Meyer, M., Peng, C. K., & Stanley, H. E. (1999). Statistical properties of the volatility of price fluctuations. Physical Review E, 60, 1390–1400. Tseng, J. J., & Li, S. P. (2011). Asset returns and volatility clustering in financial time series. Physica A, 390, 1300–1314. Henry, O. (1999). Modelling the asymmetry of stock market volatility. Applied Financial Economics, 8, 145–153. Peiró, A. (2004). Asymmetries and tails in stock index returns: Are their distributions really asymmetric? Quantitative Finance, 4, 37–44. Chen, J., Hong, H., & Stein, J. C. (2001). Forecasting crashes: Trading volume, past returns, and conditional skewness in stock prices. Journal of Financial Economics, 61, 345–381. Pelagatti, M. M. (2009). Modelling good and bad volatility. Studies in Nonlinear Dynamics and Econometrics, 13 Article 2.
Contents lists available at ScienceDirect
International Review of Financial Analysis
Quantifying volatility clustering in financial time series Jie-Jun Tseng ⁎, Sai-Ping Li Institute of Physics, Academia Sinica, Nankang, Taipei 115, Taiwan
a r t i c l e
i n f o
Available online 13 July 2011 Keywords: Econophysics Volatility clustering Financial stylized facts
a b s t r a c t A quantitative method is introduced in this work to quantify and compare the volatility clustering behavior among various financial time series. In addition to financial markets, our approach can also be applied to other complex systems and we take the earthquake as an example to demonstrate the applicability of our approach. We further propose a toy model which can mimic the stylized facts in financial markets. This model could be interpreted as the accumulation effect of the news impact on the price fluctuation in a financial market and can be viewed as a first step towards understanding the complex market behavior. © 2011 Elsevier Inc. All rights reserved.
1. Introduction With the tremendous amount of information obtained over the past decade, researchers have now come to agree on several stylized facts about financial markets, i.e., heavy tails (or fat tails in the terminology of finance) in asset return distributions, absence of autocorrelations of asset returns, volatility clustering, aggregational normality and asymmetry between rises and falls (Bouchaud & Potters, 2003; Chakraborti, Tokea, Patriarca, & Abergel, 2009; Cont, 2001; Cristelli, Pietronero, & Zaccaria, 2011; Engle & Patton, 2001; Mantegna & Stanley, 2000). Fig. 1 (a) plots the historical daily returns of NASDAQ Composite index from February 8, 1971 through June 30, 2009.1 By definition, the price return Rτ(t) at time t is the difference between the price p(t) of a financial asset (here it is the index value of NASDAQ) at time t and its price a time τ before, p(t − τ), divided by p(t− τ), pðt Þ−pðt−τ Þ Rτ ðt Þ = : pðt−τÞ
ð1Þ
Therefore, one can obtain the daily returns R1(t) by setting τ = 1 trading day and these returns reflect the price fluctuations in this time series. We will use daily returns to define fluctuations in a financial price series throughout this work. In Fig. 1 (b), we depict the probability density function of normalized daily returns of NASDAQ index. The normalized daily return is defined as (R1(t) − μR)/σR, where μR and σR denote the average and the standard deviation of R1(t). One can clearly see that there are heavy tails at the two ends of the distribution. For comparison, a Gaussian fit with μ = 0 and σ = 1 is also
⁎ Corresponding author. E-mail address: [email protected] (J.-J. Tseng). 1 In this work, we adopt the business time scale, therefore only the trading-day data are considered. 1057-5219/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.irfa.2011.06.017
included. This is one of the stylized facts that were discovered back in 1960s (Fama, 1965; Mandelbrot, 1963). In addition to those heavy tails in return distributions, large fluctuations in prices seem to lump together as well (Ding, Granger, & Engle, 1993; Engle, 1982). If one examines the empirical time series shown in Fig. 1 (a), it is easy to observe that large fluctuations in prices are more often followed by large ones while small fluctuations are more likely followed by small ones. This stylized fact is known as volatility clustering (Cont, 2007). In financial time series, it is not just that there are more large fluctuations than pure random processes but also these large fluctuations tend to cluster together. It is often suggested that a more quantitative way to view this property is to look at the autocorrelations of the return series (Ding et al., 1993). The autocorrelation function (ACF) C(xt, xt + τ) is defined as ðxt −hxt iÞ xt + τ − xt + τ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q C xt ; xt + τ ≡ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ; xt + τ − xt + τ 2 xt −hxt i2
ð2Þ
where 〈x〉 denotes the expectation value of the variable x. While the returns themselves do not show the evidence of temporal correlations, the absolute returns or their squares do display a positive, pronounced slowly decaying autocorrelation which indeed exhibit power-law decay behavior. The autocorrelations of the absolute value or the square, etc. of the asset returns are often known as the nonlinear autocorrelations. We will only consider the autocorrelation of the absolute returns as an example of the nonlinear autocorrelation in this paper. Fig. 1 (c) is the plot of the ACFs of NASDAQ returns and its absolute value. One can clearly see that there is no correlation among the returns since the ACF drops to the noise level (indicated by the solid line) within a couple of days. On the other hand, the ACF of the absolute returns, i.e., the nonlinear autocorrelation does exhibit a much slower decay behavior. Researchers have fitted this with a power law decay, and it is not clear at this moment whether the slow
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
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decay should imply long-term memory of the financial time series (Cont, 2007). We should remark here that if a financial time series is long enough, there usually appear clusters of large fluctuations. In this case, bumps will appear in e.g., the nonlinear autocorrelation of the time series. Fig. 2 gives an illustration of this behavior. Fig. 2 (a) is a comparison between NASDAQ returns (solid) and the simulated returns (dash) with the cluster of large fluctuations around the 9600th day replaced by Gaussian noise fluctuations. Fig. 2 (b) shows the event counts for the largest p% fluctuations in the corresponding simulated NASDAQ returns within a moving window of size of n days while (c) is the nonlinear autocorrelation for NASDAQ (line with dots) and the simulated returns (line with open circles) with large time lags. By replacing the cluster around the 9600th day by Gaussian noise fluctuations, we observe that the bump around the 2000th day in the nonlinear autocorrelation disappears. One can further verify the bumps that appear at various time lag in the nonlinear autocorrelations are actually results of the correlations between large fluctuation clusters by replacing them by Gaussian noise fluctuations. To smooth out this behavior in nonlinear autocorrelation, one can perform a moving origin method to obtain a smooth slow decay behavior of the nonlinear autocorrelation as was done in Liu et al. (1999). We note that the slow decay of nonlinear autocorrelation is mainly due to the correlation between large fluctuation clusters. The knowledge about the origin of the volatility clustering behavior in financial markets is essential for all model builders who attempt to accurately describe the market dynamics. Therefore, in this work, we attempt to propose a new method to deal with the volatility clustering so as to gain a deeper insight into this behavior. This paper is organized
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Fig. 2. (a) The comparison between NASDAQ returns (solid) and the simulated returns (dash) with the cluster of large fluctuations around the 9600th day replaced by Gaussian noise fluctuations. (b) The event counts for the largest p% fluctuations in the corresponding simulated NASDAQ returns within a moving window of size of n days. (c) The nonlinear autocorrelation for NASDAQ (line with dots) and the simulated returns (line with open circles) with large time lags.
as follows. In Section 2, an index as a quantitative measure of volatility clustering in financial time series. This would allow us to directly compare the degree of volatility clustering across different financial time series. The asymmetry between rises (gains) and falls (losses) in the time series will also be discussed. To demonstrate that our approach can indeed be applied to other complex systems, we further apply the clustering index to investigate the earthquake time series. In Section 3, a toy model which mimics the accumulation effect of the news impact on the price fluctuation in a financial market will be introduced. Section 4 will be the summary and discussion. In this work, we have carried out the analysis on seven different representative financial time series. They include (i) NASDAQ Composite Index (NASDAQ), (ii) Standard & Poor's 500 index (S&P500), (iii) Hang Seng Index (HSI), (iv) Microsoft stock price (MSFT), (v) US Dollar/New Taiwan Dollar (USD/NTD), (vi) Australian Dollar/New Taiwan Dollar (AUD/NTD) and (vii) West Texas Intermediate (WTI). While we use NASDAQ as an example throughout the paper, we will include the results of other financial time series in the Appendix. 2. Clustering index In order to discuss the volatility clustering behavior in a more quantitative way, it is better to introduce some parameters to quantitatively measure the volatility clustering of different financial time series that we can make comparison with. We here construct an index for this purpose.
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To begin with, we introduce a moving window with a certain window size to scan through the given time series. As an example, one can pick a window with size of n (where n is fixed throughout the scanning process) trading days. Similar to what we have done in previous section, we then count the total number of trading days that are among the largest p% fluctuations in returns within this window as we scan through the time series. As we will see, one can interpret this as the degree of volatility clustering of the largest p% fluctuations with respect to this particular window with size n. Fig. 3 is an illustration of the clustering of the largest 20% fluctuations in Fig. 1 (a) with a window size of 10 trading days, a span of 2 weeks in real daily life. The statistics here is obtained by using the so called moving window method. This means that we begin by putting the window on the first day of the whole series and count the number of days among largest 20% fluctuations within this 10-day window. We then move on to the second day of the whole series and again count the number of days among largest 20% fluctuations within this next 10-day window, the second step. We repeat the same procedure until we finish scanning through the whole time series. The curve with dots in Fig. 3 is a plot of the frequency distribution of the number of days among the largest 20% fluctuations within a 10-day period by using this moving window method. To make it into a quantitative measure of the degree of clustering, we need to compare it with a randomly generated time series for example, a Gaussian random noise. The curve with open circles in Fig. 3 is the frequency distribution of the number of days of the largest 20% fluctuations within a 10-day period from a simulated Gaussian random noise. From Fig. 3, one can already visually tell the difference between these two curves. To be more concise, we take the ratio of the standard deviation of the number of days of the largest p% fluctuations within the n-day window between the empirical data and the noise. Mathematically, it is defined as Rn ≡ σe/σG, where σe and σG are the standard deviation of the number of days of the largest p% fluctuations within an n-day period for the empirical data sets and simulated Gaussian noise, respectively. The larger the ratio is, the larger the degree of clustering will be. The average number of days of largest p% fluctuations within a window size of n is equal to p × n/100. This is true irrespective of whether it is the empirical data set or the noise. One can indeed see this for the simulated noise which has a peak near this value. However, if the time series displays the phenomenon of clustering of large fluctuations, there will be a higher frequency of occurrence that the number of days of the largest p% fluctuations within this window is much larger than the average value p × n/100. Similarly, there will also be a higher frequency of occurrence that the number of days of the largest p% fluctuations within this window is much smaller than the average value p × n/100. This scenario will indeed be reflected in the value of the standard deviation of the frequency distribution in Fig. 3. Thus, one can simply take the ratio of the standard deviation of the empirical data and the simulated Gaussian noise to have a quantitative
n! m n−m P ð1−P Þ ; m!ðn−mÞ!
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When the fluctuations are completely random, Rn = 1. Fig. 4 (a) gives the value of the clustering index of NASDAQ time series shown in Fig. 1 (a) as a function of window size n. We have included here the results for p = 5, 10, 15 and 20. Also included is the curve of the theoretical limit of the index. The index values all start from unity when the window size n corresponds to 1 trading day, and gradually increase as the window size increases. 2.2. Asymmetry in clustering With the clustering index in hand, one can practically study the clustering behavior of any sort of fluctuations in a financial time series, 5
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measure of the degree of clustering for the largest p% fluctuations in a financial time series. The ratio or index Rn that we introduce here has both theoretical upper and lower bounds which can be calculated analytically. Recall from above that the mean value of the average number of days of the largest p% fluctuations within a n-day window is equal to p × n/100. For a total of n days, the probability that there are m days with fluctuations among the largest p% fluctuations can be written as
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Fig. 4. (a) The clustering index, Rn, for NASDAQ return series with p = 5 (solid), 10 (dash), 15 (dot) and 20 (dash dot). The theoretical limit of the index is drawn as a thick line for comparison. (b) The asymmetry Als and A+ − with p = 15 and 20 for NASDAQ return series.
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such as the asymmetry of clustering. There exist discussions in the literature (Chen, Hong, & Stein, 2001; Henry, 1999; Peiró, 2004; Pelagatti, 2009) about the asymmetry of asset returns such as the skewness of the return distribution in Fig. 1 (b). In the study of financial time series, one can for example, ask whether there are more days that the returns are gains (rises) rather than losses (falls) as some kind of asymmetry. One can further ask whether the returns in gains would like to cluster more or the other way round, how large the difference is, and whether large fluctuations tend to cluster more, etc. These can all be viewed as asymmetries in a financial time series. To study the asymmetries in financial time series, let us first give the definitions here. In the case of the asymmetry between the largest/smallest returns, we adopt the notation that the largest and smallest fluctuations refer to the absolute returns as before. We first obtain the clustering index for the largest and smallest p% returns. The asymmetry of largest/smallest returns A1s is then defined as ð6Þ
where R1 and Rs are the indices for the largest and smallest p% fluctuations respectively. This asymmetry will give us an idea whether the large fluctuations or the small fluctuations would like to cluster more as we increase the size of the moving window. From this definition, it is clear that Als is equal to zero when the window size is equal to 1, since there are an equal number of largest and smallest fluctuations. In a similar fashion, one can define the asymmetry between the largest positive and negative returns, which we call A+ − as follows A +− =
Rþ −R− ; Rþ + R−
3. A toy model In previous sections, we introduced the stylized facts and proposed the clustering index to quantify the clustering of fluctuations in financial markets. Researchers have proposed models such as agent-based models to simulate the various stylized facts in financial markets. These agent-based models start with a certain number of artificial agents who make tradings according to the strategic rules given to them by the designer (Chakraborti et al., 2009; Cristelli et al., 2011). Many of these agent-based models impose complex trading rules in order to
ð7Þ
where R+ and R− are the indices for the largest positive and negative returns respectively. We should remind our reader here that in the case of A+ −, we first pick up the largest p% fluctuations from the absolute returns and then separate the fluctuations (returns) into positive and negative categories. In this way, we can see the asymmetry between the large positive and negative returns as well as their degree of clustering. Notice that the asymmetries as defined above are bounded by 1 and −1. Fig. 4 (b) is the results of asymmetries Als and A+ − for p = 15 and 20. From the figure, it is easy to observe that the two curves for Als are always positive, which means that the degree of clustering is more obvious for large fluctuations than for small fluctuations in NASDAQ times series. On the other hand, the two curves for A+ − are always below zero. This reflects the fact that negative returns, or big losses are likely to cluster together than big gains in the case of NASDAQ. This is in agreement with some observations (Chen et al., 2001; Engle & Patton, 2001) indicating that there are more big losses rather than big gains in financial markets since we have more big losses and these big losses are more likely to lump together. We should mention here that the window size equals to 1 corresponds to the asymmetry of distribution of returns in Fig. 1 (b). In the case of NASDAQ, the asymmetry is negative. There are however, examples of financial time series that the asymmetry for the probability density function is positive and they are included in the appendix below. By increasing the size of the moving window, one can also study the asymmetry of returns with respect to the clustering of large and small fluctuations. Therefore, the use of the index to study asymmetries in financial time series allows one to extract more information comparing to other methods which only focus on the skewness of returns.
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accompanied with its fore- and aftershocks, namely, the quakes occurring during this period might be clustered. Therefore, we would like to know that if it is also possible to study this clustering phenomenon with our index. Fig. 5 (a) shows the historical hourly data of the recorded earthquakes in Taiwan during the year 1999 (dot) and 2000 (open circle) while the magnitude of each earthquake is recorded in the Richter magnitude scale. The nonlinear ACFs and the clustering index for the above two earthquake time series are plotted in Fig. 5 (b) and (c), respectively. From Fig. 5 (b) and (c), it is interesting to note that there is a big difference between the 1999 and the 2000 data both for the nonlinear ACF and the clustering index. The main reason is because there was a big earthquake on September 21 in 1999 which is followed by many aftershocks through the subsequent weeks, as indicated in Fig. 5 (a). There is therefore a big clustering of large values in magnitude. By this example, we demonstrate that this index could also be used as an indicator to reflect the earthquake activities within a certain period of time.
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Window size,n 2.3. Application to other complex systems In addition to the financial market, the index introduced in this work could be used to study other complex systems. We here take earthquake as an example. It is observed that a large earthquake is usually
Fig. 5. (a) The historical hourly data of the recorded earthquakes in Taiwan during the year 1999 (dot) and 2000 (open circle). (b) The nonlinear autocorrelation function for the Taiwan earthquake series in 1999 (line with dots) and 2000 (line with open circles). (c) The clustering index for the Taiwan earthquake series in 1999 (solid) and 2000 (dash) with p = 2. The theoretical limit of the index is drawn as a thick line for comparison.
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imitate the trading behavior of human and many adjustable parameters are introduced. These models do have various successes in reproducing some stylized facts in financial markets. However, there are up to now no quantitative measures on how good these models are as compared to empirical data from financial markets other than autocorrelation functions. The clustering index studied here can thus be used to give a more quantitative measure of the stylized facts obtained from various agent-based models as compared to empirical data from financial markets. If one is only interested in simulating dataset that mimics financial times series with stylized facts, one can use very simple toy models. Instead of studying stylized facts using complex agent-based models as proposed by others, we here suggest a very simple way to generate time sequence that can show similar behavior as that of financial markets. The basic philosophy of this model is to mimic the effect of the news impact on the price fluctuation in a financial market. We would like to see if a model with randomly generated incoming news can also display similar behavior as observed in real financial markets. The news impact on price fluctuations of individual stocks and market indices in financial markets has been studied by researchers in the field. People have also analyzed the effect of news impact when there is an accumulation of financial news on financial markets over periods of time. We here assume that these are random incoming news and want to observe how that can affect the price fluctuations. To begin with, we generate a time sequence with random fluctuations from a Gaussian distribution function. This will be a series of Gaussian random noise. For comparison, we generate a time series of say, 5000 data points or so. In the next step, we randomly pick a segment of N (between 50 and 500 in our simulation) data points within this sequence and multiply this segment by a random factor between 1 and 2. This selected N-point segment will therefore be magnified by this factor. We then randomly pick another segment of data and again multiply this segment by another randomly chosen factor (between 1 and 2). One should keep in mind that some of the data points in this newly chosen segment might overlap with the previous one. The above procedures continue to iterate until the maximum interval between two adjacent clusters, formed by the largest p% fluctuations (p = 20 in this simulation), exceeds the maximum interval obtained from the empirical data. One will then arrive at a time series which looks like the solid line in Fig. 6, while the envelope function, which mimics the response to some exogenous news impact over this time horizon, is plotted as the dash line in the same figure. In a similar fashion, we can find the return distribution, the nonlinear autocorrelation function and the clustering index of this simulated return series. Fig. 7 shows the comparison between the results from our simulation and from NASDAQ return series. One can observe that the simulated series can generate stylized facts very close to that of the empirical dataset. The simulated series is indeed equivalent to the product of a Gaussian random noise sequence and an envelope function as shown in Fig. 6. This function is the result of the procedure that we used in the above by repeatedly magnifying random segments within the time series. One can now see that the simulated stylized facts are actually coming from the large values
Probability density function
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
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around day 600, 2500, 4000, etc. When the values of this function are large at some time points, clustering effect will occur. As we suggested above, one might interpret this function as a response to some exogenous news impact such as economic news, government policy, etc. This simple model indeed has no input parameters other than from the empirical market data for compared. The maximum segment size (between 50 and 500 in our simulation) is suggested by the maximum interval between two adjacent clusters from empirical data. In the case of the NASDAQ data, this maximum interval size is 200 days and so we pick the maximum segment size to be of this order. One can also pick the upper bound of the magnification factor to be some other values as long as it is not too large. A natural question to ask is how realistic this toy model would be. There is some indication that this simple way of generating a time series could indeed imitate what is happening in financial markets. Fig. 8 is a plot of the historical daily closing value for CBOE (Chicago Board Options Exchange) Volatility Index (VIX) which, is a key measurement of market expectations of near-term volatility conveyed by S&P500 option prices and is believed to be a major barometer of investor sentiment and market volatility. High VIX values represent investors see significant risk that the market will move sharply, either downward or upward. The fluctuation in this index could be a reflection of human expectation or response due to the accumulation of some exogenous news, government policy, etc. If one now multiplies the VIX time series by a Gaussian random noise, the resulting series will be very similar to Fig. 6. Since VIX is a measurement of the near-term volatility related to S&P500 option prices as suggested by market practitioners, one can do a
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reverse procedure to obtain a new time series by dividing the S&P500 return series by VIX series for the same period. Fig. 9 is a plot of the return distribution, the nonlinear autocorrelation function and the clustering index of this S&P500/VIX series. In Fig. 9 (a), the return distribution of this new series is now very close to a Gaussian distribution. Furthermore, the nonlinear autocorrelation and the clustering index shown in Fig. 9 (a) and (b) also indicate that this S&P500/VIX series behaves very much like a Gaussian random noise. This result suggests that S&P500 might, to a good approximation, be represented by a volatility function multiplied by a Gaussian random noise. Naively, this Gaussian noise can be thought of as an intrinsic noise function of the market itself while the volatility function (which can be viewed as the envelope function in our toy model) might be identified as the response of the market (and its traders) to some exogenous shocks 1
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such as economic news, government policy, etc. Whether this volatility function is an instantaneous response to such shocks or from the response of traders' experience (some kind of long-term memory) is not clear to us. Even though the toy model could simulate most of the empirical stylized facts in real markets, there are still some differences between the simulated time series and the empirical time series. One such difference is the clustering of big losses and big gains. Since the simulated time series is the product of an envelope function and a Gaussian random noise, the probability that a big gain or a big loss after a big gain (or loss) should be equally likely. This can be checked easily when one looks at the asymmetry of the dataset. On the other hand, the asymmetry still persists even after we divide S&P500 by VIX. The result is shown in Fig. 10. A possible interpretation of such a difference is due to the psychological effect of traders. Other effects are also possible and should be investigated. 4. Summary and discussion In this paper, we introduced an index to quantitatively measure the clustering behavior of fluctuations in financial time series and have given examples to demonstrate its advantages over the conventional methods. This index has both theoretical lower and upper bounds. It is equal to unity if the fluctuations are independent, identically distributed within the financial time series. On the other hand, its upper bound can also be analytically calculated and in the limit when the time scale of the given series is much longer than the window size n, the index Rn is pffiffiffi simply equal to n. With this index in hand, one can study not only the asymmetry of the asset returns but also the effect of clustering on the asymmetry properties in financial time series. One can see that the larger fluctuations tend to cluster more than the smaller ones. Similarly, big losses tend to lump together more severely than big gains. These findings should be helpful to people who make investments in financial markets. As a matter of fact, one can use this index to look at the financial time series at various periods of time to see how the clustering behaves over time. The clustering index studied here can also provide a good quantitative measure of the stylized facts obtained from various agentbased models as compared to empirical data from financial markets. Our approach can further be applied to other complex systems and we have taken the earthquake as an example to demonstrate the applicability of our approach. A toy model is also proposed here which can mimic the stylized facts in financial markets. This model could be interpreted as the effect of the news impact on the price fluctuation in a financial market and can be viewed as a first step towards understanding the complex market behavior. Other effects should be investigated to further improve this simple toy model.
Window size, n Acknowledgments Fig. 9. (a) The return distribution of S&P500/VIX series (dot) and S&P500 series (open circle). (b) The nonlinear autocorrelation function for S&P500/VIX series (line with dots), S&P500 series (line with open circle) and a Gaussian random noise (line with triangles). (c) The clustering index for S&P500/VIX series (solid) and S&P500 series (dash) with p = 20. The theoretical limit of the index is drawn as a thick line for comparison.
This work was supported in part by the National Science Council of Taiwan under grants NSC#98-2120-M-001-002 and NSC#97-2112M-001-008-MY3.
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
Appendix For comparison, we here show the results of other financial time series as mentioned in Section 1 with the same procedures introduced in this work. They include (i) Standard & Poor's 500 index (S&P500, from January 4, 1950 through June 30, 2009), (ii) Hang Seng Index
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(HSI, from January 2, 1987 through June 30, 2009), (iii) Microsoft stock price (MSFT, from March 4, 1986 through June 30, 2009), (iv) US Dollar/New Taiwan Dollar (USD/NTD, from July 2, 2001 through June 30, 2009), (v) Australian Dollar/New Taiwan Dollar (AUD/NTD, from July 2, 2001 through June 30, 2009) and (vi) West Texas Intermediate (WTI, from January 6, 1986 through June 30, 2009).
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Fig. A.1. The left panels are plots of the historical daily closing value for S&P500, HSI, MSFT, USD/NTD, AUD/NTD and WTI (from top to bottom), while the right panels show the daily returns for each of these time series.
J.-J. Tseng, S.-P. Li / International Review of Financial Analysis 23 (2012) 11–19
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