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An Exotic Long-Term Pattern in Stock Price Dynamics

  • Jianrong Wei,

    Affiliation Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai, China

  • Jiping Huang

    jphuang@fudan.edu.cn

    Affiliation Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai, China

Abstract

Background

To accurately predict the movement of stock prices is always of both academic importance and practical value. So far, a lot of research has been reported to help understand the behavior of stock prices. However, some of the existing theories tend to render us the belief that the time series of stock prices are unpredictable on a long-term timescale. The question arises whether the long-term predictability exists in stock price dynamics.

Methodology/Principal Findings

In this work, we analyze the price reversals in the US stock market and the Chinese stock market on the basis of a renormalization method. The price reversals are divided into two types: retracements (the downward trends after upward trends) and rebounds (the upward trends after downward trends), of which the intensities are described by dimensionless quantities, and , respectively. We reveal that for both mature and emerging markets, the distribution of either retracements or rebounds shows two characteristic values, 0.335 and 0.665, both of which are robust over the long term.

Conclusions/Significance

The methodology presented here provides a way to quantify the stock price reversals. Our findings strongly support the existence of the long-term predictability in stock price dynamics, and may offer a hint on how to predict the long-term movement of stock prices.

Introduction

Predicting the movement of stock prices has been regarded as one of the most challenging topics for modern scientists and economists. To understand the behavior of stock prices, a lot of properties of real stock markets have been reported and classified as so-called stylized facts [1], [2], such as volatility properties [3][5], correlation properties [5][9], scaling behavior [10], and fat tails in the probability distributions of log-returns [11], [12]. Also, numerous methods have been introduced to predict the price movement. Schoneburg (1990) analyzed the possibility of predicting prices of German stocks based on neural network approach [13]. Wang (2002) used fuzzy grey prediction system to predict the stock prices in Taiwan stock market [14]. Mittermayer (2004) showed that news textual analysis method can be used to forecast stock price trends [15]. Pai and Lin (2005) suggested a hybrid autoregressive integrated moving average model and support vector machines model in stock price forecasting [16]. It is worthy mentioned that, Kenett et al. (2012) recently reported that the average market correlation could be used as precursor to the changes in the stock market index [17].

On the other hand, there are also several existing theories which tend to render us the belief that the time series of stock prices are almost unpredictable on a long-term timescale. The efficient-market hypothesis developed by Fama (1970) believes that financial markets are informationally efficient. Even in its weak-form, the efficient-market hypothesis asserts that future prices cannot be predicted by analyzing prices from the past and thus no profitable information about future movement can be obtained in stock price series [18][20]. In the fields of physics and econophysics, the financial markets are sometimes considered as chaotic systems, in which a long-term prediction is impossible [21][24].

The question arises whether the long-term predictability exists in the stock price dynamics. In this work, we investigate the price reversals, the changes in the direction of price trends, in all 500 stocks of the S&P (Standard & Poor’s) 500 index. We divide the price reversals into two basis types: retracements and rebounds, which represent the downward trends after upward trends and the upward trends after downward trends, respectively. On the basis of a renormalization method, the intensities of retracements and rebounds are described by dimensionless quantities, and , respectively. We reveal a bimodal distribution of both and , which indicates a long-term pattern in the stock price dynamics. In addition, we randomly reshuffle the price time series to test the robustness of the pattern. We also perform a parallel analysis in the Chinese stock market and obtain the similar pattern. This long-term pattern, which can be considered as one of the stylized facts in both mature and emerging markets, strongly supports the existence of the long-term predictability in the stock price dynamics, and may offer a hint on how to predict the long-term movement of stock prices.

Methods

The direction of a price trend could be upward or downward. An upward trend means a series of increasing prices, while a downward trend means a series of decreasing prices. Naturally, the trend reversals, the changes in the direction of price trends, can be divided into two basis types: retracements (the downward trends after upward trends) and rebounds (the upward trends after downward trends). Here, both retracements and rebounds in discrete time series of stock prices are analyzed on the basis of a renormalization method.

Denote the price at time t as , where . Each is defined to be a local minimum of () if there is no lower price in the time interval [25], [26]. In this way, all the local minima of a given can be determined for the price time series [Fig. 1(a)]. These local minima are denoted by , where . The highest price between two adjacent minima, and , is represented by . We introduce a renormalized retracement, , to describe the intensity of retracements as(1)

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Figure 1. (Color online) Schematic graphs of (a) a retracement and (b) a rebound in a time series of stock prices. Here is taken as an example.

https://doi.org/10.1371/journal.pone.0051666.g001

Here, to investigate the retracements of main trends, we focus on , thus yielding . Clearly, a larger corresponds to a stronger retracement.

Analogously, to analyze the intensity of rebounds, each price is defined to be a local maximum of if there is no higher price in the time interval [25], [26]. We determine the local maxima of for the price time series [Fig. 1(b)], which are denoted by . The lowest price between two adjacent maxima, and , is denoted by . Then, the intensity of rebounds can also be described by a renormalized rebound, , namely,(2)Similarly, we focus on , thus yielding . Here, a larger corresponds to a stronger rebound.

Results

To investigate the stock prices in the US stock market, we analyze the time series of daily closing prices of all 500 stocks of the S&P 500 index. Totally, the time series comprise 2,768,341 closing prices which have been adjusted for dividends and splits. The oldest closing prices date back to January 2, 1962. The latest closing prices were recorded on April 24, 2012. It is noted that the lengths of price time series are different for different stocks in the analyzed database (see Table 1). The longest price series, IBM (International Business Machines Corporation), were recorded from January 2, 1962 to April 24, 2012, which contains roughly 12,000 prices. The shortest price series, NSM (Nationstar Mortgage), were recorded from March 8, 2012 to April 24, 2012, which contains only 33 prices.

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Table 1. Samples of the analyzed database of the S&P 500 stocks.

https://doi.org/10.1371/journal.pone.0051666.t001

In Fig. 2(a,c), the overall probability density functions (PDFs) of renormalized retracements and renormalized rebounds are calculated over from 1 day to 100 days and over all trend reversals in the price series of all 500 stocks. Each of them shows a bimodal distribution with symmetrical peaks located at two characteristic values of (or Rb) = 0.335 and (or Rb) = 0.665. These two peaks indicate that the probabilities of (or Rb) = 0.335 or (or Rb) = 0.665 are significantly higher than those of other values, suggesting a long-term pattern in the stock price reversals.

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Figure 2. (Color online) PDF of (a,b) renormalized retracements and (c,d) renormalized rebounds of stock prices.

(a,c) Overall PDF calculated over from 1 day to 100 days and over all trend reversals in the price series of all 500 stocks of the S&P 500 index. Each of them shows a bimodal distribution with two peaks located at two characteristic values of (or Rb) = 0.335 and (or Rb) = 0.665, which are symmetrical with respect to (or Rb) = 0.500. Also shown are the results obtained from randomly reshuffled price series. (b,d) The colored PDF profiles. The color represents the probability density for each given . Ridges can be found at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665. The ridge of Rt (or Rb) = 0.335 stretches from day to days. The ridge of Rt (or Rb) = 0.665 stretches from day to days.

https://doi.org/10.1371/journal.pone.0051666.g002

The colored PDF profiles are displayed in Fig. 2(b,d) where the color represents the probability density of retracements or rebounds for each given . Two ridges located at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665 can be clearly found. Evidently, these two ridges mainly contribute to the two peaks in Fig. 2(a,c). As shown in Fig. 2(b,d), the ridges are sharper in the region of smaller , indicating this pattern in stock price dynamics attenuates as time span increases. This result is consistent with the common view that it is much harder to predict the stock prices on a longer time scale. It is noted that, in spite of their attenuation, the ridges can still be found on a rather long-term time scale. The ridge of (or Rb) = 0.335 stretches from day to days. The ridge of (or Rb) = 0.665 stretches from day to days.

To show the robustness of the pattern, we randomly reshuffle the price time series. Firstly, a log-return series is calculated from the original concrete time series of stock prices, , according to(3)Then we randomly reshuffle the log-return series. The reshuffled log-return series is denoted as . The new price time series is reproduced from the reshuffled log-return series according to(4)with . As shown in Fig. 2(a,c), the reshuffling yields the disappearance of the two peaks. Thus, the long-term pattern we have found is a consequence of price time series in the US stock market. It should be noted that, only one reshuffled PDF curve is shown in Fig. 2(a,c). The reason why we focus on one reshuffled PDF curve is two-folded. First, during the reshuffling process, all the price time series of 500 stocks of the S&P 500 index have been independently reshuffled, which means 500 independent reshuffles have been completed during one reshuffling process. Second, the overall PDF in Fig. 2(a,c) is calculated over from 1 day to 100 days and over all trend reversals in the price series of all 500 stocks of the S&P 500 index, and thus the overall PDF itself is a kind of statistical average.

We also perform a parallel analysis in the Chinese stock market, which is known as an emerging market. The database contains the daily closing prices of all 300 stocks of the Shanghai Shenzhen CSI (China Securities Index) 300 index from Jan 29, 1991 to May 7, 2012 (see Table 2). Similarly, Fig. 3(a,c) shows bimodal distributions of reversals in the Chinese stock market. Also, the peaks located at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665 can be found in the PDFs of renormalized retracements and renormalized rebounds. It should be noted that, because of the data limitation, a small range of from 1 day to 20 days is taken during the analysis of the Chinese stock market. Consequently, the peaks located at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665 in Fig. 3(a,c) are not so sharp as those in Fig. 2(a,c). However, comparing with the PDF curves of reshuffled price series in Fig. 3(a,c), these two characteristic values can still be clearly obtained. Thus we suggest that the bimodal distribution of reversals is a universal pattern in both mature and emerging stock markets.

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Figure 3. (Color online) The same as Fig.2, but for the Chinese stock market.

(a,c) Overall PDF calculated over from 1 day to 20 days and over all trend reversals in the price series of all 300 stocks of the Shanghai Shenzhen CSI 300 index. Peaks can also be found at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665. (b,d) The colored PDF profiles. Ridges can also be found at Rt (or Rb) = 0.335 and Rt (or Rb) = 0.665.

https://doi.org/10.1371/journal.pone.0051666.g003

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Table 2. Samples of the analyzed database of the Shanghai Shenzhen CSI 300 index stocks.

https://doi.org/10.1371/journal.pone.0051666.t002

Discussion

This long-term pattern can be considered as one of the stylized facts in both mature and emerging stock markets. The reason for the emergence of the long-term pattern remains unknown. However, we suggest that this phenomenon should be due to the collective behavior of all traders in the stock markets. We also believe that agent-based modeling may have the merit to offer detailed insights into this exotic pattern [27][30]. This might serve as a future agenda of our research.

Furthermore, the result can be considered as a quantitative verifying of the empirical rules of so-called Fibonacci levels in stock markets [31], [32]. The technical traders, who use the Fibonacci levels in stock and future markets, believe that the intensity of a retracement (rebound) tend to be 0.382 or 0.618 times the intensity of the previous upward trend (downward trend). However, the existence of Fibonacci levels in real market has not been well verified. Here, we conclude that the specific levels do exist in the stock price reversals, although they are not exactly equal to the Fibonacci levels.

In summary, we have analyzed the reversals of daily stock prices in the US stock market and the Chinese stock market, and revealed that the distribution of either retracements or rebounds shows two symmetrical characteristic values, both of which are robust over long term. The peaks in the distributions of Rt and Rb indicate that the probabilities of Rt (or Rb) = 0.335 or Rt (or Rb) = 0.665 are significantly higher than those of other values, suggesting a long-term predictability in the stock price reversals. We also find that the ridges attenuate as time span increases, which is consistent with the common view that it is much harder to predict the stock prices on a longer time scale. Our findings suggest a long-term pattern, which strongly support the existence of the long-term predictability in the stock price dynamics and might offer a hint on the long-term prediction of stock prices. The methodology presented in this work also provides a way to quantify the price reversals in stock price dynamics.

Author Contributions

Analyzed the data: JRW JPH. Contributed reagents/materials/analysis tools: JRW JPH. Wrote the paper: JRW JPH.

References

  1. 1. Cont R (2001) Empirical properties of asset returns: stylized facts and statistical issues. Quanti-tative Finance 1: 223–236.
  2. 2. Johnson NF, Jefferies P, Hui PM (2003) Financial market complexity. New York: Oxford University Press.
  3. 3. Lux T, Marchesi M (2000) Volatility clustering in financial markets: a microsimulation of interacting agents. International Journal of Theoretical and Applied Finance 3: 675–702.
  4. 4. Shapira Y, Kenett DY, Raviv O, Ben-Jacob E (2011) Hidden temporal order unveiled in stock market volatility variance. AIP ADVANCES 1: 022127.
  5. 5. Kenett DY, Raddant M, Lux T, Ben-Jacob E (2012) Evolvement of uniformity and volatility in the stressed global financial village. PLoS ONE 7(2): e31144.
  6. 6. Shapira Y, Kenett DY, Ben-Jacob E (2009) The index cohesive effect on stock market correlations. Eur. Phys. J. B 72: 657–669.
  7. 7. Kenett DY, Shapira Y, Madi A, Bransburg-Zabary S, Gur-Gershgoren G, et al. (2010) Dynamics of stock market correlations. AUCO Czech Economic Review 4: 330–340.
  8. 8. Kenett DY, Shapira Y, Madi A, Bransburg-Zabary S, Gur-Gershgoren G, et al. (2011) Index cohesive force analysis reveals that the US market became prone to systemic collapses since 2002. PLoS ONE 6(4): e19378.
  9. 9. Mantegna RN, Stanley HE (1996) Turbulence and financial markets. Nature 383: 587–588.
  10. 10. Mantegna RN, Stanley HE (1995) Scaling behaviour in the dynamics of an economic index. Nature 376: 46–49.
  11. 11. Mantegna RN, Palagyi Z, Stanley HE (1999) Applications of statistical mechanics to finance. Physica A 274: 216–221.
  12. 12. Fuentes MA, Gerig A, Vicente J (2009) Universal behavior of extreme price movements in stock markets. PLoS ONE 4(12): e8243.
  13. 13. Schoneburg E (1990) Stock price prediction using neuralnetworks: a project report. Neurocomput-ing 2: 17–27.
  14. 14. Wang YF (2002) Predicting stockprice using fuzzy grey prediction system. Expert Systems with Applications 22: 33–38.
  15. 15. Mittermayer MA (2004) Forecasting intraday stock price trends with text mining techniques. Proceedings of the 37th Hawaii International Conference on Social Systems, Hawaii.
  16. 16. Pai PF, Lin CS (2005) A hybrid ARIMA and support vectormachinesmodel in stock price forecasting. Omega 33: 497–505.
  17. 17. Kenett DY, Preis T, Gur-Gershgoren G, Ben-Jacob E (2012) Quantifying meta-correlations in financial markets. EPL 99: 38001.
  18. 18. Fama EF (1970) Efficient capital markets: a review of theory and empirical work. Journal of Finance 25: 383–417.
  19. 19. Fama EF (1965) Random walks in stock market prices. Financial Analysts Journal 21: 55–59.
  20. 20. Malkiel BG (1989) Is the stock market efficient? Science 243: 1313–1318.
  21. 21. Hsieh DA (1991) Chaos and nonlinear dynamics: application to financial markets. Journal of Finance 46: 1839–1877.
  22. 22. Trippi RR (1995) Chaos and nonlinear dynamics in the financial markets: theory, evidence, and applications. Chicago: Irwin Professional Publishing.
  23. 23. Peters EE (1994) Fractal market analysis: applying chaos theory to investment and economics. New York: J. Wiley & Sons.
  24. 24. Caraiani P (2012) Evidence of multifractality from emerging european stock markets. PLoS ONE 7(7): e40693.
  25. 25. Preis T, Stanley HE (2010) Switching phenomena in a system with no switches. J. Stat. Phys. 138: 431C446.
  26. 26. Preis T, Schneider JJ, Stanley HE (2011) Switching processes in financial markets. Proc Natl Acad Sci USA 108: 7674–7678.
  27. 27. Bonabeau E (2002) Agent-based modeling: methods and techniques for simulating human systems. Proc Natl Acad Sci USA 99: 7280–7287.
  28. 28. Farmer JD, Foley D (2009) The economy needs agent-based modelling. Nature 460: 685–686.
  29. 29. Wang W, Chen Y, Huang JP (2009) Heterogeneous preferences, decision-making capacity, and phase transitions in a complex adaptive system. Proc Natl Acad Sci USA 106: 8423–8428.
  30. 30. Song KY, An KN, Yang G, Huang JP (2012) Risk-return relationship in a complex adaptive system. PLoS ONE 7(3): e33588.
  31. 31. Prechter RR, Frost AJ (1998) Elliott Wave Principle: key to market behavior. Gainesville, GA: New Classics Library.
  32. 32. Bhattacharya S, Kumar K (2006) A computational exploration of the efficacy of Fibonacci Sequences in technical analysis and trading. Annals of Econmics and Finance 1: 219–230.