A Persistence Probability Analysis in Major Financial Indices

Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007, pp. 249 253 A Persistence Probability Analysis in Major Financial Indices I-Chun Chen Department of Physics, National Chung Hsing University, Taiwan, R.O.C. and Nan Kai Institute of Technology, Nantou, Taiwan, R.O.C. Hung-Jung Chen and Hsen-Che Tseng Department of Physics, National Chung Hsing University, Taiwan, R.O.C. (Received 15 August 2006) We analyzed twenty world stock market indices and compared the persistence properties among these countries. Two methods were used in the analyses: (1) the price-price correlation analysis technique and (2) the persistence analysis technique. Our studies are based on numerical estimates of the persistence exponent θ p and the Hurst exponent H 2. The relation θ p = 1 H 2 among these countries is also discussed. PACS numbers: 05.10.-a, 05.30.Pr Keywords: Persistence probability, Hurst exponent, Stock market, Power law I. INTRODUCTION Janssen et al. [1] were among the first to introduce the relaxation of a system with purely dissipative dynamics, starting from an initial state characterized by non-equilibrium values for both order parameters and correlations. Such a technique is based upon short-time critical dynamics. Persistence is the phenomenon defined as the probability that a fluctuating non-equilibrium system has never changed its sign up to time t [2]. The persistence exponent θ p was first introduced in the context of the non-equilibrium coarsening dynamics of systems at zero temperature [3]. In many systems, persistence decays algebraically, P (t) t θp, with a nontrivial persistence exponent θ p. While this phenomenon was introduced for magnetic systems [4,5], simple diffusion [6, 7], disordered systems [8,9], surface growth [10], fluctuating interfaces [11], etc, much of the recent theoretical effort has gone into obtaining the numerical value of θ p for different systems. Financial stocks are complex, nonlinear, open systems characterized by a large number of parameters. Recent generalizations of the persistence probability may bring added insight into the dynamics of complicated stochastic processes. Many multifractal models [12 14] have been developed over the last decade. The Hurst exponent H q [15, 16] and the persistence exponent in these financial time series have been investigated in numerical and analytical ways [17 20]. Krug et al. [21] proved E-mail: ichun.nancy@msa.hinet.net; Fax: +886-4-2320-8357 the relation H 2 = 1 θ p for symmetric signals with zero mean. This concept of persistence exponent θ p is close to 0.5, which agrees with the relation θ p = 1 H 2. In this paper, we analyze the persistence properties of universal scaling behavior for twenty world stock market indices. Furthermore, we compare the values of the Hurst exponent H 2 and the persistence exponent θ p among these countries and calculate the H 2 + θ p value. This paper is organized as follows: In Sec. II, we introduce the methodology of the persistence analysis technique and price-price correlation analysis technique. In Sec. III, we present our results. Our concluding remarks are given in Sec. IV. II. METHODS 1. Persistence Analysis Technique We denote the value of the daily stock exchange index at a certain time t as y(t ). P + (t)(p (t)) is the probability that the value of the daily stock exchange index has never gone down (up) to the value y(t ) in time t, i.e., y(t + Nδt) > y(t ) (y(t + Nδt) < y(t )) for N = 1, 2,, t. We analyzed daily data (hence, δt = 1) and observed that the best power law appears for the average persistence probability from twenty world stock markets. The average persistence probability P (t) is defined as [P + (t) + P (t)]/2, which is expected to have a -249-

-250- Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007 power-law behavior P (t) t θp. (1) 2. Price-price Correlation Analysis Technique We considered a set of data recording the daily stock exchange index. The generalized qth-order price-price correlation function is defined as G q (t) =< y(t 0 + t) y(t 0 ) q > 1/q, (2) where y(t) is the stock price and the average is over all the initial times t 0. G q (t) also has a power-law behavior G q (t) t Hq, (3) where H q is called the generalized Hurst exponent. For the best fit value, the variation function is defined as var = t=t 1 n (ln P (ln t) p b (ln t)) 2, (4) t n t 0 + 1 t=t 0 where p b (t) = a + bt, a, b are constants. The fitted time interval is denoted by [t 0, t n ]. III. RESULTS We first discuss the results concerning the persistence probability. Our numerical results for the averaged persistence probability P (t) obey a power-law behavior (e.g., Milan stock market), as shown in Fig. 1 (a). From linear fits, we calculated the best fit values by using Eq. (4) for the time interval [t 0, t n ], as shown in Table 1. For low-frequency stocks, one observes that the best power law appears for the average persistence probability, but a power law does not appear for the persistence probability P (t) or P + (t). For high-frequency stocks, we calculated the Taiwan stock exchange index with the 5-min intraday data from 1996 to 2005 [22]. For the Taiwan stock exchange index, one observes that the persistence probabilities P ± (t) appear very close in the early time period and that the average persistence probability P (t) appears to follow the best power-law behavior. We observed an average persistence probability for low-frequency data that was consistent with high-frequency data. This result agrees with previous studies of the persistence probability for the German stock index [17]. Furthermore, we present the results of the generalized qth-order price-price correlation function G q (t) for q = 1, 2, 4, 6, 8 (e.g., the S&P 500 stock index), as shown in Fig. 1 (b). The Hurst exponent H q varies with time, and the scaling of the correlation functions suffers many transient regions. Here, we present the results of G q (t) for low-frequency data and observe that the best power Fig. 1. (a) The best fit to Eq. (4) gives var = 0.007, where the time interval is [0, 380], for the persistence probability with the daily Milan stock index from 1990 to 2005. (b) The generalized price-price correlation function G q(t) vs. t corresponding to the S&P 500 stock index from 1971 to 2005 (for q = 1, 2, 4, 6, 8). law appears for q = 2. For the Taiwan stock exchange index, one observes that high-frequency data has the same result. For illustration purposes, we have fitted certain portions of these log-log plots to obtain a qualitative view of the dependence of H q on 1/q (e.g., Sydney, NZSE-40, and Johannesburg stock indices), as shown in Fig. 2(d). In this figure, we have used a large range of values for q (i.e., q = 1/10, 1/9,, 1, 2,, 10). It is intriguing to notice that H q depends on q for q > 1 and saturates for q < 1. Finally, our results presented here follow for twenty major financial markets, the following contribution being divided into three regions: 1. Asian Stock Markets Fig. 2(a) shows the persistence probabilities of six Asian stock market indices (Tokyo NK-225, Hong Kong, Singapore, Bangkok, Manila, and South Korea). The

A Persistence Probability Analysis in Major I-Chun Chen et al. -251- Fig. 2. (a) Persistence probability with the daily Asian stock market indices. (b) Persistence probability with the daily Western - American stock market indices. (c) Persistence probability with the daily European stock market indices. (d) Plot of the generalized Hurst exponent H q vs. 1/q for various stock market indices. persistence exponent θ p is associated with the power-law decay of the average probability. From linear fits, the values of the persistence exponent θ p range from 0.39 to 0.54, and the values of the Hurst exponent H 2 from 0.55 to 0.61 (as shown in Table 1). A Hurst exponent of H 2 < 0.5 involves an antipersistent behavior while that of H 2 > 0.5 means a persistent signal. Fractional Brownian motion is characterized by H 2 = 0.5. This is an important distinction that can be explained by the persistent and antipersistent trends in random walks with H 2 > 0.5 and H 2 < 0.5. By persistent, we mean that a positive step tends to follow a positive step. For antipersistent walks, the opposite is true. 2. Western - American Stock Markets Fig. 2(b) shows the persistence probabilities of eight Western - American stock market indices (D. J. Indus. Average, S&P 500, NASDAQ, Russell 2000, Toronto 300, Sydney, Wellington NZSE-40, and Johannesburg). From linear fits, the values of the persistence exponent θ p range from 0.39 to 0.58, and the values of Hurst exponent H 2 from 0.48 to 0.60 (as shown in Table 1). 3. European Stock Markets Fig. 2(c) shows the persistence probabilities of six European stock market indices (CAC, Frankfurt, London- 100, Stockholm, Milan, and Zurich). From linear fits, the values of the persistence exponent θ p range from 0.36 to 0.47, and the values of the Hurst exponent H 2 from 0.56 to 0.65 (as shown in Table 1). From Fig. 2(a) to Fig. 2(c), the persistence probability has a persistence exponent θ p ranging from 0.36 to 0.58 and a Hurst exponent H 2 ranging from 0.48 to 0.65 (as shown in Table 1). One observes from the table that while the relation H 2 = 1 θ p holds for two stock market indices (D. J. Indus. Average and NASDAQ Composite), it only holds approximately (i.e., H 2 = (1 θ p ± 0.1)) for eighteen stock market indices (Tokyo NK-225, Hong Kong, Singapore, Bangkok, Manila, South Korea, S&P 500, Russell 2000, Toronto 300, Sydney, Welling-

-252- Journal of the Korean Physical Society, Vol. 50, No. 1, January 2007 Table 1. Comparison of the Hurst exponent H 2 and the persistence exponent θ p value for major stock indices. The best fit value was calculated using Eq. (4) for the time interval [t 0, t n]. The H 2 + θ p value was also calculated. Stock market Time period (daily) θ p var H 2 var H 2+θ p NK-225 4 Jan. 1982 1 Dec. 2005 0.42 0.009 [0,160] 0.61 0.026 [0,500] 1.03 Hong Kong 4 Jan. 1982 1 Dec. 2005 0.53 0.007 [50,140] 0.55 0.026 [0,500] 1.08 Singapore 4 Jan. 1982 15 Nov. 2005 0.39 0.008 [0,310] 0.56 0.017 [0,410] 0.95 Bangkok 2 Jan. 1992 1 Dec. 2005 0.44 0.010 [5,190] 0.59 0.020 [0,400] 1.03 Manila 2 Jan. 1990 1 Dec. 2005 0.50 0.006 [50,150] 0.55 0.019 [0,500] 1.05 South Korea 3 Jan. 1990 15 Nov. 2005 0.54 0.006 [50,110] 0.56 0.016 [0,340] 1.1 D. J. Indus. 4 Jan. 1971 1 Dec. 2005 0.46 0.009 [0,220] 0.54 0.029 [0,390] 1 S&P 500 4 Jan. 1971 1 Dec. 2005 0.48 0.014 [0,110] 0.53 0.022 [0,190] 1.01 NASDAQ 11 Oct. 1984 1 Dec. 2005 0.40 0.008 [0,230] 0.60 0.018 [0,500] 1 Russell 2000 10 Sep. 1987 1 Dec. 2005 0.51 0.004 [50,140] 0.48 0.012 [0,500] 0.99 Toronto 300 2 Jan. 1992 16 Sep. 2004 0.39 0.009 [0,310] 0.60 0.025[0,410] 0.99 Sydney 2 Jan. 1992 25 Nov. 2005 0.52 0.010 [50,460] 0.56 0.018 [0,500] 1.08 NZSE-40 6 Jan. 1992 31 Mar. 2004 0.58 0.008 [50,110] 0.49 0.020 [0,340] 1.07 Johannesburg 2 Jan. 1992 1 Dec. 2005 0.51 0.006 [50,170] 0.55 0.025 [0,500] 1.06 CAC 2 Jan. 1992 16 Sep. 2004 0.37 0.012 [5,440] 0.65 0.028 [0,500] 1.02 Frankfurt 26 Nov. 1990 1 Dec. 2005 0.40 0.010 [0,220] 0.61 0.020 [0,500] 1.01 London-100 11 Feb. 1985 1 Dec. 2005 0.47 0.014 [0,100] 0.56 0.031 [0,300] 1.03 Stockholm 2 Jan. 1992 30 Apr. 2001 0.36 0.011 [0,390] 0.62 0.028 [0,390] 0.98 Milan 3 Jan. 1990 1 Dec. 2005 0.40 0.007 [0,380] 0.61 0.028 [0,500] 1.01 Zurich 9 Nov. 1990 16 Sep. 2004 0.41 0.012 [0,100] 0.60 0.025 [0,500] 1.01 ton NZSE-40, Johannesburg, Milan, CAC, Frankfurt, London-100, Stockholm, and Zurich). In our numerical estimates, however, a value of Hurst exponent greater than 0.5, H 2 > 0.5, was obtained on all of the Asian and European stock markets and on six of the eight Western - American stock markets, which means, except for two markets, a persistent signal exists for all of the markets under discussion. IV. CONCLUSION Our motivation is to provide a better understanding of stock markets through the concepts of multifractals, the Hurst exponent H 2, and the persistence exponent θ p. First, we present an estimate for the persistence exponent extracted from twenty world stock market indices. We also use a standard dynamical scaling analysis, inspired from surface growth phenomena, to show the multifractal behavior of the financial stocks, which was extracted quantitatively from the qth-order priceprice correlation functions. Our results go in parallel with earlier analyses of other groups, which succeeded in showing multifractality in twenty stock markets. The price evolution is multifractal if the exponent H q varies with q; otherwise, it is fractal (corresponding to multi-affine and self-affine, respectively, in the theory of surface dynamical scaling). In particular, for q = 2, we recover the fractional Brownian motion case described by the well-known Hurst exponent 0 < H 2 < 1. The second-order Hurst exponent H 2 is associated with the correlation function of the stock price, which has been shown to be simply related to the persistence exponent through H 2 = 1 θ p (for symmetric signals with zero mean). We find that numerical financial stock indices follow a power-law behavior and that the values of the persistence exponent θ p are not all close to 0.5. The relation H 2 = 1 θ p is only valid for certain processes studied in Ref. 21 (for Gaussian processes, this is limited to the fractional Brownian motion). In some stock market indices, the persistence exponent is close to 0.5, which agrees with the relation θ p = 1 H 2, but in some stock market, the indices do not agree with the relation θ p = 1 H 2. The results have shown that the values of the persistence exponent θ p do not agree with 1 H 2 for all stock markets. ACKNOWLEDGMENTS The authors thank Professor Ping-Cheng Li for many discussions in the early stages of this work. This work was supported in part by the National Chung Hsing University. The data used in this paper were obtained from the library of National Chung Hsing University.

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