ARTICLE IN PRESS
Physica A 344 (2004) 272–278 www.elsevier.com/locate/physa
Multifractal features of financial markets Kyungsik Kima,, Seong-Min Yoonb a
Department of Physics, Pukyong National University, Pusan 608-737, Republic of Korea Division of Economics, Pukyong National University, Pusan 608-737, Republic of Korea
b
Received 19 December 2003 Available online 20 August 2004
Abstract We study the tick dynamical behavior of three assets in financial markets (the KOSPI, the won–dollar and yen–dollar exchange rates) using the rescaled range (R/S) analysis. The multifractal Hurst exponents with long-run memory effect can be obtained from those assets, and we discuss whether there exists the crossover or not for the Hurst exponents at characteristic time scales. Particularly, we find that the probability distribution of returns approaches to a Lorentz distribution, different from Gaussian properties. r 2004 Elsevier B.V. All rights reserved. PACS: 02.50.r; 02.60.x; 02.70.c Keywords: Yen–dollar exchange rate; Won–dollar exchange rate; KOSPI; Hurst exponent; Price–price correlation; R/S analysis
The prominent problems in econophysics [1,2] have primarily included the price changes in open market [3,4], the distribution of income of companies, the financial analysis of foreign exchange rates [5–7], the tick data analysis of bond futures [8,9], the herd behavior of financial markets [10], the self-organized segregations [11], and minority game theories [12]. Mantegna and Stanley [4] reported that price changes in a stock market scales as a power law of the index change. Takayasu et al. [5] have showed the stretched exponential distribution from the results of analysis observed by tick data of yen–dollar exchange rates and investigated extensively Corresponding author.
E-mail address:
[email protected] (K. Kim). 0378-4371/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2004.06.131
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several hyper-inflations [13] in which the rapid growth of price changes is approximated by a double exponential function. The power law distribution, stretched exponential distribution, and fat-tailed distribution have really elucidated the general mechanism in diverse scientific fields. Furthermore, several previous works [5–7,14] have presented that the price fluctuations follow the anomalous power law from the stochastic time evolution equation, which is mainly represented in terms of the Langevin-type equation. Recently, Kim and Kong [15,16] have used the box-counting method [17,18] to analyze generalized dimensions and scaling exponents for mountain heights and seabottom depths. Since there existed no statistically significant correlations between observations, the R/S analysis can be extended to distinguish from random time series from correlated ones. Scalas et al. [9] have discussed that the continuous-time random walk theory, formerly introduced by Montroll and Weiss [19], is applied to the dynamical behavior by tick-by-tick data in financial markets. Mainardi et al. [10] have also observed the waiting-time distribution for bond futures traded at LIFFE. The argument for the volume of Korean treasury bond futures traded at Korean Futures Exchange market was analytically and numerically presented in the previous paper [20]. The purpose of this paper is to study the multifractal measures from the tick dynamical behavior of prices using the rescaled range (R/S) analysis for three assets (the KOSPI, the won–dollar and yen–dollar exchange rates). Particularly, the quantities such as the multifractal Hurst exponents, the price–price correlation function, and probability distribution of the prices are also evaluated with long-run memory effects. We introduce the R/S analysis and the Hurst exponents and present in detail the price–price correlation function and probability distribution of prices. To the best of our knowledge, such an attempt in this direction has not been made in Korean and Japanese financial markets. We estimate the generalized qth-order Hurst exponents in the price–price correlation function and find numerically the form of the probability distribution of prices. To quantify the Hurst exponents, we employ the R/S analysis developed by Hurst [21]. First of all, let us consider a price time series Pn of length n given by Pn ¼ fpðt1 Þ; pðt2 Þ; . . . ; pðtn Þg ;
(1)
and the statistical quantity rðtÞ which is the so-called price t-returns having length n and time scale t represented in terms of rðtÞ ¼ fr1 ðtÞ; r2 ðtÞ; . . . ; rn ðtÞg ;
(2)
where ri ðtÞ ¼ ln pðti þ tÞ ln pðti Þ; i ¼ 1; 2; . . . ; n: The average value ðR=SÞM ðtÞ; i.e., the rescaled/normalized relation between RM;d ðtÞ and SM;d ðtÞ; becomes ðR=SÞM ðtÞ ¼
N 1 X RM;d ðtÞ N d¼1 SM;d ðtÞ
/ M HðtÞ ;
ð3Þ
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where the subseries RM;d ðtÞ is given by RM;d ðtÞ ¼ maxfD1;d ðtÞ; D2;d ðtÞ; . . . ; DM;d ðtÞg minfD1;d ðtÞ; D2;d ðtÞ; . . . ; DM;d ðtÞg ; and the standard deviation SM;d ðtÞ for each subseries is defined by " #1=2 M 1 X 2 SM;d ðtÞ ¼ ðrk;d ðtÞ rM;d ðtÞÞ : M k¼1
ð4Þ
(5)
For simplicity, we can calculate the mean of the prices contained in the subseries E M;d ðtÞ ¼ fr1;d ðtÞ; r2;d ðtÞ; . . . ; rM;d ðtÞg (with d ¼ 1; 2; . . . ; M), i.e., rM;d ðtÞ ¼
M 1 X rM;d ðtÞ : M k¼1
(6)
Until now, several methods have been suggested for more than one decade in order to investigate the multifractal properties for tick data. Baraba´si et al. [17] have recently reported the multifractality of self-affine fractals and also studied the multiaffine function and the multifractal spectrum. The qth price–price correlation function F q ðtÞ can be expressed as F q ðtÞ ¼ hjpðt þ tÞ pðtÞjq i / tqH q ;
ð7Þ
where H q and pðtÞ is, respectively, the generalized qth-order Hurst exponent and the bond future price, and the angular bracket denotes an average over t. In reality, a nontrivial multiaffine spectrum can be found as H q varies with q, and the large fluctuation effects in the dynamical behavior of the price pðtÞ can be explored from Eq. (7). Lastly, since we consider the probability distribution of prices, the probability distribution of returns can be represented in terms of a Lorentz distribution, i.e., PðrÞ ¼
2b a ; p r 2 þ a2
(8)
where a and b are constants. The returns r existing in the time series in Eq. ð8Þ is defined by r ¼ rn hrn iM ;
(9)
where rn ¼ frðt1 Þ; rðt2 Þ; . . . ; rðtn Þg; and the corresponding quantity hrn iM is given by hrn iM ¼
iþM=2 1 X rj : M j¼iM=2
(10)
In our scheme, we will make use of Eqs. (4), (7), and (8) to describe the multifractal features of prices. The mathematical techniques discussed above lead us to more general results, and our result will be numerically compared with that of recent calculations.
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To analyze three financial assets, i.e., the KOSPI, the won–dollar and yen–dollar exchange rates, we will present in detail numerical data of Hurst exponents from the R/S analysis. Although we can find other statistical quantities via computer simulation for studies of returns, we restrict ourselves to estimate the generalized qth-order Hurst exponents in the price–price correlation function and the form of the probability distribution of returns. In this paper, we introduce the price time series for three financial assets as follows: first, we used the tick data of the KOSPI transacted for 23 years from April 1981. Second, the tick data for the won–dollar exchange rate (the yen–dollar exchange rate) were taken from April 1981 to December 2002 (from January 1971 to June 2003). In our case, the time step between ticks for three assets is evoluted for one day. The Hurst exponents are obtained numerically from the results of R/S analysis given by Eq. (2), as summarized in Table 1. We find that this process is located in the persistence region, similar to that of the crude oil prices [22]. On the other hand, it may be expected that the Hurst exponent is taken anomalously to be near one as the time series proceeds with long-run memory effects. There is no crossover for the Hurst exponent of the yen–dollar exchange rate, while HðtÞ from our tick data is similarly found to have the existence of crossovers at characteristic times t ¼ 7 and 35 (t ¼ 9) for the KOSPI (the won–dollar exchange rate) [23] in Fig. 1. Since the numerical simulation from Eq. (7) is performed, we obtain the values of the generalized qth-order Hurst exponent H q in the price–price correlation function for three assets, and these numerical data are also summarized in Table 1. Especially, the generalized Hurst exponent for the won–dollar exchange rate is taken to be near 0:6535 as q ! 1: The probability distribution of returns is well consistent with a Lorentz distribution different from Gaussian properties, and it is found that the constants in Eq. (8) are a ¼ 3:0 104 ð5:0 103 Þ and b ¼ 9:4 105 ð1:1 104 Þ for the KOSPI (the won–dollar exchange rate) [23], and a ¼ 7:0 103 and b ¼ 9:0 103 for the yen–dollar exchange rate, as shown in Fig. 2.
Table 1 Summary of values of the Hurst exponent HðtÞ and the generalized qth-order Hurst exponent H q for the KOSPI, the won–dollar and yen–dollar exchange rates HðtÞ
Hq
KOSPI
Hðt ¼ 1Þ ¼ 0:6238 Hðt ¼ 5Þ ¼ 0:6575 Hðt ¼ 24Þ ¼ 0:7278
H 1 ¼ 0:7791 H 2 ¼ 0:5426 H 3 ¼ 0:5215
H 4 ¼ 0:4990 H 5 ¼ 0:4767 H 6 ¼ 0:4357
Won–dollar
Hðt ¼ 1Þ ¼ 0:6886 Hðt ¼ 5Þ ¼ 0:7283 Hðt ¼ 24Þ ¼ 0:7332
H 1 ¼ 0:6535 H 2 ¼ 0:5614 H 3 ¼ 0:4859
H 4 ¼ 0:4307 H 5 ¼ 0:3914 H 6 ¼ 0:3629
Yen–dollar
Hðt ¼ 1Þ ¼ 0:6513 Hðt ¼ 5Þ ¼ 0:5710 Hðt ¼ 24Þ ¼ 0:5870
H 1 ¼ 0:6216 H 2 ¼ 0:5688 H 3 ¼ 0:4956
H 4 ¼ 0:4148 H 5 ¼ 0:3486 H 6 ¼ 0:2996
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0.9
τ 35
H(τ)
0.8
0.7
0.6
0
10
10
1
2
10
log τ Fig. 1. Hurst exponents and existence of two crossovers at characteristic times t ¼ 7 and 35 for the KOSPI.
0.10
Gaussian
P(r)
Lorentz
0.05
0.00 -0.02
0
0.02
r Fig. 2. Probability distribution of returns for the yen–dollar exchange rate. The dot line is represented in 3 a and b ¼ 9:0 103 for the terms of a Lorentz distribution, i.e., PðrÞ ¼ 2bp r2 þa 2 ; where a ¼ 7:0 10 yen–dollar exchange rate.
In conclusion, we have presented the multifractal measures from the dynamical behavior of prices using the R/S analysis for three financial assets, i.e., the KOSPI, the won–dollar and yen–dollar exchange rates. The multifractal Hurst exponent, the generalized qth-order Hurst exponent, and the form of the probability distribution have been discussed with long-run memory effect. Since Hurst exponents are larger
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than 0:5 through R/S analysis, our time series of prices is persistent. It is apparent that the existence of crossovers in the cases of the KOSPI and the won–dollar exchange rate is similar to that of other results [22]. It is in particular found that the probability distribution for all returns is well consistent with a Lorentz distribution. Since it supports to carry out the dynamical behavior in our stock and foreign exchange markets, the present analysis would assure that it is able to capture essentially multifractal properties from our results. In the future, our results will be applied to extensively investigate the other tick data in Korean financial markets and compared in detail with other assets transacted in other nations. This work was supported by the Korea Research Foundation Grant (KRF-2004002-B00026).
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