We propose a novel method to quantify the clustering behavior in a complex time series and apply it to a high-frequency data of the financial markets. We find that regardless of used data sets, all data exhibits the volatility clustering properties, whereas those which filtered the volatility clustering effect by using the GARCH model reduce volatility clustering significantly. The result confirms that our method can measure the volatility clustering effect in financial market.
Deep Dive into Measuring Volatility Clustering in Stock Markets.
We propose a novel method to quantify the clustering behavior in a complex time series and apply it to a high-frequency data of the financial markets. We find that regardless of used data sets, all data exhibits the volatility clustering properties, whereas those which filtered the volatility clustering effect by using the GARCH model reduce volatility clustering significantly. The result confirms that our method can measure the volatility clustering effect in financial market.
Recently, financial markets have been known as representatives of complex system, which changes the property of system dynamically according to inflows of various information from outside and interactions between heterogenous agents [1]. In order to understand the complexity of financial market, the methods of interdisciplinary research have been achieving in physics and economics fields. The various stylized facts such as the long-term memory of volatility [2], volatility clustering [3], fat tails [4], multifractality [5,6] are observed.
The obvious properties among the stylized facts are the long-term memory property and clustering effects of the volatility data. In previous studies, clustering behaviors are shown in the return time interval statistics of the climate records [7], medical data, extreme floods [8], and economics [9].
The various models which reflects the volatility clustering effect in order to predict exactly the volatility in the econometrics field are introduced. The autoregressive conditional heteroskedasticity (ARCH) [10] and the generalized autoregressive conditional heteroskedasticity (GARCH) model [11] are the representatives. Namely, the many researches to understand the micro-mechanism of market has been processed. However, the study to quantify the volatility clustering effects is not sufficient yet. If we observe quantitatively the volatility clustering effect in financial markets, we will understand micro phenomena of the market.
In this paper, we propose the novel method to quantify the volatility clustering effect in the financial time series.
We find that all data sets analyzed in this paper exhibit the volatility clustering property, whereas the data which filters the volatility clustering effect by the GARCH(1,1) model reduces the degree of volatility clustering significantly.
In the next section, we describe the data sets and methods used in this paper. In section III, we preset our results of this study. Section IV concludes.
We investigate quantitatively the volatility clustering behaviors using financial time series including the following market data sets: the 5 minute S&P 500 index from 1995 to 2004 and the 5 minute 28 individual stocks traded in the NYSE with the largest liquidity from 1993 to 2002. The return time series r(t) is calculated by the log-difference of high-frequency prices as follows: r(t) = ln P (t) -ln P (t -1) where P (t) represents the stock price at time t.
In this subsection, we propose a novel method to quantify the volatility clustering effect.
We estimate and analyze quantitatively the volatility clustering effect existed in the financial time series. The process is explained by the following.
Step 1 (The symbolized process): We transfer the return time series r(t) to the symbolic data s(t) in order to quantify the volatility clustering effect in the financial data using the control parameter, such as the number of bins which is defined as
where N b is the number of bins. The conditional distribution with statistical significance is calculated by the symbolized process.
Step 2 (Calculating the conditional distribution): We estimate the conditional distribution using the symbolized time series generated in step 1. In other words, the conditional distribution corresponds to the next value of a specific symbol S T in the symbolic data.
Next, we calculate repeatedly the conditional distribution P (S j |S T ) for all symbolic data in the proper regime. The conditional distribution of each symbolic data has a non-trivial property like the conditional value S T if there is a volatility clustering behavior.
Step 3 (The average value of conditional distribution): The step 3 is the calculation of the average value of the conditional distribution estimated in the step 2. We only consider the conditional distribution of symbolic data in the proper range because the extreme symbolic data are rare. By the average value of the conditional distribution, we observe the volatility clustering effect defined as
where N T is the element numbers of the conditional distribution P (S j |S T ) in terms of a specific symbol S T . If the average value is not dependent on the symbolic value S T , there is no volatility clustering effect because the time series shows the volatility clustering effect only when it has a positive (negative) relation with the positive (negative) values of S T .
Next, we calculate the relation between the specific symbolic values S T and average values S T of conditional distribution. In other words, we observe the degree of volatility clustering (DVC) behavior according to the relationship between S T and S T . The average value S T is definded as
where DV C P,N is the degree of volatility clustering effect for the positive and negative cases respectively. When DV C P,N = 0, there is no clustering effect. However, when the value of DV C P,N is nonzero, the degree of volatility clustering effect according to the relative magnit
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