Realized volatility of index constituent stocks in Hong Kong

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1 Available online at Mathematics and Computers in Simulation 79 (2009) Realized volatility of index constituent stocks in Hong Kong Ying-Foon Chow, James T.K. Lam, Hinson S. Yeung Department of Finance, The Chinese University of Hong Kong, Shatin, Hong Kong Received 27 July 2008; received in revised form 21 September 2008 Available online 8 November 2008 Abstract High-frequency financial data are useful for studying the statistical properties of asset returns at lower frequencies, and they have been widely used to study various market microstructure related issues. However, most studies to date have been concentrated on markets in developed economies such as the stock markets in US or UK. This article aims to investigate the statistical properties of stock return volatility in Hong Kong. Using the sample of constituent stocks of Hang Seng Index () and Hang Seng China Enterprises Index ( or H-shares Index ), we found that the mean daily realized volatilities of stocks to be significantly higher than their counterpart, while the correlations between H-shares stay relatively lower than that of stocks. A long-memory effect is also reported for the logarithmic standard deviations of all shares, with most of them showing slow decay over the series IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Equity markets; High-frequency data; Realized volatility; Correlation 1. Introduction The joint distributional characteristics of asset returns are essential to the analysis of financial issues such as asset pricing, portfolio allocation and risk management. In particular, the increased role played by risk and uncertainty in models of economic decision making lead to further considerations of the second moments. Many traditional financial models have assumed constant volatilities and correlations among returns, but empirical evidence have shown them to vary significantly over time, and much effort has been put into the modeling and forecasting of their properties and structure. Over time, the availability of data for increasingly shorter return horizons has enabled practitioners to narrow the frequencies of their volatility models. However, progress has stalled over the modeling of daily return volatilities, partly due to standard volatility models general inability to handle intraday data, while few high-dimensional models have been suggested for practical use as well. As for several notable multivariate autoregressive conditional heteroskedasticity (ARCH) and stochastic volatility models being proposed, namely Bollerslev et al. [9] and Ghysels et al. [11], have also been challenged for being unable to deal with more than a few assets simultaneously. At the same time, others have relied on ad hoc, model-free approaches like simple exponential smoothing based on the counterfactual assumption of conditionally normally distributed returns, or through squared returns over the relevant return horizon, which in turn does not account for the presence of noise. Corresponding author. address: yfchow@cuhk.edu.hk (Y.-F. Chow) /$ IMACS. Published by Elsevier B.V. All rights reserved. doi: /j.matcom

2 2810 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) The limitations of the above approaches have led researchers to seek an alternative framework of (conditional) variance of financial asset returns. Among others, Andersen, Bollerslev, Diebold and Ebens (henceforth ABDE) [3] and Barndorff-Nielsen and Shephard [8] turned to a new realized volatility approach. See McAleer and Medeiros [12] for an excellent review of the rapidly expanding literature on this issue. Notwithstanding the potential statistical issues induced by various microstructure effects, ABDE [3] proposed using continuously recorded transaction prices and by summing squares and cross-products of intraday high-frequency returns, they constructed estimates known as ex post realized daily volatilities. Under the theory of quadratic variation, the realized daily volatility obtained is an unbiased and highly efficient estimator of return volatility. The advantages of such a measure are that the volatility measures are model-free, and are free from measurement error as the sampling frequency of the returns approaches infinity. Focused on 30 stocks in the Dow Jones Industrial Average (DJIA) index, ABDE [3] settled down on 5-min intervals as the effective continuous time record so as to tackle microstructure frictions such as bid ask bounce effects and price discreteness. On the other hand, Andersen et al. [4] adopted a similar approach to estimate exchange rate volatilities while choosing 30-min intervals. In general, we observed that similar studies on high-frequency financial data have mainly concentrated on developed markets while relatively little has been done on emerging markets like Hong Kong. This article is thus a preliminary attempt to study the volatility structure of shares listed on the Hong Kong Exchanges and Clearing Limited (HKEx) using transaction data. Due to the need for high-frequency price observations, we would focus on the constituent stocks of Hang Seng Index () and Hang Seng China Enterprises Index ( or H-shares Index ). 2. Intuition of realized volatility To set out the basic idea and intuition of realized volatility, let us first consider the case of no microstructure frictions such as price discreteness, infrequent trading, and bid ask bounce effects. Assume that the logarithmic N 1 vector price process, p t, follows a multivariate continuous time stochastic volatility diffusion, dp t = μ t dt + Ω t dw t, where W t denotes a standard N-dimensional Brownian motion, the process for the N N positive definite diffusion matrix, Ω t, is strictly stationary, and we normalize the unit time interval, or h = 1, to represent one trading day. Conditional on the sample path realization of μ t and Ω t, the distribution of the continuously compounded h-period returns, r t+h,h = p t+h p t, is then ( h ) h r t+h,h σ{μ t+τ,ω t+τ } h τ=0 N μ t+τ dτ, Ω t+τ dτ, (2) 0 0 where σ{μ t+τ,ω t+τ } h τ=0 denotes the σ-field generated by the sample paths of μ t+τ and Ω t+τ for 0 τ h. The integrated diffusion matrix thus provides a natural measure of the true latent h-period volatility. By the theory of quadratic variation, we have that under weak regularity conditions, h r t+jδ,δ r t+jδ,δ Ω t+τ dτ 0, (3) j=1,...,[h/δ] 0 almost surely for all t as the sampling frequency of the returns increases, or Δ 0. Thus, by summing sufficiently finely sampled high-frequency returns, it is possible to construct ex post realized volatility measures for the integrated latent volatilities that are asymptotically free of measurement error. ABDE [3] obtained the realized daily covariance matrix as Cov t r t+jδ,δ r t+jδ,δ (3a) with Δ referring to each incremental time interval, which is small enough to be considered continuous in the model. In practice, prices are observed at discrete and irregularly spaced intervals. For instance, ABDE [3] reported that while the DJIA stocks are among the most actively traded U.S. equities, the median inter-trade duration for all stocks across the full sample is 23.1 s, ranging from a low of 7 s for Merck & Co. Inc. (MRK) to a high of 54 s for United (1)

3 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Technologies Corp. (UTX). As such, it is not practically feasible to push beyond this level the continuous record asymptotics and the length of the observation interval Δ. Furthermore, it is known that intraday quotes and transaction prices are subject to various regulatory measures and bid ask bounce effects. For example, even if all traders knew the true value of a stock with complete certainty, bid ask bounce is the term used to describe the transitory bouncing back and forth of the transaction price between bid ask prices depending on whether the trade was initiated by a buyer or a seller. These changes could be in fact spurious, and will cause r t+δ,δ to be an autocorrelated process so that the sum of all available intraday high-frequency squared returns will be a biased estimator of the latent true daily volatility, leading to a breakdown of Eq. (3). In general, such market microstructure features are not important when analyzing longer horizon interdaily returns but can seriously distort the distributional properties of high-frequency intraday returns. Thus, an alternative and much more complicated approach would be to utilize all of the observations by explicitly modeling the high-frequency frictions. See McAleer and Medeiros [12] for further discussion or alternative procedures in this context. 3. Data Our empirical analysis is based on selected stocks in and HKCEI with data from HKEx database. Specifically, we focus on 25 firms in each index with highest 90-day average trading volume to ensure a reasonable degree of liquidity. Following ABDE [3], we resort to artificially constructed 5-min returns, which are believed to be a balance between reducing the effect of various market microstructures and finding appropriate discrete approximations of quadratic variations. In case there are no transactions in our predetermined time intervals, the price of the one immediately before the mark will be selected to replace the missing data. With daily transactions from 10:00 HKT to 12:30 HKT and 14:30 HKT to 16:00 HKT, a total of 48 records can be collected each day, i.e., Δ = 1/48. Our sampling starts from 2 January 2004 to 31 December 2005, with a total of 496 trading days and hence we have a total of 23,808 samples for each stock. As mentioned above, market microstructure characteristics could potentially distort the distributional properties of intraday returns thereby distorting the results of statistical inference. More specifically, as pointed out in McAleer and Medeiros [12], such microstructure effects can introduce a severe bias on the daily volatility estimation rendering the approach along the lines of ABDE [3] sub-optimal in comparison with the more recent approaches as in Aït-Sahalia et al. [1,2], Barndorff-Nielsen et al. [6,7] and Zhang et al. [14]. In this regard, we should view the following results as preliminary examinations on the data and will consider alternative sampling scheme in future investigations. 4. Univariate unconditional return and volatility distributions 4.1. Returns The summary statistics in Table 1 show that the daily return of and HKCEI stocks, r i,t, have fatter tails than the normal distribution and for majority of stocks, they are skewed to the right. In addition, they are in line with previous empirical evidence about the fat-tail characteristics of returns. The average values of skewness coefficients are about 0.14/0.01 (/) for raw returns and 0.10/0.07 (/) for standardized returns, while the average values of excess kurtosis are 2.33/2.28 (/) for raw returns and 0.49/0.55 (/) for standardized returns. Fig. 1 plots the standardized returns for Esprit (0330.HK) as an example. It can be seen that normal distribution may be reasonable but not necessarily a very fit description of the data. This result is in line with the leptokurtic distributions for standardized high-frequency returns implicitly assumed in ARCH and stochastic volatility model Variances and log standard deviations Table 2 provides the same set of summary statistics for the unconditional realized daily return variances and logarithmic standard deviations. In Panel A, the averages of daily variance (column 1) are 4.81 and for and stocks, which translate to an annualized volatility of about 34.5% and 53.7%. This shows that H-shares are more volatile than the blue chips in general. The standard deviation of the realized volatility (column 2) indicates that the realized volatility of different assets vary significantly over time. The skewness (column 3) and excess kurtosis (column 4) of daily realized variances show that the distributions are highly skewed to the right and leptokurtic with

4 2812 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Table 1 Unconditional daily return distributions. Mean Standard deviation Skewness Excess kurtosis Panel A: r i,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Panel B: r i,t /v i,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Fig. 1. Unconditional distribution of the standardized daily returns on Esprit (0330.HK) compared to normal distribution. The sample period is 2 January 2004 to 30 December 2005 for a total of 496 daily observations. average skewness of 3.33/3.05 (/) and average excess kurtosis of 25.6/28.47 (/), respectively. Such results show that normality assumption is not appropriate for return variances. Panel B reports the summary statistics of logarithmic realized volatility, lv i,t. It can be seen that the average values of the sample skewness are close to zero for the realized variances albeit being slightly negative. The excess kurtosis coefficients are significantly lower and hence the assumption of normality may be reasonable but not necessarily a very fit description of the data. As an illustration, the standardized daily realized return standard deviations for Wharf Holdings (0004.HK) are plotted in Fig. 2.

5 Table 2 Unconditional volatility distributions. Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Mean Standard deviation Skewness Excess kurtosis Panel A: v 2 i,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Panel B: lv i,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Covariances and correlations Table 3 reports the summary statistics of the covariances and correlations between log realized volatilities. The averages of covariances and correlations across all stocks are about 0.41/0.81 (/) and 0.09/0.06 (/) with substantial variations over time indicated by their standard deviations (column 2). The skewness and excess kurtosis of realized covariances and correlations are reported in columns 3 and 4, respectively. The average skewness and excess Fig. 2. Unconditional distribution of the standardized daily realized logarithmic standard deviations for Wharf Holdings (0004.HK) compared to normal distribution. The realized volatilities are calculated from 5-min intraday returns.

6 2814 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Table 3 Unconditional covariances and correlations distributions. Mean Standard deviation Skewness Excess kurtosis Panel A: Cov i,j,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Panel B: Corr i,j,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum kurtosis for realized covariances are 2.2/1.4 (/) and 15.0/9.0 (/), compared to almost zero for realized correlations. It seems that the distributions of the realized covariances are right skewed, but the correlations appear to be normally distributed. The results in Panel B of Table 3 indicate that the correlations between H-shares in general are lower than the blue chips despite their higher volatilities. Fig. 3 shows the unconditional realized correlation between Cheung Kong Holdings (0001.HK) and Hutchison Whampoa (0013.HK), the two considerably large stocks in in terms of market capitalization. It appears that normal distribution can serve as a good approximation to the daily realized correlations. 5. Dynamic dependence of volatilities and correlations After exploring the unconditional distributions of the return generating process, we extend our analysis to the conditional distributions of the volatility processes. Previous studies have shown the existence of long-term dependence in volatility. Here we focus on logarithmic volatilities and correlations since both can be approximated well by normal distributions, with results in Table Log standard deviations In Fig. 4, we present the time series plot of the logarithmic standard deviations of Cheung Kong Holdings (0001.HK), which may exhibit positive autocorrelations as Fig. 5 indicates a slowly declining autocorrelation coefficient. We report the values of standard Ljung-Box portmanteau test for the joint significance of the first 22 autocorrelations of lv i,t in column 1 of Panel A of Table 4. The null hypothesis is significantly rejected for all stocks. In fact, the autocorrelations are systematically above the conventional 95% confidence band, even at a long horizon of 120 trading days (approximately half year). This slow decline in autocorrelations has led us to perform augmented Dickey-Fuller tests to detect the existence of unit roots in each of the volatility series. The results are reported in column 2, which shows that not all stocks can reject the null hypothesis at 5% level.

7 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Fig. 3. Unconditional distribution of the standardized realized daily correlations between Cheung Kong Holdings (0001.HK) and Hutchison Whampoa (0013.HK) compared to normal distribution. The realized correlations are calculated from 5-min intraday returns. With such findings, we extend the analysis to examine the potential long-memory effect modeled by a FIGARCH model as in Baillie et al. [5]. Using the approach developed in Geweke and Porter-Hudak [10], the degree of fractional integration for the realized logarithmic volatilities, denoted as d GPH, are shown in the third column in Panel A of Table 4. The average values are 0.58/0.56 (/) with respective standard deviations of 0.14/0.12 (/). Thus, Table 4 Dynamic volatility dependence in the sample. Q 22 ADF d GPH Panel A: lv i,t Average Standard deviation Minimum Maximum Average Standard deviation Minimum Maximum Panel B: Corr i,j,t Average standard deviation Minimum Maximum Average Standard deviation Minimum Maximum

8 2816 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Fig. 4. Time series plot for the logarithmic standard deviation of Cheung Kong Holdings (0001.HK). The figure shows the time series plot of the daily realized logarithmic standard deviations for Cheung Kong Holdings (0001.HK). The realized volatilities are calculated from 5-min intraday returns. our findings are consistent with most of the literature in that the dynamics of the log-realized variance is a highly persistent but stationary time series process Correlations Panel B of Table 4 reports the results on the temporal behavior of the daily realized correlations Corr i,j,t. Compared to the analysis in Section 5.1, the null hypothesis of no autocorrelation is not rejected for quite a number of stocks Fig. 5. Sample autocorrelations for daily logarithmic standard deviations. The figure shows the sample autocorrelations for the daily realized logarithmic standard deviations for Cheung Kong Holdings (0001.HK), lv 0001.HK,t log(v 0001.HK,t ), out to a displacement of 100 days. The realized volatilities are calculated from 5-min intraday returns.

9 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Fig. 6. Time series of daily correlations. The figure shows the time series of daily realized logarithmic standard deviations between Cheung Kong Holdings (0001.HK) and Hutchison Whampoa (0013.HK). The sample period is 2 January 2004 to 30 December 2005 for a total of 496 daily observations. The realized correlations are calculated from 5-min intraday returns. using Ljung-Box portmanteau test. Similarly, we test for the existence of unit root by augmented Dickey-Fuller test, and we found that again quite a number of correlation pairs cannot reject the null hypothesis. We then fit a FIGARCH model to investigate the long-memory effect in the processes. The averages of d GPH are about 0.23/0.27 (/) with standard deviations of 0.18/0.17 (/). Thus, the stationarity assumption for correlations does not seem to hold for some elements in the correlation matrix, although the evidence is weaker than that of logarithmic standard deviations. As an example, Fig. 6 shows the time series plot of the realized correlations between Cheung Kong Holdings (0001.HK) and Hutchison Whampoa (0013.HK). Despite the variability, the daily realized correlations seem to exhibit possible autocorrelations over time. It should be noted, however, the Geweke and Porter-Hudak [10] estimator may not yield the optimal long-memory parameter estimate as reported in Rea et al. [13]. Therefore, while the results seem to suggest the existence of long-memory effects in logarithmic realized standard deviations and daily realized correlations, we need to interpret the estimates with care and further investigations are warranted. 6. Concluding remarks In this article, we analyzed the high-frequency transaction data for selected stocks in and over the period from 2004 to 2005 based on ABDE [3] approach. The univariate unconditional distributions of daily returns, variances, logarithmic standard deviations, covariances and correlations are examined. It is found that the mean daily realized volatilities of H-shares to be significantly higher than their counterparts, while the correlations between H-shares stay relatively lower than that of stocks. We also investigated the possible long-memory effect in realized logarithmic standard deviations and correlations through standard time series tests and FIGARCH model. It is found that realized daily logarithmic standard deviations demonstrated strong evidence of long-memory, while the long-memory property seems to be weaker in correlations for large portion of stocks in our study. As mentioned in McAleer and Medeiros [12], however, it should be noted that microstructure noise could cause severe problems in terms of consistent estimation of the daily realized volatility estimator along the lines of ABDE [3]. Therefore, further investigation on the volatility structures in Hong Kong stock market will be pursued in near future.

10 2818 Y.-F. Chow et al. / Mathematics and Computers in Simulation 79 (2009) Acknowledgements The research described herein was partially supported by the Center for Chinese Financial Development and Reform of the Chinese University of Hong Kong. We would also like to thank Tao Li, Patrick Hui, seminar participants at the Chinese University of Hong Kong and the MODSIM 2007 for their helpful comments and suggestions on earlier drafts. Any remaining errors are our own. References [1] Y. Aït-Sahalia, P.A. Mykland, L. Zhang, How often to sample a continuous time process in the presence of market microstructure noise, Review of Financial Studies 18 (2005) [2] Y. Aït-Sahalia, P.A. Mykland, L. Zhang, Ultra high frequency volatility estimation with dependent microstructure noise, NBER Working Paper 11380, [3] T.G. Andersen, T. Bollerslev, F.X. Diebold, H. Ebens, The distribution of realized stock return volatility, Journal of Financial Economics 61 (2001) [4] T.G. Andersen, T. Bollerslev, F.X. Diebold, P. Labys, Modeling and forecasting realized volatility, Econometrica 71 (2003) [5] R.T. Baillie, T. Bollerslev, H.O. Mikkelsen, Fractionally integrated generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 74 (1996) [6] O.E. Barndorff-Nielsen, P.R. Hansen, A. Lund, N. Shephard, Designing realized kernels to measure the ex-post variation of equity prices in the presence of noise, Econometrica 76 (2008) [7] O.E. Barndorff-Nielsen, P.R. Hansen, A. Lund, N. Shephard, Subsampled realized kernels, Journal of Econometrics (in press). [8] O.E. Barndorff-Nielsen, N. Shephard, Econometric analysis of realized volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society: Series B 64 (2002) [9] T. Bollerslev, R.F. Engle, D.B. Nelson, ARCH models, in: R.F. Engle, D.L. McFadden (Eds.), Handbook of Econometrics, vol. 4, North-Holland, Amsterdam, 1994, pp [10] J. Geweke, S. Porter-Hudak, The estimation and application of long memory time-series models, Journal of Time Series Analysis 4 (1983) [11] E. Ghysels, A. Harvey, E. Renault, Stochastic volatility, in: G.S. Maddala, C.R. Rao (Eds.), Handbook of Statistics, vol. 14, North-Holland, Amsterdam, 1996, pp [12] M. McAleer, M.C. Medeiros, Realized volatility: a review, Econometric Reviews 27 (2008) [13] W. Rea, L. Oxley, M. Reale, J. Brown, The empirical properties of some popular estimators of long memory processes, Working Paper 13/2008, Department of Economics, University of Canterbury, [14] L. Zhang, P.A. Mykland, Y. Aït-Sahalia, A tale of two time scales: determining integrated volatility with noisy high frequency data, Journal of the American Statistical Association 100 (2005)