forecasting intraday volatility and value-at-risk with high-frequency data

Asia-Pacific Finan Markets (2013) 20:83–111DOI 10.1007/s10690-012-9160-1

Forecasting Intraday Volatility and Value-at-Riskwith High-Frequency Data

Mike K. P. So · Rui Xu

Published online: 9 November 2012© Springer Japan 2012

Abstract In this paper, we develop modeling tools to forecast Value-at-Risk andvolatility with investment horizons of less than one day. We quantify the market riskbased on the study at a 30-min time horizon using modified GARCH models. Theevaluation of intraday market risk can be useful to market participants (day tradersand market makers) involved in frequent trading. As expected, the volatility featuresa significant intraday seasonality, which motivates us to include the intraday sea-sonal indexes in the GARCH models. We also incorporate realized variance (RV) andtime-varying degrees of freedom in the GARCH models to capture more intradayinformation on the volatile market. The intrinsic tail risk index is introduced to assistwith understanding the inherent risk level in each trading time interval. The proposedmodels are evaluated based on their forecasting performance of one-period-ahead vol-atility and Intraday Value-at-Risk (IVaR) with application to the 30 constituent stocks.We find that models with seasonal indexes generally outperform those without; RVcan improve the out-of-sample forecasts of IVaR; student GARCH models with time-varying degrees of freedom perform best at 0.5 and 1 % IVaR, while normal GARCHmodels excel for 2.5 and 5 % IVaR. The results show that RV and seasonal indexesare useful to forecasting intraday volatility and Intraday VaR.

Keywords GARCH · Intraday market risk · Intrinsic tail risk index ·Realized volatility · Risk management · Seasonality · Value at Risk

M. K. P. So (B)Department of Information Systems, Business Statistics and Operations Management,School of Business and Management, The Hong Kong University of Scienceand Technology, Clear Water Bay, Hong Konge-mail: [email protected]

R. XuDepartment of Economics, Stanford University, Stanford, CA, USA

123

84 M. K. P. So, R. Xu

1 Introduction

High frequency data has become readily available to all as a result of recentadvances in trading technology. Grasping the opportunities that this presents, a newgroup of active market participants, such as high-frequency traders, has emergedin the major financial markets. These practitioners are characterized by their veryshort investment horizons, and their (heavy) reliance on tools for market risk mea-surement. As risk must be estimated in intraday intervals, traditional risk mea-sures, such as Value at Risk (VaR) and return volatility, have to be extended tocater for shorter than one-day time intervals. As demonstrated during the globalfinancial crisis in late 2008, even the price of blue-chip stocks can fall sub-stantially within a day. This kind of dramatic intraday price movements in theequity market can cause huge losses for many active traders and market makers.1

Here, we investigate the intraday forecasts of volatility and VaR using tick-by-tick transactions with an aim to improve the tools for measuring the market risk.The use of high-frequency data, defined as intraday market information includ-ing transaction prices, bid-ask prices and intraday trading volume, etc, has beenstudied for more than ten years. Early researches have concentrated on exchangerates, e.g. Zhou (1996), Taylor and Xu (1997) and Beltratti and Morana (1999).More recent works with a focus on equity data include Andersen et al. (2001a),Giot and Laurent (2004) and Fuertes et al. (2009). The above-mentioned stud-ies have centered on the calculation of daily risk measures, whereas this paperdevelops methodologies for forecasting risk measures with a horizon of less thana day.

The major contributions of our research rest on four aspects. First, we examine thedistribution of intraday equity returns and realized volatilities with high-frequencydata of Hang Seng Index’s (HSI) stocks. We choose HSI because it is represen-tative of the statistical properties of stock intraday returns in Hong Kong. To thebest of our knowledge, this is also the first comprehensive study of intraday stockreturn distribution in Asia. Our research on the 30-min return distributions presents agood comparative case to similar studies on the daily return distributions of the S&P500 index (Andersen and Bollerslev 1998), the Dow Jones Industrial Average stocks(Andersen et al. 2001a), the NYSE stocks (Fuertes et al. 2009) and the Spanish StockExchange (Coroneo and Veredas 2006). From the summary statistics, we find thatthe distribution of intraday equity returns and realized volatilities of HSI constituentstocks share similar properties with the daily equity return and realized volatility dis-tribution presented in Andersen et al. (2001a), but the intraday distributions exhibitclear seasonality which should be accommodated in forecasting exercises. Specif-ically, we identify a significant W-shaped intraday volatility pattern, which servesas the major motivation for us to incorporate seasonal indexes in intraday volatilitymodeling.

Second, we propose simultaneous modeling of seasonality in volatility and tailrisk under GARCH. Our approach explains heterogeneous volatility levels and

1 For example, CITIC Pacific, which is a blue-chip constituent of thein Hong Kong Stock Exchange,dropped by as much as 28.5 % on October 27th, 2008 to close at 3.66 HKD.

123

Forecasting Intraday Volatility and Value-at-Risk 85

different tail characteristics in different intraday time periods. In our volatilityforecasting models, intraday seasonal indexes are incorporated to account for the sig-nificant seasonal patterns demonstrated in the descriptive statistics of 30-min returns.In past literature, the intraday seasonal patterns were usually filtered before modelingvolatility as in Giot (2005), Coroneo and Veredas (2006) and Dionne et al. (2009);while in our study, we include seasonal indexes in the conditional variance predictionmodel, together with time-dependent tail characteristic parameters, to improve out-of-sample forecasting performance. As it turns out, models with seasonal indexes out-perform models without in terms of the ability to predict intraday volatility and VaR.In addition, we introduce the intrinsic tail risk index to capture the part of VaRdue to the seasonality and the inherent risk in each time interval. This index canhelp investors to pre-assess the intraday trading risk with the most recent marketcondition.

Third, we incorporate 30-min realized variance (RV) into the volatility forecastingmodel. RV, a sum of squared returns with moderate frequency data, is a natural mea-sure of variability of the price path (see Andersen et al. 2001b; Barndorff-Nielsen andShephard 2002; Zhang et al. 2005; Hansen and Lunde 2006). The nonparametric natureof RV and the simplicity of its calculation have made it popular among practitioners.It has been used for asset allocation (Fleming et al. 2003), the forecasting of Value atRisk (Giot and Laurent 2004), evaluation of volatility forecasting models (Andersenand Bollerslev 1998) and the building of stochastic volatility models (Takahashi et al.2009), among other purposes. Unlike past studies where daily RV is used as a proxyfor daily latent return volatility (Martens 2001, 2002; Koopman et al. 2005; Brown-lees and Gallo 2010), we incorporate intraday RV in the conditional volatility modelto capture the most recent information and help provide better forecasts of intradayvolatility. RV is considered in GARCH models in two ways: one is to replace 30-minsquared return with RV, the other is to add an additional term of RV into GARCHmodels.

Finally, we derive a forecasting formula for |Rt+1|c under conditional t distri-butions to assess volatility predictive performance. Based on the prediction perfor-mances of powered absolute returns, our novel use of RV significantly improves thevolatility forecasting performance for a 30-min investment horizon. We also use theconditional VaR framework to study the performance of VaR prediction by modifiedGARCH-t models with time-varying degrees of freedom. Each model is evaluatedand scored according to its ranking in forecasting accuracy. The overall performanceof the models, obtained by averaging the ranking scores, shows that seasonal indexesand time-varying degrees of freedom can improve out-of-sample forecasts of intradayVaR.

The remainder of this paper is organized as follows. Section 2 describes the datasets used in the study and the statistical properties of 30-min stock return and realizedvolatility. In Sect. 3, we present the construction of three groups of modified GARCHmodels to forecast intraday volatility. The intrinsic tail risk index is also introduced.In Sect. 4, volatility forecasting and VaR forecasting performances of all models areevaluated with application to the empirical data of 30 HSI constituent stocks. Outper-forming models are identified for volatility and intraday VaR forecasting. Section 5concludes the paper.

123

86 M. K. P. So, R. Xu

2 Data and Descriptive Statistics

2.1 Data Description and Cleaning

Our data set, obtained from the History Tool of Bloomberg Station, contains tick-by-tick transaction information (time, price and volume) of 30 constituent stocks of theHSI. The 30 stocks are selected as follows: we start by including all 42 constituentstocks of the HSI; then we eliminate those that are not the major constituents of theHang Seng HK LargeCap Index, the HK MidCap Index or the Mainland Compos-ite Index. Finally, stocks with erroneous data are excluded. The sample time periodstarts on March 25th 2008 and ends on May 31st 2009, which covers the most vola-tile period of the subprime financial crisis. The 30 stocks are Cheung Kong HoldingsLtd (CK), CLP Holdings Ltd (CLP), Wharf Holdings Ltd (Wharf), HSBC Holdingsplc (HSBC), Hong Kong Electric Holdings Ltd (HK Electric), Hang Seng Bank Ltd(HS Bank), Henderson Land Development Co. Ltd (Henderson), Hutchison Wham-poa Ltd (Hutchison), Sun Hung Kai Properties Ltd (SHK), New World Development(New World), Swire Pacific Ltd ‘A’ (Swire A), MTR Corporation Ltd (MTR), SinoLand Co. Ltd (Sino Land), Hang Lung Properties Ltd (Hang Lung), CITIC PacificLtd (CITIC Pacific), Cathay Pacific Airways Ltd (Cathay Pacific), Sinopec Corp (Sin-opec), HKEx Limited (HKEx), Li & Fung Ltd (Li & Fung), China Unicom (ChinaUnicom), PetroChina Co. Ltd (PetroChina), CNOOC Ltd (CNOOC), China Construc-tion Bank (CCB), China Mobile Ltd (China Mobile), Industrial and Commercial Bankof China (ICBC), Foxconn International Holdings Ltd (FIH), Ping An Insurance (PingAn), Aluminum Corporation of China Ltd (CHALCO), China Life (China Life) andBank of China Ltd (BOC).

To help with better understanding the intraday data of the 30 stocks, we brieflydescribe some distinguished features of the Hong Kong Stock Exchange (HKEx). AsAsia’s third largest stock exchange in terms of market capitalization, the HKEx hasmore than 1,145 listed companies with a combined market capitalization of 17,769,271as of December 2009. The most widely quoted indicator of the HKEx’s performanceis the Hang Seng Index (HSI). The 42 constituent stocks of the HSI are classified intoone of four sub-indexes: Finance, Utilities, Properties, and Commerce and Industry.The 30 stocks studied in this paper cover all four sub-indexes; 12 of them are alsocomposites of the Hang Seng HK LargeCap Index (HSHKLI), 6 are composites of theHang Seng HK MidCap Index (HSHKMI) and 12 are composites of the Hang SengMainland Composite Index (HSMLCI). The trading hours of HKEx are different fromother major stock markets. A normal trading day consists of four sessions: a pre-open-ing auction session from 9:30 a.m. to 9:50 a.m.; a morning continuous trading sessionfrom 10:00 a.m. to 12:30 p.m.; an afternoon continuous trading session from 14:30 to16:00; and a closing auction session from 16:00 to 16:10.2 The high-frequency trans-action data we use include information for all trading sessions of each day. Before

2 The closing auction session was implemented in May 2008, but it was removed in March 2009 due tosignificant fluctuations in the closing prices of stocks. News releases of the implementation and suspensioncan be found at the following links: http://www.hkex.com.hk/news/hkexnews/0805192news.htm and http://www.hkex.com.hk/news/hkexnews/090320news.htm.

123

http://www.hkex.com.hk/news/hkexnews/0805192news.htm




computing the 30-min equity return and the realized volatility, we take two steps toclean the data sets. First, trades in the closing auction session are ignored, because thesession has been removed in March 2009. Second, transactions with odd lot, whosetrading volume is not exact multiples of the lot size, are removed because they usuallyhave big jumps compared to their neighbor transaction prices. After data cleaning, wecalculate the 30-min return and the realized volatility of the 30 constituent stocks.

To enable the study of intraday risk in a 30-min horizon, we define Pt,i , wherei = 1, . . ., 8, as the closing price in the i th interval on day t , which is the last transac-tion price in each of the eight half-hour intervals [10:00, 10:30), . . ., [12:00, 12:30),[14:30, 15:00), . . ., [15:30, 16:00). Then, the logarithm return of an asset in the i th30-min intraday interval on day t is taken to be rt,i = (log Pt,i − log Pt,i−1)× 100 %.In our study, t refers to the trading days between March 25th 2008 and May 31st 2009and i takes on the value of 1–8. In particular, we define Pt,0 as the last transactionprice in the pre-opening auction session in order to get rt,1. As for the calculation ofrealized volatility, we follow the simple practice of summing up the frequently sam-pled squared returns. This approach is justified under the assumption of a continuousstochastic model. However, in the presence of microstructure noise, such as bid-askbounce, asynchronous trading, infrequent trading, and price discreteness, the morefrequently stock returns are sampled, the more volatility will be overestimated. Todetermine the optimal sampling frequency, we follow the argument of Andersen andBollerslev (1998) and use the calendar time sampling scheme to obtain the RV bysumming up the six 5-min squared returns in the 30-min intraday interval. We definethe 30-min RV rv2

t,i as follows.

rv2t,i =

6∑

n=1

[log Pt,i,n − log Pt,i,n−1

]2, i = 1, . . . , 8,

where Pt,i,n is the price just before the end of the nth 5-min interval in the i th 30-mininterval on day t .

2.2 Unconditional Distributions

After calculating the 30-min return and realized volatility series for 30 stocks, we dis-play their unconditional distributions in Tables 1 and 2 in a way similar to Andersenet al. (2001a), yet for a 30-min intraday horizon. Daily return and realized volatilitydistributions are shown in Tables 3 and 4 for comparison. The left panel of Table 1refers to the 30-min unconditional distributions of the 30 mean returns, standard devi-ations, skewnesses and excess kurtosis. Most mean returns in our data set are negativedue possibly to the effect of the financial tsunami since late 2007. The majority ofthe 30-min return skewnesses are positive. Compared with the daily return distribu-tion in Table 3, the skewnesses of the 30-min data are more dispersed, ranging from−1.150 to 4.470. The excess kurtosis of rt,i indicates more severe fat-tailed behaviorthan the daily data. The median excess kurtosis is 8.265, which is greater than 4.285of the daily return, suggesting more severe fat-tailed behavior for intraday returns.

123

88 M. K. P. So, R. Xu

Table 1 Unconditional 30-min return distributions (%)

Stock rt,i rt,i /rvt,i

Mean SD Skewness Kurtosis Mean SD Skewness Kurtosis

Min −0.085 0.654 −1.150 5.420 −0.081 0.798 −0.090 −0.780

0.100 −0.047 0.807 −0.375 6.327 −0.049 0.878 −0.023 −0.701

0.250 −0.027 0.902 −0.205 6.653 −0.038 0.884 −0.008 −0.665

0.500 −0.008 1.051 0.105 8.265 −0.017 0.901 0.020 −0.620

0.750 −0.002 1.243 0.383 15.245 −0.005 0.914 0.030 −0.573

0.900 0.008 1.385 0.500 20.449 0.008 0.939 0.041 −0.468

Max 0.026 1.931 4.470 106.740 0.033 1.179 1.600 26.240

Mean −0.015 1.106 0.198 14.183 −0.020 0.907 0.060 0.286

The table summarizes the 30-min return distributions for the 30 HSI constituent stocks, rt,i . The samplecovers the period from March 25, 2008 through May 31, 2009 for a total of 2,300 observations. The 30-minrealized volatilities, rvt,i , are calculated from 5-min intraday returns, as detailed in the main text

Table 2 Unconditional 30-min realized volatility distributions (%)

Stock rvt,i log(rvt,i )


Min 0.519 0.404 2.430 9.580 −0.887 0.527 −2.830 −0.180

0.100 0.690 0.518 2.735 12.096 −0.620 0.570 −0.610 0.160

0.250 0.820 0.606 2.885 14.995 −0.447 0.618 −0.498 0.783

0.500 0.929 0.705 3.135 18.440 −0.256 0.670 −0.160 1.925

0.750 1.075 0.829 3.425 23.538 −0.165 0.717 −0.020 3.258

0.900 1.177 0.973 4.528 50.793 −0.055 0.763 0.110 4.289

Max 1.613 1.324 9.830 220.510 0.215 0.802 0.360 19.340

Mean 0.949 0.733 3.662 34.190 −0.302 0.668 −0.295 2.579

The table summarizes the distributions of the 30-min realized volatilities for the 30 HSI constituent stocks,rvt,i . The realized volatilities, rvt,i and logarithmic standard deviations, log(rvt,i ), are calculated from5-min intraday returns, as detailed in the main text

In the panel on the right, the mean standardized returns, rt,i/rvt,i , are close to zero.The standard deviations of rt,i/rvt,i are close to one as in the daily data. After stan-dardization, the distribution is closer to but still not exactly Gaussian according to theAnderson-Darling test (the histograms for HSBC before and after standardization areshown in Fig. 1a, b). It is in fact quite different from the daily standardized return,which is approximately normal as shown in Fig. 1c. The major difference, again, liesin the excess kurtosis of rt,i/rvt,i , which is smaller and mostly negative for 30-minstandardized returns.

In Table 2, we present unconditional distributions of the summary statistics of the30-min realized volatility. Despite having smaller mean rvt,i than the daily data, thestandard deviations of the 30-min log(rvt,i ) are generally larger than that of the dailydata. The results suggest higher volatility in 30-min realized volatility than in daily

123


Table 3 Unconditional daily return distributions (%)

Stock rt,i rt,i /rvt,i


Min −0.425 1.855 −1.590 1.280 −0.135 0.768 −0.430 −0.620

0.1 −0.263 2.207 −1.237 1.749 −0.088 0.832 −0.228 −0.442

0.25 −0.142 2.677 −0.260 2.293 −0.060 0.863 0.015 −0.410

0.5 −0.037 3.036 0.105 4.285 −0.035 0.890 0.075 −0.100

0.75 0.043 3.477 0.408 6.765 −0.002 0.983 0.153 0.253

0.9 0.089 3.942 0.603 12.553 0.018 1.006 0.270 0.634

Max 0.255 6.016 0.930 17.660 0.082 1.021 0.390 3.410

Mean −0.056 3.145 −0.076 5.781 −0.033 0.904 0.063 0.102

Table 4 Unconditional daily realized volatility distributions (%)

Stock rvt,i log(rvt,i )


Min 1.642 0.963 1.390 2.780 0.351 0.338 −0.170 −0.410

0.1 2.246 1.106 1.811 4.895 0.703 0.372 0.228 0.035

0.25 2.625 1.305 2.028 5.793 0.847 0.403 0.305 0.200

0.5 2.867 1.566 2.410 8.415 0.959 0.434 0.475 0.550

0.75 3.424 1.772 2.705 12.075 1.124 0.478 0.675 1.225

0.9 3.734 2.236 4.532 37.438 1.232 0.531 0.800 1.709

Max 5.132 4.522 14.570 234.440 1.514 0.579 1.020 3.890

Mean 3.024 1.679 2.981 19.882 0.970 0.442 0.491 0.799

realized volatility after eliminating the scale effect. The third and fourth columnsdemonstrate that the distributions of the 30-min realized volatilities are extremelyright-skewed and leptokurtic. After logarithm transformation, interestingly, the medianvalue of the sample skewness across all of the 30 stocks is reduced to -0.160, com-pared to 3.135 before transformation. In the panel on the right, the skewness andexcess kurtosis of log (rvt,i ) indicate that log(rvt,i ) is roughly symmetric and has lowleptokurtosis, though all but one of the stocks’ excess kurtosis exceeds the normalvalue of zero. As shown in Fig. 2b, the Gaussian assumption made on the distributionis plausible. This evidence is consistent with the findings in Andersen et al. (2001a,b).

2.3 Return and Realized Volatility Distributions in Eight 30-min Intervals

After studying the unconditional distributions, we conduct a separate analysis forthe eight 30-min intervals, [10:00, 10:30), . . ., [12:00, 12:30), [14:30, 15:00), . . .,[15:30, 16:00). Table 5 displays the distribution of returns, rt,i , and standardizedreturns, rt,i/rvt,i , for i = 1-8. First, by comparing the median statistics of the returndistributions in the left panel, we can identify significant W-shaped intraday seasonal

123

90 M. K. P. So, R. Xu

(a)

5.253.501.750.00-1.75-3.50-5.25

700

600

500

400

300

200

100

0

30- min return of HSBC

Fre

qu

ency

NormalHistogram of 30-min return - HSBC

(b)

2.41.81.20.60.0-0.6-1.2-1.8

250

200

150

100

50

0

Standardized return

Fre

qu

ency

NormalHistogram of standardized 30-min return

(c)

2.251.500.750.00-0.75-1.50-2.25

40

30

20

10

0

standardized return

Fre

qu

ency

NormalHistogram of standardized daily return - HSBC

Fig. 1 Distribution of 30-min return for HSBC. a Unconditional 30-min raw return distribution,b standardized 30-min return distribution, c standardized daily return distribution

123


(b)

(a)

Fig. 2 Distribution of 30-min realized volatility for HSBC. a Unconditional 30-min realized volatilitydistribution, b logarithm realized volatility distribution

patterns: the standard deviation of rt,i shows that stock prices are most volatile in thefirst 30 min of trading at 10:00 a.m.–10:30 a.m., followed by the 30 min of trading at14:30–15:00 after the lunch break, and 15:30–16:00 before the market closes. Thisseasonal pattern of the volatility is reasonable, as news released overnight or duringthe lunch break may lead to significant price movement right after the market reopens.At 15:30–16:00, day-traders would liquidate any open positions at closing, in order topre-empt any adverse overnight moves resulting in large gap openings of the next trad-ing day. Also, the mean kurtosis of return peaks at 12:00–12:30 indicates extremelylarge fluctuations in price before the lunch break. Moreover, we identify three inter-vals with interesting uniform positive/negative returns: at 11:30–12:00, all but two(CITIC Pacific and FIH, which are also the most volatile stocks amongst all) stockshave positive mean returns; at 14:30–15:00 right after lunch break, all 30 stocks havenegative mean returns; at 15:30–16:00 before the market closes, all but two (Wharfand CITIC Pacific) stocks have positive mean returns. The first of these intervals isespecially worth attention, because positive return is not common during the financial

123

92 M. K. P. So, R. Xu

Table 5 Median of the four statistics (mean, standard deviation, skewness and excess kurtosis) of 30-minreturn in eight intraday intervals

Time interval rt,i rt,i /rvt,i


10:00–10:30 −0.110 1.584 0.120 2.840 −0.062 0.948 0.060 −0.830

10:30–11:00 0.003 0.904 −0.170 6.020 −0.026 0.896 −0.025 −0.535

11:00–11:30 −0.050 0.765 −0.210 4.230 −0.083 0.887 0.030 −0.560

11:30–12:00 0.046 0.780 0.690 6.780 0.040 0.871 −0.060 −0.495

12:00–12:30 0.019 0.848 1.820 12.480 −0.018 0.884 0.020 −0.590

14:30–15:00 −0.091 1.258 −0.050 5.700 −0.062 0.928 0.065 −0.740

15:00–15:30 0.003 0.995 0.020 5.740 −0.013 0.919 0.025 −0.585

15:30–16:00 0.069 1.041 −0.050 4.690 0.076 0.874 −0.120 −0.500

Bold values refer to the W-shaped seasonal patternItalic values refer to the extreme fluctuation in price before lunch break

crisis period and traders may make use of this pattern in their day trading algorithm tomake profits. The negative return interval occurs right after the lunch break, indicatingthat the significant price movements in this interval are mainly downside. The positivemean return before the closing probably indicates that buying between 15:00–15:30and liquidating the position between 15:30–16:00 may result in profit in the longrun. After standardizing the returns, it is striking to see that similar seasonal patternsremain: the standard deviation of standardized return still peaks at the market openingin the morning and the market reopening right after the lunch break. This indicatesthat seasonal patterns of intraday return cannot be fully explained by the seasonalityin realized volatility, thus we propose to incorporate seasonal indexes in our con-ditional volatility modeling. In addition, the kurtosis of standardized returns rangesfrom −0.830 at 10:00–10:30 to −0.495 at 11:30–12:00. This motivates us to considerseasonality in the tail heaviness for the intraday volatility and VaR estimation.

The findings in Table 6 also reflect strong seasonality in intraday volatility. Inaccordance with the seasonal patterns of the intraday return series, realized volatilitydemonstrates similar seasonal patterns, with the first peak at 10:00–10:30, and thesecond peak at 14:30–15:00 right after the lunch break, as shown in column 1 ofTable 6. Fat-tailed behavior is evident in rvt,i for all i = 1–8, with the largest kurtosisoccurring at 14:30–15:00. After logarithm transformation, it is evident in the rightpart of Table 6 that the seasonal patterns are retained (as logarithm is a monotonetransformation) and the distributions of log(rvt,i ) are much closer to Gaussian.

In summary, a significant seasonal pattern is present in the level of volatility andthe kurtosis of 30-min intraday returns. We should take it into account when construct-ing volatility models. Seasonal indexes and time-varying degrees of freedom will beincorporated in the volatility forecasting models to be introduced in Sect. 3.

3 GARCH Modeling with Seasonality in Volatility and the Tail Risk

In the past, various studies (Giot 2005; Hansen and Lunde 2005) have been conductedto compare the performance of different GARCH-type models in forecasting daily

123


Table 6 Median of the four statistics (mean, standard deviation, skewness and excess kurtosis) of 30-minrealized volatility in eight intraday intervals

Time interval rvt,i log(rvt,i )


10:00–10:30 1.448 1.013 2.350 8.525 0.185 0.583 0.070 0.215

10:30–11:00 0.873 0.528 2.035 6.190 −0.291 0.572 −0.030 0.360

11:00–11:30 0.774 0.465 1.960 6.045 −0.411 0.589 −0.185 0.215

11:30–12:00 0.735 0.460 2.060 6.440 −0.452 0.645 −0.705 3.020

12:00–12:30 0.740 0.534 2.500 9.615 −0.460 0.650 −0.050 0.650

14:30–15:00 1.135 0.846 2.930 12.600 −0.064 0.648 0.160 0.495

15:00–15:30 0.859 0.591 2.765 12.280 −0.331 0.646 −0.125 0.830

15:30–16:00 0.914 0.608 2.755 11.275 −0.243 0.609 −0.020 0.260

Bold values refer to the strong seasonality in intraday volatility

volatility. The ordinary GARCH(1,1) model was shown to provide satisfactory results.Therefore, we use the framework of the GARCH(1,1) model in all three groups ofmodels. One of our major modifications is to incorporate RV in GARCH(1,1) models,either by replacing r2

t,i with rv2t,i , or by adding rv2

t,i as an extra term. As mentioned atthe end of Sect. 2, we also consider seasonal patterns in one group of models, and bothseasonal patterns and changing degrees of freedom in the third group. The detailedmodel structures of different groups are introduced below.

3.1 Modified GARCH Models

The first group of modified GARCH(1,1) models focuses on the use of RV as one ofthe predictors of the conditional variance of the 30-min return rt,i . In all the modifiedGARCH models we develop in this paper, we assume that rt,i is generated by

rt,i = σt,i × εt,i , i = 1, 2, . . . , 8. (1)

The major differences among various models lie in the formulation of σt,i , the con-ditional variance of rt,i given the previous information up to the (i −1)th interval onday t,�t,i−1, and in the specification of the distribution of εt,i . The first group of sixmodels is defined as follows.

σ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1, εt ∼ N (0, 1), (Model 1Gn)

σ 2t,i = α0 + α1rv2

t,i−1 + β1σ2t,i−1, εt ∼ N (0, 1), (Model 1Rn)

σ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1 + γ1rv2

t,i−1, εt ∼ N (0, 1), (Model GRn)

σ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1, εt ∼ T (0, 1, ν), (Model 1Gt)

123

94 M. K. P. So, R. Xu

σ 2t,i = α0 + α1rv2

t,i−1 + β1σ2t,i−1, εt ∼ T (0, 1, ν), (Model 1Rt)

σ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1 + γ1rv2

t,i−1, εt ∼ T (0, 1, ν),

(Model 1GRt)

where rv2t,i is the RV defined in Sect. 2; T (0, 1, ν) refers to a t distribution with mean

0, variance 1 and a degree of freedom ν which is assumed to be greater than 4. Asusual, we also restrict α0 > 0 and α1 +β1 < 1. From Eq. (1), we model the returns ofthe first to the eighth 30-min interval using previous returns. In particular, we estimatethe first 30-min return rt,1 using rt−1,8 and rvt−1,8 of the previous trading day. Of theabove six GARCH specifications, Models 1Gn and 1Gt are the basic GARCH-normaland student GARCH models. In Models 1Rn and 1Rt, we replace r2

t,i−1 with rv2t,i−1,

the RV, which is thought to be a good volatility proxy, especially for intraday marketmovement. Models 1GRn and 1GRt are formed by adding rv02

t,i−1 as an exogenousvariable to Models 1Gn and 1Gt, respectively. In essence, the first group of modifi-cations aims to investigate the power of RV for predicting future volatility and VaR.The model parameters, α0, α1, β1, γ1 and ν are estimated using the Maximum Like-lihood (MLE) approach. The log-likelihood function for models with normal-errordistribution and models with t-error distribution are as follows:

log L =N∑

t=1

8∑

i=1

[−1

2log(2π) − log(σt ) − r2

t,i

2σ 2t,i

], for εt,i ∼ N (0, 1)

log L =N∑

t=1

8∑

i=1

[−1

2log

(ν + 1

2

)− log

(ν

2

)− 1

2log(π)

−1

2log(ν − 2) − log(σt,i )

−1

2(ν + 1) log

(1 + r2

t,i

σ 2t,i (ν − 2)

)], for εt,i ∼ T (0, 1, ν) (2)

where the initial value of σ 21 is set to be the variance of rt ; (.)is the gamma function;

N is the total number of 30-min returns for each stock from March 25th 2008 to May31st 2009.

In the second group of modified GARCH models, we introduce multiplicative sea-sonal indexes to reflect the characteristics of the intraday variation due to the 30-minsegmentation of the 4-h trading period and any possible intraday periodicity. Similarto the model structure of models in group 1, a multiplicative seasonal index S(i) isadded to the models in group 2 in the volatility equation of σ 2

t,i . Specifically,

σ 2t,i = S(i) × τ 2

t,i , (3)

123


where τ 2t,i is called the adjusted conditional variance and is taken as the right-hand side

of any of Models 1Gn − 1GRt and S(i)(i = 1, 2, . . ., 8) are positive seasonal indexesfor various 30-min changes. To identify the model in (3), we set

∏8i=1 S(i) = 1. The

log-likelihood functions are the same as those in the models in group one. We labelthe six models with seasonal indexes as Model 2Gn, 2Rn, 2GRn, 2Gt, 2Rt and 2GRt.For instance, Model 2Gn is defined as

τ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1,

σ 2t,i = S(i) ×

(α0 + α1r2

t,i−1 + β1σ2t,i−1

), εt,i ∼ N (0, 1), (Model 2Gn)

and Model 2GRt as

τ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1 + γ1rv2

t,i−1,

σ 2t,i = S(i) ×

(α0 + α1r2

t,i−1 + β1σ2t,i−1 + γ1rv2

t,i−1

), εt ∼ T (0, 1, ν).

(Model 2GRt)

The novelty of the group 2 models defined in (3) is that the conditional varianceσ 2

t,i is decomposed into two parts; one is the adjusted conditional variance τ 2t,i and the

other is the seasonal index. The decomposition implies that the conditional variance ofadjusted return, rt,i/

√S(i), is equal to τ 2

t,i . In other words, var(rt,i/√

S(i)|�t,i−1) =τ 2

t,i . From Eq. (3) and the above constraint, the geometric mean of the σ 2t,1, . . ., σ

2t,8 is

identical to the geometric mean of τ 2t,1, . . ., τ

2t,8, which is independent of S(i). We can

interpret τ 2t,i as the explained return variation attributed to previous return movement

and S(i) as an intrinsic scale factor independent of the market movement.The third group of modified GARCH models combines seasonal indexes and time-

varying degrees of freedom together. As documented in Sect. 2, there is evidence ofintraday variation in the kurtosis or more generally of a tail risk, which can be quan-titatively described by the degrees of freedom of the t distribution characterizing theerror εt,i . We define Models 3Gt, 3Rt and 3GRt, from Models 2Gt, 2Rt and 2GRt toincorporate both seasonal indexes and to allow ν, the degrees of freedom, to vary indifferent intraday intervals. We also relax the assumption for ν by reducing its lowerbound from 4 to 2. For instance, Model 3Gt is defined as

τ 2t,i = α0 + α1r2

t,i−1 + β1σ2t,i−1,

σ 2t,i = S(i) ×

(α0 + α1r2

t,i−1 + β1σ2t,i−1

), εt ∼ T (0, 1, νi ). (Model 3Gt)

The major advantage of the group 3 models is that we can have flexible tail struc-tures in 30-min returns, which enables financial analysts to monitor tail risk on anintraday basis. On top of the GARCH parameters, we have eight pairs of S(i) and νi

to be estimated by maximizing the log-likelihood function below

123

96 M. K. P. So, R. Xu

log L =N∑

t=1

8∑

i=1

[−1

2log

(νi + 1

2

)− log

(νi

2

)− 1

2log(π) − 1

2log(νi − 2)

− log(σt,i ) − 1

2(νi + 1) log

(1 + r2

t,i

σ 2t,i (νi − 2)

)],

where νi is the time-varying degrees of freedom in the i th intraday interval.

3.2 Intraday Seasonality Results

We compute the parameters of the 15 models in the three groups introduced in Sect. 3.1using the training data set from March 25th 2008 to October 31st 2008. The estimatedparameters for the 15 models are summarized in Table 7a–c. In Table 7a, the medianvalues, the MLE of α0, α1, β1, γ1 and ν are given. We find that α0 is very close to0 in all 15 models and the α1 in Models Rn and Rt (associated with the RV, rv2

t,i−1)is generally larger than the α1 in Models Gn and Gt (associated with the squaredreturn, r2

t,i−1). The difference in α1 among the models in Group 2 and Group 3 is morepronounced when we incorporate seasonal indexes. These findings have an importantindication: RV may have greater explanatory power than return in the prediction ofvolatility. The estimates of β1 in Group 2 are smaller than those in Group 1, indicatingthat part of the effect of σ 2

t,i−1 on σ 2t,i can be attributed to the seasonal variation, which

is captured by S(i) in Group 2 models. A comparison between Group 2 and Group3 results shows that the other parameter estimates do not change much even thoughwe allow time-varying degrees of freedom in Group 3. Consistent with past findingsusing daily return, we also find that α1 + β1 is very close to 1 in all models. However,the degrees of freedom are much smaller than those obtained with daily return seriesand they are very close to the lower bound of 4 in Group 2. The small degrees offreedom correspond to the large kurtosis of the intraday return and realized volatility,suggesting the presence of dramatic price movements during the day.

Table 7b shows the median value of seasonal indexes in eight intraday intervals.Consistent with the most volatile interval at 10:00–10:30 (i = 1) indicated by thereturn and the realized volatility distribution in Sect. 2, seasonal index peaks after themarket opening, followed by a second peak at 14:30–15:00 (i = 6) after the lunchbreak. The overall intraday volatility pattern follows a significant W-shape, as dis-played in Fig. 3. The second peak of seasonal index at interval 6 can be attributedto the after-lunch 30-min return, which accounts for the information released and themarket movements in Shenzhen and Shanghai during lunchtime in Hong Kong. On thecontrary, the median seasonal indexes have the lowest values at 10:30–11:00 (i = 2)and 15:00–15:30 (i = 7). The change in variation from 15:00–15:30 to 15:30–16:00,which is expressed as S(8)–S(7), is substantial. The market seems to enter from avery volatile 30-min period right after lunch, to a ‘quiet’ 30-min period at 15:00–15:30, and then to a volatile 30-min period before closing. The change from S(7) toS(8) may be due the typical trading pattern of investors, especially day traders, whotend to execute a large number of trades in the last 30 min to rebalance their portfo-lios, to take profit or cut losses and to incorporate all the daily information in their

123


Table 7 Results of parameter estimation for modified GARCH(1,1) models

Model α0 α1 β1 γ 1 ν

a. Median values of basic GARCH parameters for 30 stocks (for Groups 1, 2 and 3)

Group 11Gn 0.006 0.052 0.946 / /

1Rn 0.005 0.054 0.935 / /

1GRn 0.007 0.011 0.927 0.048 /

1Gt 0.009 0.050 0.946 / 4.000

1Rt 0.003 0.071 0.914 / 4.000

1GRt 0.008 0.020 0.909 0.049 4.000

Group 22Gn 0.010 0.086 0.903 / /

2Rn 0.005 0.113 0.865 / /

2GRn 0.008 0.030 0.854 0.101 /

2Gt 0.015 0.110 0.884 / 4.000

2Rt 0.006 0.139 0.832 / 4.000

2GRt 0.009 0.044 0.810 0.113 4.492

Group 33Gt 0.013 0.110 0.886 / /

3Rt 0.008 0.156 0.825 / /

3GRt 0.010 0.039 0.818 0.127 /

i 1 2 3 4 5 6 7 8

b. Median values of the seasonal indexes S(i) for 30 stocks (for Groups 2 and 3)Group 2

2Gn 2.732 0.278 0.751 0.912 0.961 2.563 0.660 1.196

2Rn 2.997 0.280 0.737 0.897 0.959 2.546 0.659 1.197

2GRn 2.970 0.276 0.741 0.905 0.954 2.501 0.652 1.184

2Gt 2.942 0.265 0.749 0.880 1.044 2.491 0.614 1.350

2Rt 2.978 0.267 0.705 0.822 1.028 2.453 0.626 1.309

2GRt 3.001 0.267 0.714 0.845 1.029 2.499 0.635 1.307

Group 33Gt 2.584 0.287 0.813 0.887 0.923 2.339 0.673 1.185

3Rt 2.705 0.298 0.759 0.923 0.993 2.172 0.733 1.219

3GRt 2.701 0.294 0.776 0.916 0.952 2.162 0.680 1.173c. Median values of the time-varying degrees of freedom, Vi (for Group 3)

3Gt 8.429 4.331 3.752 3.518 4.216 4.503 4.039 4.837

3Rt 9.212 4.633 4.038 3.330 3.876 5.012 3.864 5.867

3GRt 9.693 4.544 4.059 3.344 3.811 4.784 4.159 6.360

i = 1, 2, 3, 4, 5, 6, 7, 8 correspond to the intraday intervals of 10:00–10:30, 10:30–11:00, 11:00–11:30,11:30–12:00, 12:00–12:30, 14:30–15:00, 15:00–15:30, 15:30–16:00, respectively

strategies, etc. In summary, the seasonal indexes can be arranged in the followingorder: S(1) > S(6) > S(8) > S(5) > S(4) > S(3) > S(7) > S(2). This order holdsin all the Group 2 and Group 3 models that we have investigated.

123

98 M. K. P. So, R. Xu

Fig. 3 W-shaped seasonal pattern for intraday conditional variance

In Table 7c, we report the median values of νi for the 30 stocks in Group 3 models.From the table, most of the ν4 and ν5 in Models 3Gt, 3Rt and 3GRt are smaller than 4,which explain that the kurtosis of the errors, εt,i , does not exist. Intervals 3–5, despitehaving relatively small S(3), S(4) and S(5), are more fat-tailed than the other inter-vals. This indicates extreme tail behavior of 30-min returns even after standardizationby the conditional variance. With the strong intraday seasonal effect reflected by S(1)

and S(8) in Table 7b, interval 1 has the highest degrees of freedom, or is the leastfat-tailed, followed by interval 8. This suggests that much of the tail risk in intervals 1and 8 has been accounted for by the intraday volatility. In short, the seasonal dynamicin the volatility and in the tail behavior give us a lot of insights into understanding therisk of short-term portfolios.

3.3 Intrinsic Tail Risk Index

To further understand the impact of S(i) and νi on the evolution of risk in the eight30-min intervals, we introduce the concept of the tail risk index below using Value atRisk. Intraday VaR (IVaR) with confidence level 1 − α is defined by

Pr(rt,i < −IVaRt,i (α)|�t,i−1) = α,

where rt,i is the realized 30-min return defined in Sect. 2; common confidence levelsused in the literature (e.g. Giot 2005) are 1−α = 95, 97.5, 99 and 99.5 %. We considerall four confidence levels in our empirical study. Various approaches can be used toforecast VaR, including parametric, nonparametric and semi-parametric. We use theparametric calculation formula defined as follows:

IVaRt,i (α) = −σt,i T−1νi

(α), when εt,i ∼ T (0, 1, νi ), (4)

123


where T −1νi

(α) is the αth percentile of T (0, 1, νi ). The formula in (4) allows thedegrees of freedom to vary and so is applicable to Group 3 models. In particular, ifeither εt,i ∼ N (0, 1) or νi is static, we can set νi to ∞ and ν, respectively. Fromthe decomposition in (3), i.e. σ 2

t,i = S(i) × τ 2t,i , we interpret the first factor, τ 2

t,i , asthe explained return variation attributed to previous return movement and the secondfactor, S(i), as an intrinsic scale factor independent of the market movement. TheIVaR in (4) can be rewritten as

IVaRt,i (α) = −√S(i)τt,i T −1

νi(α),

from which the αth intrinsic tail risk index (TRI) is defined as

TRI(α) = −√S(i) T −1

νi(α). (5)

The TRI represents the IVaR per unit of ‘explained price variation’, i.e. τt,i = 1. Sincethe index does not depend on previous price information, it measures how bad thereturn can be regardless of the market movement under extreme market stress. Theintraday tail risk, which is quantified by IVaR, is determined by the product of TRI in(5) and τt,i , reflecting the latest market condition. Market participants can learn par-tially from the TRI when the best time to trade is even without the most updated marketinformation. Table 8 presents the TRI(α) for Group 2 and 3 models at α = 0.5, 1, 2.5and 5 %. The TRIs are obtained using the median S(i) and νi in Table 7. For the fourα we investigate, the ratio of the highest TRI recorded in interval 1 to the lowest TRIrecorded in interval 2 is about three. That means that the maximum loss in interval 1as given by the IVaR can be three times as much as that in interval 2, provided that theexplained price variations τt,i in the two intervals are similar. For α = 1 %, the orderof TRI with respect to the intervals 1–8 is 1 > 6 > 8 > 5 > 4 > 3 > 7 > 2. Theorder is stable across different α, meaning that the intrinsic tail risk is robust to theconfidence level we fix. From the order, the intrinsic tail risk increases from interval2 (10:30–11:00) to interval 1 (10:00–10:30). The order of TRI exhibits a W-shapedstructure throughout the typical trading day. The W-shaped pattern for the model 3GRtat four levels of α is representative of the three models in group 3 and displayed inFig. 4. In summary, our intrinsic tail risk index, which integrates the seasonality ofvolatility and the tail behavior of returns, can give risk analysts a prior idea of theintraday risk in specific trading intervals.

4 Intraday Value at Risk and Volatility Forecast Evaluation

4.1 Predicting Power of Absolute Returns

After estimating the parameters, we conduct out-of-sample forecasting of one-period-ahead volatility using the validation data set from November 1st 2008 to May 31st2009. Given that volatility is a latent variable, past literature use squared return, r2

t,i , to

proxy for actual volatility rvt,i . Yet, as shown in Lopez (2001), while r2t,i is an unbiased

estimator of σ 2t,i under (1) because of E[r2

t,i |�t,i−1] = σ 2t,i , it is very imprecise due to

123

100 M. K. P. So, R. Xu

Table 8 Intrinsic tail risk index of intervals 1–8 and α = 0.5, 1, 2.5 and 5 % obtained using the medianS(i) and in Table 7

i 1 2 3 4 5 6 7 8

α = 0.5 %Group 2

2Gn 4.257 1.359 2.232 2.460 2.525 4.123 2.092 2.818

2Rn 4.459 1.364 2.211 2.441 2.522 4.110 2.091 2.819

2GRn 4.439 1.352 2.217 2.451 2.515 4.074 2.080 2.803

2Gt 5.584 1.656 2.788 3.054 3.272 5.102 2.549 3.772

2Rt 5.617 1.649 2.682 2.935 3.191 4.997 2.488 3.564

2GRt 5.594 1.631 2.682 2.911 3.166 5.016 2.495 3.573

Group 3

3Gt 4.572 1.685 2.984 3.058 3.043 4.717 2.735 3.381

3Rt 4.604 1.739 2.764 3.140 3.067 4.663 2.719 3.133

3GRt 4.599 1.693 2.764 3.090 3.059 4.646 2.606 3.074

α = 1 %

Group 2

2Gn 3.845 1.228 2.016 2.222 2.280 3.724 1.890 2.545

2Rn 4.027 1.232 1.997 2.204 2.278 3.712 1.889 2.546

2GRn 4.009 1.222 2.002 2.213 2.272 3.679 1.879 2.531

2Gt 4.544 1.365 2.293 2.485 2.695 4.182 2.074 3.078

2Rt 4.572 1.358 2.212 2.388 2.642 4.108 2.064 2.976

2GRt 4.589 1.354 2.231 2.420 2.668 4.133 2.077 2.992

Group 3

3Gt 4.022 1.384 2.356 2.482 2.452 3.935 2.174 2.802

3Rt 4.093 1.433 2.231 2.514 2.548 3.859 2.248 2.725

3GRt 4.030 1.403 2.249 2.506 2.553 3.856 2.158 2.679

α = 2.5 %

Group 2

2Gn 3.239 1.035 1.699 1.872 1.921 3.138 1.593 2.144

2Rn 3.393 1.038 1.682 1.857 1.919 3.128 1.591 2.145

2GRn 3.378 1.029 1.687 1.865 1.914 3.100 1.582 2.133

2Gt 3.367 1.012 1.699 1.841 2.007 3.099 1.550 2.281

2Rt 3.415 1.021 1.649 1.798 2.009 3.115 1.557 2.252

2GRt 3.408 1.019 1.661 1.819 2.008 3.121 1.589 2.270

Group 3

3Gt 3.182 1.055 1.678 1.822 1.885 2.995 1.613 2.154

3Rt 3.281 1.078 1.681 1.840 1.935 2.931 1.632 2.162

3GRt 3.230 1.065 1.677 1.853 1.887 2.919 1.623 2.146

α = 5 %Group 2

2Gn 2.719 0.868 1.426 1.571 1.612 2.634 1.336 1.799

2Rn 2.847 0.871 1.412 1.558 1.611 2.625 1.335 1.800

123


Table 8 continued

i 1 2 3 4 5 6 7 8

2GRn 2.834 0.864 1.416 1.565 1.607 2.602 1.328 1.790

2Gt 2.586 0.777 1.304 1.414 1.541 2.380 1.201 1.7522Rt 2.671 0.799 1.266 1.399 1.556 2.424 1.226 1.749

2GRt 2.666 0.802 1.281 1.432 1.547 2.427 1.237 1.770

Group 3

3Gt 2.598 0.804 1.306 1.402 1.431 2.333 1.227 1.708

3Rt 2.642 0.840 1.285 1.388 1.517 2.306 1.232 1.692

3GRt 2.650 0.836 1.291 1.379 1.462 2.302 1.217 1.704

Fig. 4 W-shaped structure of intraday tail risk index for 2GRt

its asymmetric distribution. Christodoulakis and Satchell (1998) have also shown thatthe mis-estimation of forecast performance is likely to be worsened by non-normalitywhich is known to be present in financial data. Hence, the use of r2

t,i as a volatilityproxy may undermine the inference regarding forecast accuracy. Therefore, we pro-pose two more measures to evaluate volatility forecast: |rt,i | and |rt,i |1.5. According toForsberg and Ghysels (2007), an estimation of the variance function that is based onabsolute returns is more robust than if it is based on squared returns against asymmetryor non-normality. The use of |rt,i |1.5, however, is novel in our study. The motivation isto draw a comparison between |rt,i | and r2

t,i and make the evaluation more robust. Wealso forecast one-period-ahead IVaR with the predicted volatility and conduct uncon-ditional coverage test to evaluate IVaR forecasts. The calculation for the prediction ofpowered return is specified below. For GARCH-normal models with rt,i = σt,i × εt,i

and εt,i ∼ N (0, 1), it can be shown that for c > 0, E[|rt,i |c | �t,i−1] = σ ct,i E[|Z |c],

where Z is a standard normal random variable. To predict |rt,i |1.5, it is natural to useσ c

t,i E[|Z |c], where σ ct,i E[|Z |c] can either have an explicit formula or can be computed

123

102 M. K. P. So, R. Xu

Fig. 5 Boxplot of 30-min predicted Conditional Variance (HSBC)

via numerical integration. For the popular choices of c = 1, 1.5 and 2, we determinethe following predictors for |rt,i |, |rt,i |1.5 and |rt,i |2 as

E[|rt,i | | �t,i−1] = σt,i

√2

π,

E[|rt,i |1.5 | �t,i−1] = σ 1.5t,i (0.86004), and

E[r2t,i | �t,i−1] = σ 2

t,i .

For GARCH-t models with rt,i = σt,i × εt,i and εt,i ∼ T (0, 1, ν), we can derive

E[|rt,i |c | �t,i−1] = σ ct,i (ν − 2)c/2 E[|Z |c]

(ν−c

2

)

(

ν2

) 2−c/2. (6)

The proof of (6) can be found in the “Appendix”. In particular, we have

E[|rt,i | | �t,i−1] = σt,i

√ν − 2

π

(

ν−12

)

(

ν2

) ,

and

E[|rt,i |1.5 | �t,i−1] = σ 1.5t,i (ν − 2)0.75 E

[|Z |1.5

] (

ν−1.52

)

(

ν2

) 2−0.75,

where E[|Z |1.5] = 0.86004 and E[r2t,i | �t,i−1] = σ 2

t,i is the same as in the GARCH-normal model.

123


4.2 Forecasting Performance of Powered Intraday Returns

We show the boxplots of the eight 30-min interval conditional variances, σ 2t,i , i =

1, 2, . . ., 8, for HSBC stock returns in the out-sample period, November 1st 2008 toMay 31st 2009 in Fig. 5. From the box plots, a clear seasonal pattern is revealed. Asreflected in S(1), the median conditional variance at 10:00–10:30 (i = 1) is the larg-est among the eight intervals, followed by that at 14:30–15:00 after the lunch break.Furthermore, the interquartile range of the conditional variance is also the largest at10:00–10:30, indicating the volatility of volatility is at the highest right after the mar-ket opening. Comparatively, the last interval, 15:30–16:00, has smaller median σ 2

t,ithan the after-lunch interval, 14:30–15:00, and the latter interval contains more wide-spread extreme values making the conditional variance distribution highly skewedto the right. On the other hand, the four intervals before lunch have smaller medianand spread, and the four distributions look quite similar. The conditional variance hasrelatively low values and is relatively more stable at 11:00–12:30. Using the forecastformula in Eq. (6), we calculate the predicted |rt,i |, |rt,i |1.5, r2

t,i and their mean abso-lute error (MAE) to evaluate the 30-min volatility forecasting performance. The meanabsolute error of the predicted |rt,i |1.5 of the 30 stocks using the 15 models is pre-sented in Table 9.3 Within each model, we observe large variations in MAE becauseof the different volatility levels in different stocks. In many of the stocks we investi-gate, the MAE from different models is quite stable. We highlight the best-performingmodel(s) for each stock with bold formatting to identify consistently outperformingmodels. It turns out that a majority of the best-performing models contain realizedvolatility as an exogenous variable in the conditional variance equations (e.g. mod-els 1Rt, 1GRt, 2Rt, 2GRt, 3Rt and 3GRt). Most of the best models belong to class 2,meaning that S(i) plays a more important role than time-varying degrees of freedom,νi , in predicting |rt,i |1.5.

In order to summarize the forecasting performance of |rt,i |, |rt,i |1.5, and r2t,i in terms

of MAE, we follow So and Yu (2006) to produce a table of performance rankings.For each stock, we rank the 15 models from 1 (smallest MAE) to 15 (largest MAE).The performance rankings are then obtained by averaging the 30 ranks of each model.By this construction, the lower the performance ranking, the smaller the MAE, whichindicates a better forecasting model. Table 10 presents the ranking results. Values lessthan five are highlighted in grey with the smallest ranking in bold. For the forecastingof |rt,i | and |rt,i |1.5, t-error models are preferred, whereas for predicting r2

t,i , normal-

error models do better. In terms of the prediction accuracy of |rt,i | and |rt,i |1.5, model2GRt performs best; for |rt,i |2, model 2GRn gives the best prediction. Models withrealized volatility in building σ 2

t,i usually give smaller MAE. This finding shows thatRV and seasonal index possess great explanatory power and can provide more accuratevolatility forecasts.

3 Tables for |rt,i | and r2t,i are also available upon request.

123

104 M. K. P. So, R. Xu

Tabl

e9

Mea

nab

solu

teer

ror

ofth

epr

edic

ted

|r t,i|1.

5of

the

30st

ocks

usin

gth

e15

mod

els

1Gn

1Rn

1GR

n1G

t1R

t1G

Rt

2Gn

2Rn

2GR

n2G

t2R

t2G

Rt

3Gt

3Rt

3GR

t

CK

0.78

0.76

0.76

0.76

0.76

0.76

0.76

0.73

0.73

0.75

0.76

0.73

0.76

0.72

0.73

CL

P0.

390.

440.

400.

350.

410.

370.

340.

370.

350.

320.

350.

340.

330.

350.

34

Wha

rf1.

321.

221.

251.

151.

111.

101.

211.

111.

121.

171.

071.

101.

171.

091.

11

HSB

C0.

520.

480.

490.

470.

460.

470.

480.

430.

440.

460.

430.

440.

460.

430.

43

HK

Ele

ctri

c0.

480.

510.

490.

430.

450.

440.

440.

460.

440.

420.

440.

420.

430.

490.

43

HS

Ban

k0.

740.

810.

810.

690.

730.

730.

670.

720.

710.

650.

690.

690.

670.

700.

72

Hen

ders

on1.

151.

051.

051.

050.

990.

991.

110.

971.

001.

090.

981.

001.

090.

971.

00

Hut

chis

on0.

760.

740.

740.

730.

730.

730.

730.

710.

710.

720.

730.

710.

740.

710.

71

SHK

0.91

0.89

0.89

0.84

0.84

0.84

0.82

0.80

0.80

0.82

0.80

0.81

0.81

0.82

0.82

New

Wor

ld1.

191.

251.

231.

101.

161.

131.

091.

101.

091.

041.

121.

071.

061.

101.

09

Swir

eA

0.85

0.90

0.87

0.82

0.85

0.83

0.84

0.89

0.87

0.83

0.89

0.88

0.81

0.85

0.85

MT

R0.

640.

650.

640.

580.

600.

590.

570.

570.

570.

570.

570.

570.

560.

570.

57

Sino

Lan

d1.

211.

211.

211.

151.

161.

161.

161.

121.

121.

141.

121.

111.

131.

101.

10

Han

gL

ung

1.37

1.45

1.30

1.19

1.18

1.24

1.18

1.15

1.16

1.13

1.11

1.11

1.11

1.15

1.11

CIT

ICPa

cific

1.59

1.56

1.69

1.39

1.45

1.39

1.38

1.59

1.39

1.28

1.33

1.29

1.28

1.34

1.34

Cat

hay

Paci

fic1.

111.

121.

121.

061.

000.

990.

930.

980.

980.

920.

950.

940.

900.

960.

97

Sino

pec

0.89

0.91

0.91

0.85

0.87

0.86

0.87

0.90

0.90

0.87

0.88

0.88

0.86

0.88

0.88

HK

Ex

0.89

0.91

0.90

0.86

0.87

0.86

0.84

0.86

0.85

0.81

0.82

0.82

0.84

0.81

0.87

Li&

Fung

1.30

1.34

1.34

1.08

1.03

1.10

1.15

1.12

1.12

1.06

1.07

1.07

1.10

1.11

1.15

Chi

naU

nico

m0.

920.

930.

930.

840.

850.

850.

880.

880.

880.

860.

860.

850.

830.

870.

85

Petr

oChi

na0.

760.

770.

770.

740.

740.

740.

740.

730.

740.

740.

720.

720.

720.

720.

72

CN

OO

C0.

890.

850.

850.

870.

850.

840.

850.

800.

800.

840.

790.

790.

830.

790.

79

123


Tabl

e9

cont

inue

d 1Gn

1Rn

1GR

n1G

t1R

t1G

Rt

2Gn

2Rn

2GR

n2G

t2R

t2G

Rt

3Gt

3Rt

3GR

t

CC

B0.

820.

890.

890.

770.

880.

860.

760.

770.

770.

760.

770.

770.

760.

790.

79

Chi

naM

obile

0.55

0.54

0.54

0.51

0.55

0.53

0.51

0.52

0.52

0.50

0.51

0.51

0.49

0.50

0.50

ICB

C0.

590.

590.

590.

580.

590.

580.

550.

540.

540.

550.

540.

540.

560.

540.

54

FIH

1.99

2.17

2.08

1.83

1.78

1.90

1.92

2.07

1.98

1.88

2.05

1.96

1.90

2.01

2.01

Ping

An

1.04

1.11

1.07

0.97

1.06

1.00

0.97

1.04

1.01

0.95

0.99

0.96

0.93

0.98

0.95

CH

AL

CO

1.54

1.39

1.40

1.46

1.38

1.40

1.38

1.31

1.32

1.41

1.32

1.33

1.36

1.30

1.32

Chi

naL

ife

0.57

0.59

0.58

0.56

0.58

0.58

0.55

0.57

0.56

0.56

0.57

0.57

0.55

0.57

0.57

BO

C0.

580.

580.

580.

560.

570.

560.

540.

550.

550.

540.

540.

540.

530.

560.

54

Bol

dva

lues

refe

rto

the

best

perf

orm

ing

mod

elin

term

sof

MA

E

123

106 M. K. P. So, R. Xu

Tabl

e10

Vol

atili

tyan

dIV

aRfo

reca

stin

gpe

rfor

man

cera

nkin

gsfo

rth

e15

mod

els

1Gn

1Rn

1GR

n1G

t1R

t1G

Rt

2Gn

2Rn

2GR

n2G

t2R

t2G

Rt

3Gt

3Rt

3GR

t

Vol

. R

ank

|rt,I

|13

.67

10.1

313

.97

9.47

10.5

010

.20

9.23

9.20

8.33

4.00

4.83

3.80

3.83

4.70

4.13

|rt,I

|1.5

13.0

313

.87

13.5

77.

879.

638.

707.

577.

476.

905.

105.

734.

574.

705.

905.

40

r t,i2

8.77

9.67

9.27

7.10

9.23

8.03

4.87

4.63

3.73

9.57

8.97

7.20

9.80

9.87

9.30

IVaR

Ran

k

0.5%

12.8

310

.80

11.8

75.

376.

535.

379.

838.

078.

306.

335.

675.

104.

705.

674.

30

1%10

.83

9.47

10.3

36.

008.

276.

139.

708.

508.

336.

176.

035.

534.

576.

305.

63

2.5%

7.33

8.20

7.77

8.63

9.67

7.30

7.07

8.67

7.37

5.17

7.00

5.77

5.90

8.13

7.00

5%6.

538.

707.

238.

478.

738.

176.

207.

006.

477.

908.

107.

738.

608.

237.

30

Bol

dva

lues

refe

rto

the

smal

lest

rank

ing

mod

el

123


4.3 Intraday VaR Predictive Performance

An unconditional coverage test of the predicted IVaR (Christoffersen 1998) isalso conducted to evaluate the IVaR forecasting performance. Based on the val-ues of the likelihood ratio test statistics of the coverage test, we rank the modelsfor each stock and produce the performance rankings in Table 10 by averagingthe rankings among stocks. We highlight all rankings below seven and the best-performing models as in Sect. 4.2. For 0.5 % IVaR, both the seasonal factor S(i)and the time-varying degrees of freedom are important. Out of the best nine indi-cated (1Gt, 1Rt. 1GRt, 2Gt, 2Rt. 2GRt, 3Gt, 3Rt, 3GRt) for α = 0.5 %, models withrealized volatility are superior. Models 3GRt, 3Gt and 2Gt perform the best atα = 0.5, 1 and 2.5 %, respectively. The results show that the time-varying degreesof freedom in Models 3Gt, 3Rt and 3GRt help improve the prediction of 0.5 and 1 %IVaR. For 5 % IVaR, model 2Gn outperforms the rest. To examine the statistical prop-erties of empirical coverage, i.e. the proportion of 30-min returns smaller than theirIVaR, we display the distributions of the coverage using boxplots in Fig. 6a–d. It isclear from Fig. 6a that normal-error models substantially underestimate IVaR (empir-ical coverage > α), whereas moderate overestimation is recorded in t-error modelswhen α = 0.5 %. For the six models, 2Gt, 2Rt. 2GRt, 3Gt , 3Rt , 3GRt , the empiri-cal coverage distributions are very much in coherence. In Fig. 6b, we can see thatmost of the coverage of Models 2Gt and 3Gt (best performers) are within 1 ± 0.5%.As in α = 0.5 %, the t-error models produce less biased and more stable cover-age than normal-error models. In Fig. 6c, all models suffer from underestimationof IVaR, yet models with t-error still outperform models with normal-error. Model2Gt provides the most robust prediction. In Fig. 6d, the six normal-error modelshave median coverage smaller than the nominal value of 5 %. The best perform-ing model 2Gn gives symmetric coverage round α = 5 %. In short, the aggregatemodel rankings are different for volatility forecasting and IVaR forecasting. In IVaRprediction, there is no evidence that RV can improve the forecasting performance.Models with time-varying degrees of freedom, νi , work particularly well in 0.5 and1 % IVaR.

We can conclude from the above findings that t-error distribution works well for0.5, 1 and 2.5 % VaR, which is different from the results in Giot (2005). In their study,when α is at the 1, 2.5 and 5 % levels, the performance of models based on normaldistribution is quite satisfactory; the GARCH-t is rejected twice at the 5 % level. Thedifference at α = 1 and 2.5 % may be attributed to two reasons. First, in Giot (2005),the data set is limited to three stocks, BOEING, EXXON and IBM of the NYSE, ina shorter period of 5 months. Their results, therefore, may not be extended to otherstocks in the NYSE. In our study, the data cover 30 HSI constituent stocks in a periodof over one year. Hence our findings are more representative of individual stocks listedin Hong Kong. Second, stocks listed in these two markets may have different charac-teristics and return distributions. Compared with BOEING, EXXON and IBM, mostof the 30 stocks analyzed in our study have a larger kurtosis for 30-min return, dueto relatively lower liquidity. Therefore, it is reasonable for the t-error distribution towork better for all levels in our study.

123

108 M. K. P. So, R. Xu

(a)

(b)

3GRt3Rt3Gt2GRt2Rt2Gt2GRn2Rn2Gn1GRt1Rt1Gt1GRn1Rn1Gn


Fig. 6 Boxplots for IVaR empirical coverage of the 15 models. a 0.5 % IVaR empirical coverage of the15 models. b 1 % IVaR empirical coverage of the 15 models. c 2.5 % IVaR empirical coverage of the 15models. d 5 % IVaR empirical coverage of the 15 models

5 Conclusion

In this paper, we first investigate the (properties of) intraday equity returns and real-ized volatilities of 30 constituent stocks of the Hang Seng Index. We find that bothreturn and realized volatility demonstrate significant fat-tailed behavior. Consistentwith previous papers by Andersen and Bollerslev (1998), Giot (2000) and Giot (2005),the empirical returns of the 30 stocks also feature a strong intraday seasonality in thevolatility: the price movement is most volatile after the market opens at 10:00–10:30and after the lunch break at 14:30–15:00. This seasonal pattern is considered in ourvolatility estimation model later. After standardizing return and taking logarithm ofrealized volatility, the distributions become approximately normal in most cases.

Then we construct three groups of modified GARCH(1,1) models to predict one-period-ahead volatility and intraday VaR. Although VaR models are usually applied

123


(c)

(d)



Fig. 6 continued

to daily data to help manage the financial risks of banking and financial institutions,we use the parametric VaR models to assess market risk on 30-min intraday returns.Our time horizon is thus much shorter than what is usually considered in past VaRliterature, and is therefore more applicable to market participants involved in frequenttrading, such as market makers or day traders. In our modified GARCH(1,1) mod-els, we incorporate RV and address the seasonality in both volatility and kurtosis byintroducing multiplicative seasonal indexes and time-varying degrees of freedom. Themodels are applied to intraday data for the 30 stocks and they are ranked according totheir forecasting performance of intraday volatility and VaR. The overall performanceranking shows that the Student t GARCH model with seasonal index and an additionalRV term performs the best in volatility forecasting. This indicates that RV can helppredict intraday volatility. Yet when it comes to IVaR prediction, RV does not seemto help; Student t GARCH models with seasonal index and time-varying degrees offreedom performs best for 0.5, 1 and 2.5 % IVaR; for 5 % VaR, the normal GARCH

123

110 M. K. P. So, R. Xu

model with seasonal index outperforms the others. Further investigation is worth tounderstand why RV can help predict volatility but not IVaR. In summary, our resultsdemonstrate unequivocally that seasonal index, time-varying degrees of freedom andt-error distribution can improve both volatility and IVaR prediction.

Appendix: Proof of Equation (6)

Let X be a standard t random variable distributed as T (0,√

νν−2 , ν). We can represent

the distribution of X as a scaled mixture of normal, i.e.

X = s−1 Z ,

where νs2 ∼ χ2ν , a chi-square distribution with ν degrees of freedom, Z ∼ N (0, 1)

and s2 and Z are independent. For any c < ν,

E[s−c] = E[νc/2(νs2)−c/2

]

= νc/2∫

x−c/2 1

(

ν2

)2ν/2

xν/2−1e−x/2dx

= νc/2 1

(

ν2

)2ν/2

∫x−c/2xν/2−1e−x/2dx

= νc/2 (

ν−c2

)2(ν−c)/2

(

ν2

)2ν/2

= νc/2 (

ν−c2

)2−c/2

(

ν2

) . (A.1)

The second equality is based on the distribution νs2 ∼ χ2ν and the second last

equality is obtained from the normalization constant of a χ2ν−c distribution. Therefore,

for GARCH-t models with rt,i = σt,i × εt,i and εt,i ∼ T (0, 1, ν), we have

E[|rt,i |c | �t,i−1

] = σ ct,i E

[|εt,i |c | �t,i−1]

= σ ct,i E

[∣∣∣∣∣

√ν − 2

νX

∣∣∣∣∣

c

| �t,i−1

]

=(

ν − 2

ν

)c/2

σ ct,i E[|s−1 Z |c | �t,i−1]

=(

ν − 2

ν

)c/2

σ ct,i E[|Z |c]E

[s−c | �t,i−1

]

= σ ct,i (ν − 2)c/2 E[|Z |c]

(ν−c

2

)

(

ν2

) 2−c/2.

The last equality above is obtained by substituting (A.1).

123


References

Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models doprovide accurate forecasts. International Economic Review, 39, 4.

Andersen, T. G., Bollerslev, T., Diebold, F., & Ebens, H. (2001a). The distribution of stock returnrealized volatility. Journal of Financial Economics, 61, 43–76.

Andersen, T. G., Bollerslev, T., Diebold, F., & Labys, P. (2001b). The distribution of realized exchangerate volatility. Journal of the American Statistical Association, 96, 42–55.

Barndorff-Nielsen, O. E., & Shephard, N. (2002). Econometric analysis of realised volatility and its usein estimating stochastic volatility models. Journal of the Royal Statistical Society, Series B, 64,253–280.

Beltratti, A., & Morana, C. (1999). Computing Value at Risk with high frequency data. Journal ofEmpirical Finance, 6, 431–455.

Brownlees, C., & Gallo, G. (2010). Comparison of volatility measures: A risk management perspec-tive. Journal of Financial Econometrics, 8, 29–56.

Christodoulakis, G. A., & Satchell, S. E. (1998). Hashing GARCH: A reassessment of volatility forecastand performance. Forecasting Volatility in the Financial Markets, 168–192.

Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39, 841–862.Coroneo, L., & Veredas, D. (2006). Intradaily seasonality of returns distribution. A quantile regression

approach and intradaily VaR estimation. CORE Discussion Papers, 2006077.Dionne, G., Duchesne, P., & Pacurar, M. (2009). Intraday Value at Risk (IVaR) using tick-by-tick data

with application to the Toronto stock exchange. Journal of Empirical Finance, 16, 777–792.Fleming, J., Kirby, C., & Ostdiek, B. (2003). The economic value of volatility timing using realized

volatility. Journal of Financial Economics, 67, 473–509.Forsberg, L., & Ghysels, E. (2007). Why do absolute returns predict volatility so well? Journal of

Financial Econometrics, 5, 31–67.Fuertes, A. M., Izzeldin, M., & Kalotychou, E. (2009). On forecasting daily stock volatility. International

Journal of Forecasting, 25, 259–281.Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a

GARCH(1,1)?. Journal of Applied Econometrics, 20(7), 873–889.Hansen, P. R., & Lunde, A. (2006). Realized variance and market microstructure noise. Journal of

Business and Economic Statistics, 24, 127–161.Giot, P. (2000). Time transformations, intraday data and volatility models. Journal of Computational

Finance, 4, 31–62.Giot, P. (2005). Market risk models for intraday data. The European Journal of Finance, 11, 309–324.Giot, P., & Laurent, S. (2004). Modeling daily Value-at-Risk using realized volatility and arch type

models. Journal of Empirical Finance, 11, 379–398.Koopman, S. J., Jungbacker, B., & Hol, E. (2005). Forecasting daily variability of the S&P 100 stock index

using historical, realised and implied volatility measurements. Journal of Empirical Finance, 12,445–475.

Lopez, J. A. (2001). Evaluating the predictive accuracy of volatility models. Journal of Forecasting,20, 87–109.

Martens, M. (2001). Forecasting daily exchange return volatility using intraday returns. Journal ofInternational Money and Finance, 20, 1–23.

Martens, M. (2002). Measuring and forecasting S&P 500 index-futures volatility using high-frequencydata. Journal of Futures Markets, 22, 497–518.

So, M. K. P., & Yu, P.L.H. (2006). Empirical analysis of GARCH models in VaR estimation. Journalof International Financial Markets, Institutions and Money, 16, 180–197.

Takahashi, M., Omori, Y., & Watanabe, T. (2009). Estimating stochastic volatility models using daily returnsand realized volatility simultaneously. Computational Statistics and Data Analysis, 53, 2404–2426.

Taylor, S. J., & Xu, X. (1997). The incremental volatility information in one million foreign exchangequotations. Journal of Empirical Finance, 4, 317–340.

Zhang, L., Mykland, P., & Aït-Sahalia, Y. (2005). A tale of two time series: Determining integrated volatilitywith noisy high-frequency data. Journal of the American Statistical Association, 100, 1394–1411.

Zhou, B. (1996). High-frequency data and volatility in foreign exchange rates. Journal of Business andEconomic Statistics, 14, 45–52.

123

forecasting intraday volatility and value-at-risk with high-frequency data

Documents