Research Article GARCH-Type Model with Continuous and Jump Variation for Stock Volatility and Its Empirical Study in China

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1 Mathematical Problems i Egieerig, Article ID , 8 pages Research Article GARCH-Type Model with Cotiuous ad Jump Variatio for Stock Volatility ad Its Empirical Study i Chia Huaa Zhag ad Qiuju La Busiess School of Hua Uiversity, Chagsha , Chia Correspodece should be addressed to Qiuju La; laqiuju@hu.edu.c Received 29 October 2013; Accepted 19 December 2013; Published 12 Jauary 2014 Academic Editor: Feghua We Copyright 2014 H. Zhag ad Q. La. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. O the basis of GARCH-RV-type model, we decomposed the realized volatility ito cotiuous sample path variatio ad discotiuous jump variatio, the proposed a ew volatility model which we call the GARCH-type model with cotiuous ad jump variatio (GARCH-CJ-type model). By usig the 5-miute high frequecy data of HUSHEN 300 idex i Chia, we estimated parameters of the GARCH-type model, the GARCH-RV-type model, ad the GARCH-CJ-type model ad compared the three types of models predictive power to the future volatility. The results show that the realized volatility ad the cotiuous sample path variatio have certai predictive power for future volatility, but the discotiuous jump variatio does ot have that kid of fuctio. What is more, the GARCH-CJ-type model has a more power to predict the future volatility tha the other two types of models. Therefore, the GARCH-CJ-type model is much more useful for the research o the capital assets pricig, the derivative security valuatio, ad so o. 1. Itroductio The research o asset volatility i fiacial market is the foudatio of fiace, such as capital assets pricig, fiacial derivatives pricig, ad fiacial risk measuremet. The premiseofquatitativefiacialaalysisistoaccuratelymeasure ad predict asset volatility. Therefore, the measuremet ad predictio of asset volatility are a hotspot of research all the time. To measure ad predict asset volatility accurately, Egle [1], i view of clusterig ad persistece of volatility, proposed a autoregressive coditioal heteroscedastic (ARCH) model; Bollerslev [2] built a geeralized ARCH(GARCH) model based o the ARCH model. The, GARCH model was exteded; Nelso [3] foud that the asset volatility is asymmetric. He modified the GARCH model ad built a EGARCH model; Gloste et al. [4] also examied the asymmetry ad built a TGARCH model (also called GJR model). The above models (called GARCH-style model i this paper) have bee proved to have strog power to predict the future volatility of assets [5]. Admittedly, GARCH-type models have fairly strog predictive power, but there is room for improvemet, as the accuracy pursuit for future volatility predictio is edless i fiacial operatios, such as fiacial asset pricig, fiacial derivative pricig, ad fiacial risk maagemet. Therefore, it is ecessary to improve the predictive power of the models. I order to perfect the accuracy of predictios, the realized volatility (RV) as a exogeous variable has bee itroduced by Koopma et al. [6] ito the volatility equatio of GARCH model. They built a GARCH-RV model ad foud that the GARCH-RV model has stroger predictive power tha the GARCH model. Fuertes et al. ad Frijs et al. [7, 8] also showed that the GARCH-RV model has stroger power to predict the asset volatility tha the GARCH model. But i realistic fiacial markets, the asset volatility is a cotiuous process with some jump compoets. Whe Aderse et al. ad Huag et al. [9, 10] studied the HAR-type RV model, they foudthatmodelbuiltwithcotiuoussamplepathvariatio ad discotiuous jump variatio that decomposed from RV has stroger power tha the udecomposed HAR-RV model i measurig ad predictig the asset volatility. For this reaso, i studyig the GARCH model with a itroductio of a edogeous variable RV, it is more reasoable to decompose RV ito C ad J aditroducethetwopartsito the volatility equatio of the GARCH model. O the basis

2 2 Mathematical Problems i Egieerig of the GARCH model, this paper decomposes RV ito two parts, C ad J, ad costructs a GRACH-CJ model i a attempttofurtherimprovethepredictivepowerforthefuture volatility. Similarly, this paper will also exted the EGARCH model ad GJR model to EGARCH-RV model, GJR-RV model, EGARCH-CJ model, ad GJR-CJ model. After that, we estimated parameters of the above models ad compared their predictive power for the future volatility, respectively, to idetify the volatility model with stroger power for the asset volatility measuremet ad predictio, usig the 5-miute high frequecy data of HUSHEN 300 idex i Chia. The remaider of the paper is orgaized as follows. The GARCH-CJ-type model costructio will be itroduced i Sectio 2. The empirical evidece ad predictive power of the models will be preseted i Sectio 3.Thelastpart,Sectio 4, is the coclusio. 2. Model Costructio 2.1. GARCH-CJ Model GARCH-RV Model Costructio. Stock retur volatility caot be observed directly but ca be measured i the asset retur series. The retur volatility is clusterig ad persistet. The ARCH model proposed i Egle [1] cawell capture the volatility clusterig of the retur series, but the model is rather complicated whe the regressio order gets bigger. O the basis of the ARCH model, Bollerslev [2] proposed the GARCH model to overcome the defect. GARCH(1,1) is expressed as follows: h t =ω+αε 2 t 1 +θh t 1, where R t is the retur, μ t 1 deotes the coditioal mea of R t basedoallavailableiformatio,h t is the volatility, ] t is the white oise disturbace, ad ω, α, ad θ are parameters to be estimated. I order to improve the measuremet of volatility ad the accuracy of the predictio of the model, Koopma et al. [6] itroduced the realized volatility (RV) as a exogeous variable ito the volatility equatio of GARCH(1,1) model to build a GARCH-RV model h t =ω+αε 2 t 1 +θh t 1 +λrv t 1, where λ is also a parameter to be estimated as ω, α, θ, ad RV t 1 istherealizedvolatilityatt 1period, which is defied accordig Martes [11]ad Koopma et al.[6]. With overight retur variace, realized volatility ca be expressed as RV t = N i=1 r 2 t,j +r2 t, = M r 2 t,j j=1 (1) (2), M = N+1, (3) where N is the umber of equally divided parts of a tradig day; r t,1 deotes the first retur after the opeig quotatio at Day t, r t,1 = 100(l P t,1 l P t,o ), P t,1 is the first closig price at Day t, P t,o istheopeigpriceatdayt, r t,2 is the secod retur after opeig at Day t,adr t,2 = 100(l P t,2 l P t,1 );...,sor t,n expresses the Nth retur at Day t after opeig, r t,n = 100(l P t,n l P t,n 1 ), r t, ad r t,m refer to overight retur variace, r t, =r t,m = 100(l P t,o l P t 1,c ), ad P t 1,c istheclosigpriceidayt GARCH-CJ Model Costructio. The real fiacial market reveals evidetly oliear features [12] ad the fiacial asset price volatility is ot cotiuous but shows jump volatility, sice the market is subject to the impact of some big iformatio shocks ad ivestors irratioal factors. Aderse et al. [9]showedthatithasmorepowertopredict the future volatility by decomposig the realized volatility ito cotiuous sample path variatio ad discotiuous jump variatio. I order to improve the predictive power of the model, we will itroduce the cotiuous sample path variatio C t ad the discotiuous jump variatio J t decomposedfromtherealizedvolatilityitomodel(2). To decompose the realized volatility (RV), Bardorff- Nielse ad Shephard [13, 14] proposed Realized Bipower Variatio (RBV); that is, RBV [r,s] t =( h 1 (r+s)/2 M 1 M ) r j,t r r j+1,t s, (4) j=1 where h>0is a fix time iterval, r, s 0 are costat (usually, 1 is give), ad M is the sample frequecy withi iterval h. Accordig to Bardorff-Nielse ad Shephard s research, whe M, the differece betwee RV t ad RBV t is equivalet to a cosistet estimator for discotiuous jump variatio J t RV t RBV t M J t. (5) With a limited sample size, J t calculated from (5)mayot always be oegative. I order for J t to be always oegative, we will treat J t i the followig way: J t = max [RV t RBV t,0]. (6) I calculatig discotiuous jump variatio J t,samplig itraday data at uequal frequecy will result i calculatio error. I order to improve the calculatio accuracy of J t,itis ecessary to itroduce some statistic to test the sigificace of J t.thispaperadoptsz t statistic proposed by Bardorff- Nielse ad Shephard [13, 14]based o the bipower variatio theory to test J t. Z t is expressed as follows: (RV Z t = t RBV t ) RV 1 t ((π/2) 2 +π 5)(1/M) max (1, RTQ t /RBV 2 t ) N (0, 1), (7)

3 Mathematical Problems i Egieerig 3 where RTQ t =Mμ 3 4/3 ( M M M 4 ) r t,j 4 4/3 r t,j 2 4/3 r t,j j=4 (μ 4/3 =E( Z T 4/3 )=2 2/3 Γ( 7 6 )Γ(1 2 ) 1). 4/3, The classic RBV calculatio is closely related to the samplig frequecy of the itraday data. With the icrease i the samplig frequecy, the RBV estimate caot coverge to itegral volatility because of the ifluece of factors, such as the market microstructure. So usig RBV t as the robust estimator for J t is biased, ad this paper adopts MedRV t proposed by Aderse et al. [15] as a robust estimator istead. MedRV t cabeexpressedasfollows: MedRV t = π π ( M M 2 ) M 1 i=2 Med( r t,i 1 r t,i r t,i+1 )2. Accordigly, RTQ 1,t,thestatisticforZ t i (6), is replaced by MedRTQ t, which is expressed as follows: MedRTQ t = 3πM 9π ( M M 2 ) M 1 i=2 Med( r t,i 1, r t,i, r t,i+1 )4. (8) (9) (10) After replacig RBV t with MedRV t ad replacig RTQ t with MedRTQ t i formula (7), we calculate the statistic Z t with (7) ad get the estimator for discotiuous jump variatio at the1 αsigificace level: J t =I(Z t >φ α )(RV t MedRV t ). (11) Accordigly, the cotiuous sample path variatio estimator is C t =I(Z t φ α ) RV t +I(Z t >φ α ) MedRV t. (12) I actual calculatio, we eed to select a suitable cofidece level α. Drawig o previous research, we choose 0.99 as the cofidece level α i this paper. I additio, with the test of statistic Z t ad relevat bipower variatio theory, we ca get the estimators for the cotiuous sample path variatio C t ad discotiuous jump variatio J t of the log retur volatility. Accordig to above RV decompositio method, we decompose RV t 1 of the model (2) itoc t 1 ad J t 1.Hereis the GARCH-CJ model h t =ω+αε 2 t 1 +θh t 1 +λc t 1 +γj t 1. (13) 2.2. EGARCH-CJ Modelig Buildig. I view of the asymmetriceffectofgoodadbadewsovolatility,nelsoetal. [3] costructed a EGARCH model o the basis of the GARCH model. Later, researchers built more EGARCH-type models, amog which a commoly used EGARCH(1,1) ca be preseted as R t =μ t 1 +ε t, ε t = h t υ t, ε t 1 l (h t )=ω+α h t 1 +β ε t 1 h t 1 +θl (h t 1 ). (14) Usig the method discussed i Sectio 2.1,wetakethelog of the last period s realized volatility (RV t 1 )aditroduce the log value as a exogeous variable ito EGARCH (1,1) ad thus get EGARCH-RV R t =μ t 1 +ε t, ε t = h t υ t, ε t 1 l (h t )=ω+α h t 1 +β ε t 1 h t 1 +θl (h t 1 )+λl (RV t 1 ). (15) We decompose RV t 1 ito C t 1 ad J t 1,takethelogof C t 1 ad J t 1, ad thus obtai the EGRACH-CJ model R t =μ t 1 +ε t, ε t = h t υ t, ε t 1 l (h t )=ω+α +β ε t 1 +θl (h t 1 ) h t 1 h t 1 +λl (C t 1 )+γl (J t 1 +1). (16) 2.3. GJR-CJ Model Costructio. O the basis of the GARCH model, Gloste et al. [4] costructedatgarchmodel (also called GJR model) to itroduce the leverage effect o volatility ito the ew model. GJR model (1,1) is R t =μ t 1 +ε t, ε t = h t υ t, h t =ω+(α+βn t 1 )ε 2 t 1 +θh t 1, where N t 1 is the idicator variable of the egative ε t 1 (17) N t 1 ={ 1, ε t 1 <0 (18) 0, ε t 1 0. Similarly, usig the method i Sectio 2.1, weitroduce RV t 1 as a exogeous variable ito the CJR(1,1) ad costruct the CJR-RV model: h t =ω+(α+βn t 1 )ε 2 t 1 +θh t 1 +λrv t 1. (19) We divide RV t 1 ito C t 1 ad J t 1, ad we get the CJR-CJ model h t =ω+(α+βn t 1 )ε 2 t 1 +θh t 1 +λc t 1 +γj t 1. (20)

4 4 Mathematical Problems i Egieerig Table 1: Descriptive statistics of each variable. Mea Std. dev. Skewess Kurtosis Jarque-Bera ADF-t statistic R t RV t C t J t l(rv t ) l(c t ) l(j t +1) deotes sigificace at 1% sigificace level. Table 2: Estimatio results for GARCH ad its exteded model. Residual followig Gaussia distributio Residual followig t distributio GARCH GARCH-RV GARCH-CJ GARCH GARCH-RV GARCH-CJ μ t ω α θ λ γ DOF of t distributio Log likelihood AIC ,,ad deote sigificace at the 1%, 5%, ad 10% sigificace level. 3. Empirical Study ad Comparative Aalysis of Models Predictive Power 3.1. Empirical Study Samples ad Their Statistics. For the empirical study, we take samples from the HUSHEN 300 idex for Chiese stock market, ad the data come from the Wid fiacial database.thetimespaofthesamplescoversfromapril20, 2007, to April 20, 2012, icludig 1199 tradig days. I the calculatio of the realized volatility, the samplig frequecy of itraday data has a great ifluece o the research results. O the oe had, too low samplig frequecy caot well capture the volatility iformatio; o the other had, too high samplig frequecy will produce oise which will harm the results. Therefore, i accordace with previous research of other scholars, this paper uses the 5-miute high frequecy data of HUSHEN 300 idex. Ad we use the movig average iterpolatio method to make up for the missed data, which derives valid data, 49 pieces of trasactio data for each day (icludig 1 overight trasactio data ad 48 day trasactio data). Variables eeded i this paper are R t retur, RV t realized volatility, cotiuous sample path variatio C t, ad discotiuous jump variatio J t.allare processed o Matlab 7.0 or Excel Table 1 is the descriptive statistics of retur R t,realized volatility RV t, cotiuous sample path variatio C t,ad discotiuous jump variatio J t ad log realized volatility, log cotiuous sample path variatio, ad log discotiuous jump variatio. We ca see from Table 1 that RV t series do ot follow the ormal distributio ad are leptokurtic. This implies that Chia s stock market has a big volatility. I additio, ADF test shows that all the series reject the ull hypothesis of uit root at the 99% cofidece level; it ca be cosidered that all series are statioary ad thus ca be furtherusedimodelaalysis Model Parameter Estimatio ad Aalysis. I this paper, maximum likelihood method is adopted to estimate the model i Sectio 1. Because the settig of the iitial value has a great ifluece o the result i the estimatio process, this paper adopts a approximate value from multiple fittig (also satisfyig that the likelihood score be the maximum) as the iitial parameter value. Tables 2, 3,ad4 list the estimates for GARCH ad other eight models uder the assumptios of the residuals followig Gaussia distributio ad t distributio. Comparig the log likelihood ad the AIC value for GARCH, EGARCH, ad CJR, we ca see that the goodess of fit for the asymmetric EGARCH model ad the CJR models is better tha that for the GARCH model, which idicates that the ifluece of favorable ad of ufavorable ews is asymmetric o the market volatility i Chia s stock market. I additio, comparig the log likelihood ad the AIC value

5 Mathematical Problems i Egieerig 5 Table 3: Estimatio results of EGARCH ad its exteded model. Residual followig Gaussia distributio Residual followig t distributio EGARCH EGARCH-RV EGARCH-CJ EGARCH EGARCH-RV EGARCH-CJ μ t ω α β θ λ γ DOF of t distributio Log likelihood AIC ,,ad deote sigificace at the 1%, 5%, ad 10% sigificace level. Table 4: Estimatio results of EGJR ad its exteded model. Residual followig Gaussia distributio Residual followig t distributio CJR CJR-RV CJR-CJ CJR CJR-RV CJR-CJ μ t ω α β θ λ γ DOF of t distributio Log likelihood AIC ,,ad deote sigificace at the 1%, 5%, ad 10% sigificace level. for those models, we ca see that, with the assumptio of a t distributio for the residuals, the fittig performs better tha with a Gaussia distributio assumptio. This shows that the distributio of the retur series is fat-tailed. Therefore, the assumptio of a t distributio for the residual error of the GARCH model is more reasoable. From the aalysis of Tables 2 4, the coefficiets of RV t 1 or l(rv t 1 ) of ewly added exogeous variable, λ, ithe volatility equatio of the GARCH-type models are sigificatly positive at 1% sigificace level, which shows that market volatility i Chia s stock exhibits proouced persistece ad the last period volatility may serve as a idicator for the curret period volatility. I additio, comparig the AIC values for the GARCH-RV-type model ad the GARCHtype model, we ca see that the fittig for the GARCH- RV model works better, which is cosistet with Koopma et al. [6]. Whe it comes to the estimatio results for this paper s ewly built GARCH-CJ model, the coefficiets for C t 1 (λ) are sigificatly positive at the 1% sigificace level, ad the coefficiets for J t 1 (γ) aresigificatolywhe the residual error i the GARCH-CJ model ad the CJR- CJ model is assumed to follow a Gaussia distributio, otherwise isigificat. Form this, we ca kow that, i Chia s stock market, the lagged cotiuous sample path variatio cotais relatively more iformatio for predictig the curret volatility, while the lagged discotiuous jump variatio cotais relatively less iformatio for forecastig. I additio, regardless of whether the residual error follows a Gaussia distributio or a t distributio, the AIC value for the GARCH-CJ-type model is lower tha the GARCH-RV-type ad the GARCH-type models, which fully demostrates that the fittig of the GARCH-CJ-type model has a better fittig effect The Compariso of Model Predictive Power I-Sample Predictio. I order to cofirm whether the GARCH-CJ-type model has more predictive power to future volatility tha the GARCH-type model ad the GARCH- RV-type model, this paper compares the predictive power

6 6 Mathematical Problems i Egieerig Table 5: Statistics of i-sample predictive power evaluatio idex. Residual followig Gaussia distributio Residual followig t distributio MAE HMAE RMSE HRMSE MAE HMAE RMSE HRMSE GARCH GARCH-RV GARCH-CJ EGARCH EGARCH-RV EGARCH-CJ CJR CJR-RV CJR-CJ ofthesethreetypesofmodelsusigalossfuctio.we select Mea Absolute Error (MAE), Heteroskedastic adjusted Mea Absolute Error (HMAE), Root Mea Squared Error (RMSE), ad Heteroskedastic adjusted Root Mea Squared Error (HRMSE) as 4 idexes to evaluate ad aalyze the performace of the volatility models. Geerally, the smaller the four are, the stroger predictive power the correspodig model has to predict future volatility. The formulae for gettig thevaluesofmae,hmae,rmse,adhrmseareexpressed i (21). Sice volatility caot be directly observed i the stock market, scholars ([6, 16, 17]) usually use the realized volatility (RV t ) as a substitute for the volatility i Day t. I this paper, RV t is also used as the substitute MAE = 1 σ2 t σ2 t, t=1 HMAE = 1 σ 2 t σ2 t t=1 σt 2, RMSE = 1 (σt 2 σ2 t )2, t=1 HRMSE = 1 [ σ2 t σ2 t t=1 σ 2 t 2 ], (21) where isthesizeofthepredictivesample,adσ 2 t is the real volatility; that is, RV t ; σ 2 t deotes the predicted volatility. Table 5 lists the statistics of i-sample predictive power evaluatio idex values for the GARCH type model, the GARCH-RV type model, ad the GARCH-CJ-type model whe usig lag 1 data to predict the curret volatility. Comparig the value for each evaluatio idex, we ca see that, except that the HRMSE value for the GARCH-RV-type model is greater tha that for the GARCH-type model, the RMSE for the EGARCH-CJ-type model is larger tha that for the EGARCH-RV-type model, all values for the GARCH-CJ-type model are smaller tha that for the GARCH-RV-type model, ad the value for the GARCH-RV type model is lesser tha that for the GARCH-type model. Therefore, we ca presume that i forecastig the i-sample volatility the GARCH-CJtype model has a greater i-sample predictive power tha the GARCH-RV-type model, ad the GARCH-RV-type model has greater i-sample predictive power tha the GARCH type model Out-of-Sample Predictio. Compared with the i-sample predictive power of the model, we are more cocered about the out-of-sample predictive power, sice it has more practical value. I order to effectively evaluate out-of-sample predictive power, we divide the sample (April 20, 2007 April 20, 2012) ito two parts. The first part (April 20, 2007 November 20, 2011) totals 1099 samples to be used for model estimatio; the secod part (November 21, , April 20) totals 100 samples to be used for predictio. As i the i-sample part for model estimatio, we still use the loss fuctio to compare the effectiveess of the predictio performed by the models. The results are show i Table 6. Comparig the value for each evaluatio idex, we ca see that, except that the RMSE value for the GARCH-RVtype model is greater tha that for the GARCH-type model i the case where both the models residuals are assumed to follow a t distributio, the MAE value ad the HRMSE value for the CJR-RV-type model are larger tha that for the CJR type model i the case where both the models residuals are assumed to follow a t distributio, all values for the GARCH- CJ-type model are smaller tha that for the GARCH-RVtypemodel,adthevaluefortheGARCH-RV-typemodel is lesser tha that for the GARCH type model. So we ca presume that, i terms of the out-of-sample predictive power for volatility, GARCH-CJ-type model works better tha the GARCH-RV-type model ad, i tur, the latter is superior to the GARCH-type model. Combiig the discussio i Sectio with that i Sectio 3.2.2, we ca see that amog the above three types of volatility models the GARCH-CJ-type model performs the best i predictig future volatility. Therefore, it makes sese to itroduce the realized volatility (RV) ito the GARCH-type model ad decompose it ito cotiuous sample path variatio (C) ad discotiuous jump variatio (J)toehacethe model s predictive power for volatility.

7 Mathematical Problems i Egieerig 7 Table 6: Statistics of out-of-sample predictive power evaluatio idex. Residual followig Gaussia distributio Residual followig t distributio MAE HMAE RMSE HRMSE MAE HMAE RMSE HRMSE GARCH GARCH-RV GARCH-CJ EGARCH EGARCH-RV EGARCH-CJ CJR CJR-RV CJR-CJ Coclusios This paper costructs GARCH-CJ model o the basis of the GARCH-RV model to obtai a volatility model that ca better measure ad predict asset volatility. Ad, i order to test the validity of the model, a empirical study is carried out usig the 5-miute high frequecy data of HUSHEN 300 idex i Chia (April 20, 2007, to April 20, 2012), we estimate the parameters of the GARCH-type model, the GARCH-RVtype model, ad the GARCH-CJ-type model ad evaluate all models predictive power for future market volatility usig a lossfuctio(mae,hmae,rmse,adhrmse). From the results of the estimated parameters, we ca see that favorable ad ufavorable ews have a asymmetric impact o the market volatility i Chia s stock market, ad the distributio of the market retur series is leptokurtic. At the same time, through the empirical results, we ca draw some coclusios as follows. (1) The past cotiuous sample path variatio has more predictive power for future volatility, but the past discotiuous jump variatio has less iformatio to predict. (2) The GARCH-CJ-type model has a much better fittig of the future volatility tha other two types of models (the GARCH-type model ad GARCH-RV-type model). (3) Accordig to the compariso of the predictive power of the three types of models, the GARCH-RV model performs the better i predictig the future volatility tha the GARCH-type models, which is cosistet with Koopma, Fuertes et al., ad Lehert et al. [6 8]. (4) The proposed GARCH-CJ-type model i this paper has a better ability to predict the future volatility tha the other two types of models, which meas the applicatio of GARCH-CJ model is more reasoable i measurig ad predictig volatility i fiacial practices such as capital asset pricig, fiacial derivatives pricig, ad risk measures. Although GARCH-CJ model has a greater power to predictthemarketvolatility,itisstillecessarytofurther icrease the accuracy of measurig ad predictig the market volatility. Therefore, the GARCH-CJ-type model, further improvemet i the fittig, ad predictive accuracy of the volatility models will be our emphasis for further research. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. Ackowledgmets The Research is sposored by Natioal Sciece Fud for Distiguished Youg Scholars ( ), Natural Sciece Foudatio of Chia ( ), ad Scietific Research Foudatio for the Retured Overseas Chiese Scholars (2011 [185]). Refereces [1] R. E. Egle, Autoregressive coditioal heteroskedasticity with estimates of the variace of Uited Kidom iflatio, Ecoometrica, o. 50, pp , [2] T. Bollerslev, Geeralized autoregressive coditioal heteroskedasticity, Ecoometrics,vol.31,o.3,pp , [3] D. B. Nelso, Coditioal heteroscedasticity i asset returs: a ew approach, Ecoometrica,vol.59,o.2,pp , [4] L. Gloste, R. Jagaatha, ad D. Rukle, O the Relatio betwee expected value ad the volatility of the omial excess retur o stocks, Fiace, vol. 48, o. 5, pp , [5] T. G. Aderse ad T. Bollerslev, Aswerig the skeptics: yes, stadard volatility models do provide accurate forecasts, Iteratioal Ecoomic Review, vol. 39, o. 4, pp , [6]S.J.Koopma,B.Jugbacker,adE.Hol, Forecastigdaily variability of the S&P 100 stock idex usig historical, realised ad implied volatility measuremets, Empirical Fiace, vol. 12, o. 3, pp , [7] A. Fuertes, M. Izzeldi, ad E. Kalotychou, O forecastig daily stock volatility: the role of itraday iformatio ad market coditios, Iteratioal Forecastig, vol. 25, o. 2, pp , 2009.

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