Measuring and Forecasting the Daily Variance Based on High-Frequency Intraday and Electronic Data

Transcription

1 Measuring and Forecasing he Daily Variance Based on High-Frequency Inraday and Elecronic Daa Faemeh Behzadnejad Supervisor: Benoi Perron

2 Absrac For he 4-hr foreign exchange marke, Andersen and Bollerslev use inraday reurns raher han daily reurns o obain a measure for he realized variance (RV). In equiy markes, where rading is done during a par of he day, Hansen and Lunde sugges some esimaors ha use inraday reurns during he acive par of he day and close o open reurn for he inacive par of he day. In some markes such as fuures marke for S&P500, rading is done elecronically when he real marke is closed. Using his elecronic daa, we provide a new measure for he RV and hen compare i wih he variance esimaors of Hansen and Lunde. If he measure ha uses elecronic daa (RV_oal) is considered as a reference, he opimal linear combinaion of open o close realized variance and squared close o open reurn, which is he hird esimaor of Hansen and Lunde, more corresponds o RV_oal. Having access o such measure, forecasing he fuure variance values can be done exclusive of oher variance esimaors. ii

3 Lis of Tables and Figures Table saisical properies of differen variance measures 6 Table coefficien esimaion for AR() Table3 coefficien esimaion for forecasing evaluaion regression Figure sample auocorrelaion Figure sample logarihmic auocorrelaion Figure3 ime series plos for RV_oal, es_var, es_var and es_var3 7 Figure4 auocorrelaion wih 95% confidence limi (RV_oal) 8 Figure5 auocorrelaion wih 95% confidence limi (es_var) 8 Figure6 auocorrelaion wih 95% confidence limi (es_var) 9 Figure7 auocorrelaion wih 95% confidence limi (es_var3) 9 iii

4 Table of Conens Inroducion. Definiion of Realized Variance 3. Calculaing Realized Variance by Whole Day Daa 5 3. Esimaing RV by Open Marker Hours Daa 5 3. Scaling Esimaor of IV 7 3. Incorporaing he Overnigh Reurn 7 4. Forecasing Realized Variance 9 5. Empirical applicaion o S&P500 Fuures 3 6. Conclusions References 3 iv

5 Inroducion A precise and reliable measure of variance is useful for a range of applicaions. One of hese applicaions is he evaluaion of variance models. For example Hansen and Lunde (005a) show ha a noisy measure of variance can resul an inconsisen ranking of variance models. Anoher sudy by Andersen and Bollerslev (998) shows ha in order o evaluae he performance of auoregressive condiionally heeroskedasic (ARCH)-ype model, a precise variance esimaor is necessary. One of he variance measures is realized variance (RV) which is he sum of inraday squared reurns. This measure can be used as a more precise proxy for heoreical quaniies such as inegraed variance (IV). Andersen and Bollerslev (998) showed ha daily squared reurn as a sandard variance model is exremely noisy alhough i is an unbiased esimaor. They argued ha boh heoreically and empirically, he sum of he inraday squared reurn is he bes measure for realized variance. Realized variance is consruced from high frequency inraday reurns. High-frequency daa are increasingly being used o address a wide range of problems in economerics because of he informaion hey conain abou populaion parameers. Bu high-frequency daa have been mainly used o esimae financial variance. To esimae RV for a full day one needs high-frequency daa for 4 hours of he day. Andersen and Bollerslev s resuls are for 4 hours foreign exchange marke. The difference beween exchange marke and sock marke is ha mos equiies are raded for a par of a day such as six or seven hours per day. Since a par of daily variance may ake place during

6 inacive par of he day, using only he daily inermien daa does no reflec he variance for whole day. A number of measures for sock marke variance combine inraday reurns (open o close) and overnigh reurn (close o open). For example Hansen and Lunde (005) propose hree esimaors for daily variance ha are based on he realized variance for he acive par of he day, RV,, and he squared reurn of he inacive period, r,. They characerize he assumpions ha jusify using each of hese esimaors. Their firs esimaor simply adds up he reurns of he acive par of he day and he squared overnigh reurn while he second esimaor is he scaled open o close reurn and so he overnigh reurn is no considered. Finally, he hird esimaor is he opimal linear combinaion of which is obained by mean squared error (MSE) mehod. RV, and r, In some markes, alhough he real marke is no acive during close o open period, bu elecronic ransacions are being done in his ime inerval. As an example S&P 500- index fuures are being raded on he elecronic overnigh rading sysem (GLOBEX) since 994. Making use of hese elecronic daa, we sugges anoher esimaor for he whole day variance. This new esimaor is based on realized variance for he acive par of he day, RV,, and realized variance of elecronic daa for he period ha he real marke is closed. Having access o 4 hours daa and compuing such measure of variance (named as RV_oal here) will be helpful. I can be used as a reference for comparing oher variance esimaors in order o recognize which variance measure represens beer RV_oal o be used in he case ha RV_oal is no a hand. Using our daa, he resuls of comparing he esimaors of Hansen and Lunde wih RV_oal show ha heir opimal variance esimaor more corresponds o RV_oal. I will also be ineresing o see if his variance measure is accessible, can i exclusively be used in he furher operaions on volailiy daa like forecasing, or on he conrary, he alernaive measures may improve he performance of forecasing. To do ha, we should inroduce a forecasing model ha corresponds beer

7 o our high frequency daa. Consequenly, by applying his model o differen variance measures, we can obain differen series of prediced daa ha can be used o evaluae he forecasing performance of he variance measures. To forecas one day ahead volailiy, we use AR model for each variance esimaor separaely. Applying AR on opimal esimaor of Hansen and Lunde as well as RV_oal and comparing he forecased values show a beer forecasing by RV_oal. This paper is organized as follows. Nex secion explains he concep of realized variance. Secion explains how o calculae whole day variance by using elecronic daa. Secion 3 discusses he hree esimaors of Hansen and Lunde. In secion 4, a model is presened o forecas one day ahead volailiy in he marke. The resuls of secions o 4 are applied o wo years of five minues reurns from S&P 500 index fuures followed by discussion of heir ime series, saisical properies and forecasing process.. Definiion of Realized Variance We consider * ( ) 0, p as logarihmic efficien price process which may differ from he observed price process p because of he marke microsrucure noise. So, we define p p * u where u is a noise process. If we use rading day as ime uni and as marke closing ime, close o close reurn will be r p( ) p( ). We shall assume he following coninuous ime model for he price process: * dp ( ) d dw where w is he sandard Brownian moion and and denoe o drif and volailiy erms. We shall assume ha 0 for all o simplify he problem. In fac, he drif erm 3

8 is of order d which is smaller han / (d) of he volailiy erm and so is negligible a high-frequency daa (see e.g. Andersen, Bollerslev and Diebold (00) for more deails). Afer simplificaion, he model will be: dp * ( ) dw where is in general smooh ime varying sochasic process ha is independen of Close o close reurn can be considered as: w. r p( ) p( ) dw =,, u u Dividing each day o he inervals wih lenghs h, inraday reurn for any horizon h can be defined as follow: r i i ) i ih p( ) p ( dw For i= /h (/h ineger) u ( i) h u We eliminae he dependence of r i on he horizon h. Inraday reurns are i. i. d. N(0, h) if is consan. In oher words we have: r i p( i ) p( ih i ( i) h ) udwu ( W ihw ( i) h ) ui ~ i. i. d. N(0, h) Where u i ~ i. i. d. N(0, h) for i= /h. The parameer of ineres is inegraed variance over a day ha we assume i o be finie: IV u du 0 4

9 An empirical esimaor of inegraed variance is realized variance which is sum of he squared inraday reurns: RV / h r i i When h 0 or he number of inraday observaions ends o infiniy, RV is a consisen esimaor of inegraed variance under cerain assumpions including absence of microsrucure noise. We define realized variance for an inerval [a,b] as follow: RV m a, b p i ) p( i ) i ( () Noe ha is a pariion of [a,b] and RV is he realized variance of his pariion. [ a, b]. Calculaing Realized Variance by Whole Day Daa In order o calculae realized variance for a full day, we need he daa of enire 4 hours of a day. For example Andersen and Bollerslev (998) calculaed realized variance upon 4-hours foreign exchange rae marke daa. The problem is ha fuure markes do no rade on a 4-hours basis. While i seems ha he markes are normally open for a fracion of a day such as 7 or 8 hours a day, mos equiies are elecronically raded during he hours he real marke is closed. In his siuaion ha we have access o high frequency daa during day and elecronic daa over nigh, realized variance for he whole day can be esimaed by 5

10 RV RV m a, b p( i ) p( i ) i Where [a,b] is he inerval of one complee day. p ) represens eiher he asse price during he acive par of he day or he elecronic asse price over nigh when he real marke is closed. ( i 3. Esimaing RV by Open Marke Hours Daa Hansen and Lunde (005), presened hree esimaors o calculae whole day variance if high frequency daa are available only for he acive period of a day. Noe ha here acive period represens he hours of opening he marke and so does no include elecronic rades while he marke is closed. Defining 0 as he inerval of ime in which he marke is acive, IV, IV[, ] and IV, IV[ 0, ] 0 represen inegraed variance of inacive and acive par of he day, respecively. We also wrie r r, r, ha r p( 0 ) p( ) is close o open reurn, and r p( ) p( 0) is he open o close reurn. Open o close period is exacly he, ime when he high frequency daa are available. We le RV, RV[ 0, ] he RV measure of his acive par of day. Three esimaors of Hansen and Lunde (005), are based on realized variance of acive par of he day, RV, and he square of close o open reurn, r,.their firs esimaor is a scaled value of RV, while he second one uses he value of r, and is he sum of RV, and r,. Finally, hey define he hird esimaor as r, RV, in which, are he weighs ha minimize he mean-squared error (MSE). 6

11 3. Scaling esimaor of IV The firs esimaor of IV presened by Hansen and Lunde (005) is consruced scales he RV, by consan value of scale, so hey consider RV RV,. If r n n hen ( r r RV is a consisen esimaor of E IV ] E [ ]. _ ) n,. [ RV, n r, The condiion needed o jusify his simple scaling are compleely characerized hrough heorem of Hansen and Lunde (005). 3. Incorporaing he over nigh reurn While scaling of make use of over nigh reurn, RV, seems ineresing o obain whole day variance, an alernaive is o r, r, as well. Hansen and Lunde (005) presened wo ways o combine RV, and in order o calculae daily variance. In he firs approach, is simply added o he high frequency inraday reurn and hus he esimaor is given by r, on RV r, RV, 7

12 Noe ha he wo erms of RV, and r can be considered as esimaors of IV during he, acive period (open o close) and inacive period (close o open), respecively. The second approach is o consider general linear combinaion of r, and RV,, evenually RV ( ) r, RV, () where, ). Thus he wo esimaors of ( scale RV and on RV are he special case of RV () using he weighs of (0, ) and (, ), respecively. In equaion () he opimal value of, ) is he soluion of he following opimizaion problem: ( min var( r, RV, ),, s.. (3) 0 where 0,, are defined as 0 E( IV ), E( r, ) and E( RV, ). Under some assumpions, E [ RV ( w)] IV for all ha saisfy 0 Hansen and Lunde (005) for furher deails). (see Le var( r, ), var RV, and cov( r,, RV, ). The soluion o equaion (3) is given by * 0 ( ) and * 0 (4) Where is a relaive imporance facor, defined by (5) This soluion is inuiive and is paricularly simple o inerpre if = 0. In his special 8

13 * * case we have which shows ha an increase in volailiy during he acive period (relaive o he inacive period) has a posiive impac on he relaive * * weigh ) ( ) 0, whereas he opposie is he case for an increase in he ( * * relaive noise, ( ) ( ) 0 The resul of his heorem can be easily used in pracice by replacing he quaniies of,,, and, by heir sample average. 4. Forecasing Realized Variance Financial marke volailiy is a key facor in risk managemen heory and asse pricing. As an example, invesor s assessmen of he sock variance over he life of he opion is a crucial parameer in mos pricing models. Thus accurae volailiy forecas is necessary o successfully deermine he price of derivaive securiies. Many saisical mehods have been suggesed o describe volailiy dynamic in he financial markes, including ARMA, differen versions of GARCH models and many oher models ha are based on he daily reurn. The preceding models are mosly based on he daily reurn volailiy. Andersen, Bollerslev, Diebold and Labys (ABDL) (003) have proposed a framework for forecasing he realized volailiy where high- frequency inra day reurns are available. This model is moivaed from following regulariies which are he resuls of experimenal analysis by ABDL. Firs, he disribuion of logarihms of realized volailiy is approximaely Gaussian, alhough he disribuion of realized volailiy is righ skewed. Second, a fracionally-inegraed long run process can provide a good esimaion for he long run memory of he logarihms of realized volailiy. 9

14 Regarding he menioned disribuional feaures, hey consider his simple vecor auo regressive model for he logarihm of realized variance or VAR-RV. Where ( L)( (6) d L) ( y ) y is he logarihm of realized volailiy and is a whie noise process. Afer deermining he degree of fracional differencing operaor or d, he model can be easily esimaed by applying OLS. This mehod efficienly makes use of he informaion in he inraday reurns wihou having o presen a model for his inraday daa. On he oher hand, comparing wih he oher currenly popular models ha are relied on he daily reurns, VAR-RV makes a significan improvemen in forecasing performance. ABDL (003) have compared heir resuls of forecasing he exchange rae marke volailiy wih a wide variey of models. For example, hey have compared VAR-RV wih he long memory filered daily logarihmic absolue reurn. This model is idenical o (6) excep for he volailiy proxy which is he daily absolue reurn insead of he realized volailiy. VAR-RV forecass are also compared wih he GARCH model of Engle (98) and Bollerslev (986) which is he mos popular procedure in academic applicaions. They have also considered FIEGARCH ha is a varian of he GARCH model ha incorporae long memory. Anoher model considered by ABDL is RiskMerics by J.P.Morgan s (997) which is he mos widespread model used by praciioners. The resuls of forecasing variance by all he above models and also VAR- RV have been sriking. The regressing for forecas evaluaion has he following form v b0 b. v b. v u, i i, VAR RV i, Model i If his regression includes only one variance measure, R is always he highes for VAR- 0

15 RV in he daa used by ABDL. On he oher hand, for almos none of he VAR-RV forecas, can hey rejec he hypohesis ha b 0 0 and b in he corresponding -es. They rejec he hypohesis ha b 0 0 and/or b for mos of he models. Furhermore, if he regression includes boh VAR-RV and anoher alernae variance forecas, for mos of he cases, he esimaion of b and b is close o and 0, respecively. As i was menioned, he long run dependence in financial marke volailiy can be modeled by fracionally inegraed processes such as he VAR-RV model explained previously or inegraed ARCH; see, e.g., Baillie, Bollerslev and Mikkelsen (996). In order o obain parameer d in fracionally inegraed processes, he implied hyperbolic decay rae d k can be used. Using he Geweke and Porer-Hudak (983) logpriodogram regression, called GPH echnique, he value of d can be esimaed. In essence, if we esimae he logarihm of correlaion by he lag logarihms, he esimaed linear esimaion of logarihm has he slope of d.abdl (003) applied mulivariae exension of GPH esimaor o he sample of auocorrelaion of he realized logarihmic volailiy in exchange rae marke ou o lag of 70 days which resuled on an esimaion of d equal o 0.40, which is a common value in such markes. Implemening he degree of fracional differencing equal o 0.40 o filer our daa does no resul in a good forecas. Since he size of available daa is no big enough, esimaing d by our sample does no lead o he bes value for d. However, as i can be seen in figures and i suggess ha he values of d which are closer o zero can provide a beer predicion for y. Choosing d equal o zero ransforms he model (6) o an AR process for which is he model finally applied o our daa. Here, y / )log( V ) or he ( logarihm of volailiy, where V is he variance of reurns a day and. I is assumed ha he order of lag polynomial is one day. Therefore he AR process ha is considered is as follow ( L)( ) (7) y

16 Figure sample auocorrelaion Logarihmic correlaion Fied value Figure sample logarihmic auocorrelaion

17 This simple auoregressive model can be applied o RV_oal and also o differen variance measures of Hansen and Lunde o forecas one day ahead volailiy. Consequenly, he forecasing done by each of he variance esimaors can be evaluaed by comparing differen forecass. In fac, here is no generally acceped mehod o evaluae he performance of compeing forecass and many saisical procedures have been used o do ha (see Andersen, Bollerslev, and Lange (999). Here he alernaive forecass are evaluaed by projecing he volailiy logarihm on a consan and he differen model forecass. y 0 ( y ) ( z ) u (8) The relaive weigh of coefficiens and he saisics of he regression can be used o evaluae he differen forecass. 5. Empirical applicaion o S&P 500 fuures In his secion, he resuls of he previous pars are applied o he daa of S&P 500 indexfuures ransacion prices. The esimaed RV s are calculaed for S&P 500 index fuures prices. S&P 500 index fuures have raded elecronically on GLOBEX during nigh when he sock marke is inacive since 994. The chosen period for his sudy is from January 006 o December 007. The sample period conains n=54 rading days. A Chicago Mercanile Exchange (CME), fuures floor rading is open from 8:30 a.m. o 3:5 p.m., Chicago ime, for day ime rading. GLOBEX overnigh rading begins from 3:30 p.m. and lass unil 8:5 a.m. of he nex day. 5-minues inervals are seleced o avoid marke micro srucure problem such as bid- ask bounce. In each 5-min inerval, he chosen price is he las price of he inerval or closing price. 3

18 In order o esimae daily variance accuraely, a high number of inraday reurns should be available. On he oher hand if he chosen ime inervals are oo small or he reurns are oo frequen, hen microsrucure effecs such as bid-ask spread may cause some biases. Finally, 5- minues inervals are used o eliminae such biases. Daily realized variance is calculaed based on four esimaors. The firs esimaor, RV, which was presened in secion uses boh day ime rading and elecronic nigh ime rading. Variance of each day is calculaed by replacing p ) by logarihm of 5-minue inerval prices. p ) could be eiher he real marke price (in day ime) or he elecronic ( i price (in nigh ime). The realized variance calculaed by his mehod is referred as RV_oal. ( i The hree remaining esimaors were explained in secion 3. All of hese hree esimaors use he realized variance of acive par of he day, RV, and he squared close o open reurn r. Therefore he nigh ime elecronic daa are no considered in hese esimaors., As i was explained, o obain he scaled esimaor he parameer of is needed which is calculaed in secion 3.. We call his esimaor as es_var. The second esimaor of daily variance is achieved simply by adding es_var. RV, and r,. This esimaor is referred as The hird esimaor is he opimum linear combinaion of RV, and r suggesed by, Hansen and Lunde and was explained in secion 3.. The esimaes of,, are defined from (4) and (5). The following equaions could be used o esimae he parameers needed in (4) and (5). 4

19 n 0 ( r, RV ),, n, n ( r, ) n n n r,, n RV, n n ( RV, ) n n, RV, ( r, ) n Table summarizes he saisical properies of he four esimaed variances. RV_oal is he daily variance using elecronic daa and es_var, es_var and es_var3 are he hree esimaors ha use RV, and r,. If RV_oal is considered as a reference, i is possible o compare he hree esimaors of Hansen and Lunde ogeher. Considering he average and variance of differen variance measures in he able, he average of es_var3 is much closer o RV_oal. Besides ha his measure is more sable han es_var and es_var3. Figure 3 shows he ime series plo of four esimaed daily variances. As i can be seen in he figure, if RV_oal is he reference, es_var underesimaes RV_oal and as a resul he posiion of es_var ime series is lower han RV_oal and he oher esimaors. 5

20 Table saisical properies of differen variance measures Mean 0 5 S.Dev. 0 5 Skewness kurosis RV_oal es_var es_var es_var correlaion RV_oal es_var es_var es_var3 RV_oal es_var es_var es_var This able conains he saisical properies of RV_oal, es_var, es_var and es_var3 as differen measures of variance. If RV_oal is considered as he reference, es_var3 which is he opimal linear combinaion of RV, and r,, is closer o RV_oal. 6

21 es_var es_var es_var es_var3 Figure3 Time series plo for RV_oal, es_var, es_var and es_var3 There is a bias problem in he realized variance measure of equaion (). This bias is due o he auocorrelaion in he inraday reurns which is caused by marke microsrucure effecs such as bid ask bounces, nonsynchronous rading, and rounding errors. [see, e.g., Andreou and Ghysels (00)] 7

22 The inraday reurns auocorrelaion becomes more problemaic when sample frequency increases. Bandi and Russell (004) and Zhang, Mykland and Ai Sahalia(005) found ha under independen marke microsrucure noise he opimal sampling frequency is ofen beween one and five minues. In pracice he frequency which corresponds o five minues inraday reurns is chosen. Figure 5, 6 and 7 shows he auocorrelaion plo of differen variance measures..0 s 0.5 RV_oal Lag Figure4 Auocorrelaions wih 95% Confidence Limi (RV_oal) es_var Lag Figure5 Auocorrelaions wih 95% Confidence Limi (es_var) 8

23 es_var Lag Figure6 Auocorrelaions wih 95% Confidence Limis (es_var) es_var Lag Figure7 Auocorrelaions wih 95% Confidence Limis (es_var3) 9

24 In he following par we implemen he resuls of par 4 o forecas one day ahead logarihm of volailiy or y for in-sample daa. As i was menioned in par 4, he model which is considered o is AR for y. The degree of lag polynomial, seleced by Bayesian informaion crierion (BIC), is equal o. Therefore we have ( L)( y ) or y y y This AR model can be applied o any measure of variance. We apply his model o RV_oal as he measure of variance ha uses elecronic daa during nighs and also o he es_var3 which is he opimum variance measure of Hansen and Lunde. Figure 8 shows he ime series of in sample forecas for RV_oal and es_var3. Table shows he saisical parameers of he AR() model. To evaluae he forecasing performance of each model, we use he regression (8) or y 0 ( y ) ( z ) u Where y is he logarihm of volailiy obained from elecronic daa or ( / )log( RV _ oal ) and z is he logarihm of he opimal volailiy esimaed by Hansen and Lunde or z / )log( es_ var3 ). Table 3 shows he saisics of his regression. ( The resuls show he coefficiens of y and z are close o and 0 respecively. From his resul and also negaive sign of, i appears ha including he opimal variance of Hansen and Lunde or es_var3 doesn add any new informaion in he forecasing process and RV_oal measure of variance can be effecively used for forecasing, if i is available. 0

25 In sample volailiy Figure8 in sample volailiy and day predicion -day predicion Table coefficien esimaion for AR() R RV_oal es_var Table3 coefficien esimaion for forecasing evaluaion regression p value ) p value ) ( ( R

26 6. Conclusion In his sudy he idea of using inraday reurns o measure he daily variance which was presened by Andersen and Bollerslev (998) is applied for measuring he sock marke variance. The sock markes are usually acive for a fracion of he day, bu in some markes elecronic rading is acive when he real marke is closed overnigh. In his siuaion, we can use he real marke daa for he acive par of he day and elecronic daa for he remaining par of he day in order o esimae he realized variance. Anoher alernaive which was suggesed by Hansen and Lunde (005) does no consider he elecronic daa. In heir mehod he whole day variance is declared by hree esimaors ha make use of he inraday reurns for he acive par of he day and he squared close o open reurn for he inacive par of he day. To compare he differen variance measures, he resuls of he sudy are applied o wo years S&P500 index fuures daa. In he empirical analysis i can be seen ha he esimaor ha has he form of scaled acive par variance underesimaes he daily variance if he esimaor which uses he elecronic daa is considered as reference. In addiion, we can see ha he opimal linear combinaion of inraday reurns and squared overnigh reurn declares a beer esimaion for he whole day variance comparing o he firs and second esimaor of Hansen and Lunde. An AR() model could be effecively used o forecas one day ahead logarihmic volailiy using all differen measures of variance. Forecasing evaluaion of RV_oal ha uses 4 hours daa (elecronic during nighs) and he opimal variance measure of Hansen and Lunde, shows ha RV_oal has a beer forecasing performance.

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