Comparisons Between RV, ARV and WRV

Transcription

1 Comparisos Bewee RV, ARV ad WRV Cao Gag,Guo Migyua School of Maageme ad Ecoomics, Tiaji Uiversiy, Tiaji,30007 Absrac: Realized Volailiy (RV) have bee widely used sice i was pu forward by Aderso ad Bollerslev i 998.Some scholars pu forward heir improved form for opimizig he qualiy of RV. As a resul, i is possible o compariso bewee RV, ARV ad WRV. This paper akes a corac of sock idex fuures i Chia as sample, aimig a researchig he compariso of RV, ARV ad WRV. Key Words: Realized Volailiy (RV), Adjused Realized Volailiy (ARV), Weighed Realized Volailiy (WRV), sock idex fuures Iroducio Realized Volailiy have bee widely used sice i was pu forward by Aderso ad Bollerslev i 998[]. I esimaes he volailiy usig he sum of square of reurs.i is model free ad do' eed researchers o esimaig he parameers of he model. Some scholars foud ha here exis errors whe usig RV i some siuaios, so hey pu forward heir improved forms. Xu (004) [] pu forward "Adjused Realized Volailiy" (ARV) ad Guo (006)[3] pu forward "Weighed Realized Volailiy". Boh auhors proved heir improved form is beer ha RV. Afer ha, ARV ad WRV is widely used i empirical aalysis, while here is' a lieraure does compariso of RV,ARV ad WRV. This paper akes a corac of sock idex fuures i Chia as sample, aimig a researchig he compariso of RV, ARV ad WRV. Realized Volailiy Assume r (, = p, p, =,, L T, =,LN) p, is h logarihmic price i r h day,, is he reur of logarihmic price of fiacial asse, N is he umber of sample ake from [-,] Accordig o Aderso ad Bollerslev(998) [],realized volailiy is he sum of square of reurs i a radig day, so 99

2 N = = RV r, () 3 Adjused Realized Volailiy Xu (004)[] pu forward "Adjused Realized Volailiy (ARV)"o dealig wih he measure error of RV. Xu ook he followig formula as ARV: E( RV) E( RQ) Var( RV) E( RQ) () ARV = + RV Var( RV ) Var( RV ) 3 j, ) 4 RQ r( + Where: i= Xu (004) [] proved he mea of ARV is equal o RV ad he variace of ARV is smaller ha RV. So i his opiio, ARV is more effecive ha RV. 4 Weighed Realized Volailiy Accordig o he defiiio of RV, i gives a equal weigh o each square of iraday reur. Bu here exiss "Caledar Effecs" i sock marke which meas he iraday reur was higher i he opeig ad closig period which smaller i he mid-ime of radig day. So i's ureasoable o imposig he ideical weighs. Guo (006)[3] ook "Caledar effecs" io cosideraio ad pu forward "Weighed Realized Volailiy (WRV)",I aimed a o describig he "Caledar Effecs" beer. Defiiio Weighed Realized Volailiy (WRV) is a weighed sum of he square of iraday reur of fiacial asses WRV N = = wr, (3) w is he weigh of square of iraday reurs, i mees N = w = (4) The mos impora work o compuig WRV is o compuig he weighs. Guo pu forward codiios : ()WRV is he ubiased esimaor of IV ()WRV is he esimaor wih leas variace. I compuig of codiio, Guo brough Lagrage fucio o makig he leas variace, he resul is 00

3 N = N r, w = r, (5) 5 The compariso of saisical iformaio amog RV, ARV ad WRV 5. Daa Descripio The sample came from a corac of sock idex fuures marke of Chia. I raged from Apr.6,00-Aug 3,0 which ivolved i 330 radig day.the aalysis aims a he daily realized volailiy of IF0,we compue i wih he daa of 5-miue idex price. 5. Saisical feaures of RV, ARV ad WRV () The series of RV, ARV ad WRV Table shows he saisical iformaio of 3 origial series. From able, i is easy o realizig ha 3 origial series are all "skew o righ wih high peak" which mees he firs iem i he characer of RV; heir mea is equal which is cosise wih he coclusio of Xu (004) ad Guo (006). Table. Saisical iformaio of RV, ARV ad WRV RV ARV WRV Mea Sd.Dev skewess Kurosis Jarque-Bera () The series of l RV, l ARV ad l WRV Table shows he saisical iformaio of 3 logarihmic series. Table. Saisical iformaio of l RV,l ARV ad l WRV l RV l ARV l WRV Mea Sd.Dev skewess Kurosis Jarque-Bera

4 From able, i's clear ha 3 logarihmic series improve heir resul obviously i skewess, kurosis ad J-B saisics. They are much approximae o ormal disribuio ha heir origial series which mees he secod iem of characer of RV. (3) The series of sadardized reurs r RV Se as sadardized reurs, so as o ARV ad WRV. Table 3 shows he saisical iformaio of hree series of sadardized reurs. Table 3. Table capios should always be posiioed above he ables r RV r ARV r WRV Mea Sd.Dev skewess Kurosis Jarque-Bera From able 3, i is easily o geig he followig coclusio: he series of r RV r ARV ad is close eough o ormal disribuio while he series r WRV of feaures wih "high peak ad skew o lef". 5.3 The Auocorrelaio of RV, ARV ad WRV This par researches he auocorrelaio of RV, ARV ad WRV. Image -3 shows he series of 00-order auocorrelaio coefficies..4 RVZXGX Fig.. 00-order auocorrelaio coefficies of RV 0

5 RVPZXGXS Fig.. 00-order parial auocorrelaio coefficies of RV.4 ARVZXGXS Fig order auocorrelaio coefficies of ARV From Figure -3, all of hree series have characer of obvious auocorrelaio which mees he firs iem i characer of RV. The red of auocorrelaio coefficies bewee RV ad ARV almos he same While he red of WRV's is a lile differe from hem. The reaso is obvious: he formula of ARV ca be divided io pars, he firs par is a produc of a cosa ad RV, he secod par is a cosa. As a resul, he reds of auocorrelaio coefficies bewee RV ad ARV are almos he same. Bu he fomula of WRV break ou he srucure of RV, so is red is differe from RV ad ARV. 5.4 The Summary of Saisical Iformaio of RV,ARV ad WRV Here make a coclusio abou he compariso of RV,ARV ad WRV ()Boh hree origial series are all "skew o righ wih high peak" while heir mea is equal; 03

6 ()Three logarihmic series improve heir resul obviously i skewess, kurosis ad J-B saisics. They are much approximae o ormal disribuio ha heir origial series; (3)The hree series of sadard deviaio are all "skew o righ wih high peak", bu perform beer ha origial series; (4)The red of auocorrelaio ad parial auocorrelaio coefficies bewee RV ad ARV almos he same While he red of WRV's is a lile differe from hem. 6 Coclusios Realized Volailiy (RV) has bee widely used sice i was pu forward. Some scholars pu forward he improved form of RV aimig a beer performace i describig he volailiy. This paper selec mai form of iprovemes: Adjused Realized Volailiy (ARV) ad Weighed Realized Volailiy (WRV) ad make a compariso of RV, ARV ad WRV of a corac of sock idex of fuures i Chia. The empirical aalysis icludes saisical iformaio ad he auocorrelaio series. Refereces. Aderse T.G., Bollerslev T. Aswerig he Criics: yes, ARCH Models Do Provide Good Volailiy Forecass [J]. 997, NBER Workig Paper, No.603. Xu Z.G. Aalysis, Modelig ad Applicaio of Fiacial Marke s High FrequecyUlra High Frequecy Time Series[D].Maser Degree Paper of Tiaji Uiversiy, Guo M.Y. Aalysis, Modelig ad Applicaio of volailiy of Fiacial High Frequecy Time Series[D]. Docor Degree Paper of Tiaji Uiversiy, Aderse T. G., Bollerslev T. e.al. Modellig ad Forecasig Realized Volailiy [J]. Ecoomerics, 003, Vol.7, Issue : Aderse T.G., Bollerslev T. Exchage Rae Reurs Sadardized by Realized Volailiy are (Nearly) Gaussia [J]. 000, NBER Workig Paper, No Aderse T.G. e al. The Disribuio of Realized Exchage Rae Volailiy[J]. Joural of he America Saisical Associaio, 00, Vol.96, Issue 453: Aderse T.G., Bollerslev T. e al. The Disribuio of Realized Sock Reur Volailiy [J]. Joural of Fiacial Ecoomics, 00, Vol.6, Issue : Bamdorff-Nielse O., Shephard N. Ecoomeric aalysis of realized volailiy ad is use i esimaig sochasic volailiy models[j]. Joural of he Royal Saisical Sociey: Series B (Saisical Mehodology), 00, Vol.64, Issue :