Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 Yen-Hsien Lee (Taiwan) Jui-Cheng Hung (Taiwan) Yi-Hsien Wang (Taiwan) Chin-Yen Huang (Taiwan) A sudy of dynamics in marke volailiy indices beween he US and Taiwan Absrac This sudy firs invesigaes he long-run equilibrium by coinegraion es and also explores he causaliy asymmery and jump inensiy relaionship by Correlaed Bivariae Poisson Jump (hereafer CBPJ) model beween he US and Taiwan volailiy index (hereafer VIX and TVIX). The empirical resuls found he long-run equilibrium relaionship beween VIX and TVIX. The volailiy persisence of he changes in VIX is greaer han TVIX and he changes in VIX have he volailiy asymmery. The wo volailiy indices have he individual jump and he join jump behavior and hen he change in he TVIX has highly jump risk. Keywords: volailiy index causaliy es CBPJ model. JEL Classificaion: C1 G15. Inroducion In recen years he volailiy index which represens he invesor s expecaions wih respec o price changes in he fuure as observed by Whaley (000) has played an imporan role in financial markes and has been viewed as an indicaor of invesor senimen; herefore he volailiy index is regarded as he invesor fear gauge. As far back as 1993 he Chicago Board Opions Exchange (CBOE) launched a volailiy index which was published based on he S&P100 index and hen he CBOE in 003 inroduced a new volailiy index (hereafer referred o as he VIX) which was based on opions on he S&P500 index. In 008 Fuures Indusry Associaion saisics revealed ha TAIEX Opions (TXO) amouned o 975754 conracs hereby ranking he TXO as he 15h larges commodiy conrac marke in he world. The Taiwan Fuures Exchange (TAIFEX) also creaed he Taiwan volailiy index (TVIX) ha enabled invesors o verify he accuracy and reliabiliy of he informaion 1. Whaley (000) noed ha invesigaing he issue of he volailiy indices in differen counries would become a major opic for research and i is for his reason ha his sudy examines he marke volailiy indices beween he US and Taiwan. Mos exising sudies focus on wheher he US sock dominaes he Taiwan sock marke based on he Granger causaliy es and wheher here exiss a long-run relaionship beween he wo markes based on he coinegraion es. This would imply ha his relaionship is an imporan producer of informaion affecing heir markes and ha he inernaional diversificaion beween he wo markes is effecive. Various sudies (Kwan e al. 1995; Cha and Cheung 1998; Ding 010 and Baharumshah e al. 003) have found ha in erms of he US sock marke s impac on he Taiwan sock marke here is evidence of marke inegraion beween he US and Taiwan. However oher sudies poin ou ha he Taiwan sock marke is no influenced by he US sock marke (Cheung and Mak 199; Ghosh e al. 1999; Sheng and Tu 000) and here is a lack of coinegraion beween he wo sock markes (Chan e al. 199; Cheung and Mak 199; Dunis and Shannon 005; Jeyanhi and Pandian 008). Therefore differen opinions exis over he exen of he linkages beween he US and Taiwan markes. On he oher hand we find few sudies have suggesed ha he VIX will affec oher counries volailiy indices 3 alhough i has been found ha he VIX Granger causes he VXN (Badshah 009) and ha he VDAX Granger causes boh he VSMI and VSTOXX (Äijö 008). Based on he above findings differen conclusions are reached regarding he causaliy and long-run relaionships beween he US and Taiwan sock markes. According o he above discussions and he need o respond o he dearh of research on he volailiy indices beween he US and Taiwan he purpose of he presen sudy is o explore he causaliy and long-run relaionship beween he VIX and TVIX. Wagner and Szimayer (004) Dosis e al. (007) Becker e al. (009) and Lin and Lee (010) have confirmed ha he jump behavior of he volailiy index is an imporan indicaor hereby implying ha he jump-diffusion model can measure he capaciy of he volailiy index. While mos oher sudies have researched he impac of jump behavior on he volailiy index using he univariae jump model his aricle by conras considers issues relaed o he Yen-Hsien Lee Jui-Cheng Hung Yi-Hsien Wang Chin-Yen Huang 01. 1 Wong and Tu (009) and Tzang e al. (011) found ha he VIX is superior o he VXO in Taiwan and ha he Taiwan volailiy index conains mos of he informaion; hence he new volailiy index is more precise and reliable han he old volailiy index. For example Eun and Shim (1989) Cheung and Mak (199) Cha and Cheung (1998) Liu e al. (1998) Sheng and Tu (000) and Ding (010). 3 For example Wagner and Szimayer (004) Nikkinen and Sahlsröm (004) Nikkinen e al. (006) Äjiö (008) and Badshah (009). 89
Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 jump aciviy beween he VIX and TVIX. Many exising sudies have indicaed ha he bi-variae jump model should be used o examine jump behavior in more han wo markes and Chan (003) designed he Correlaed Bivariae Poisson Jump (CBPJ) model o invesigae boh independen and join jump behavior 1. Lin and Lee (010) invesigaed he S&P500 and changes in he VIX and found evidence of he jump-diffusion process and join jump behavior. As far as he presen wrier is aware here have been no sudies on correlaed jump behavior beween volailiy indices. The major advanage of his approach invesigaes jump behavior; moreover he curren research hopes o fill he exising gap in he lieraure. To he bes of our knowledge his sudy is he firs o uilize he CBPJ model o explore he join jump behavior beween he VIX and TVIX and poins ou he advanages of such an approach for invesors seeking o esablish a jump dynamic sraegy. The remainder of his paper is organized as follows. In Secion 1 we describe he daa and discuss he CBPJ model. The empirical resuls are presened in Secion. The final secion concludes he paper. 1. Daa and mehodology This paper adops daa from wo sources. We obain VIX and TVIX daily ime series price daa from he CBOE and TAIFEX. The TAIFEX originally adoped he approach of he CBOE and i hen launched he TVIX on December 18 006. The daa used in his sudy cover he period from January 03 007 o Sepember 30 010 providing a oal of 896 observaions. This paper applies he CBPJ model by Chan (003) o invesigae he relaionships beween he VIX and TVIX. This model can adequaely capure he diffusion and jump relaionships beween he changes in VIX (r vix ) and TVIX (r vix ). The CBPJ model is described as follows: r 1 r 90 10 j 1 j 1 j 1 Z 1 1 J1 0 n j 1 r Z 1 J n c r j j m j 1 m j 1 r d r j j j 1 j (1) () where r 1 (r ) denoes he changes in he VIX (TVIX) a ime given by ln(vi ) ln(vi -1 ) VI is he volailiy index 1 and are he error erms and J 1 and J are he jump componens for he 1 For example Chan (003) Chan and Young (006) Chiu and Hung (007) Lee and Cheng (007) Chiu and Lee (007) Cheng (008) and Lin and Lee (010). changes in he VIX and TVIX. Z -1 is an error correcion erm. This paper applies he causaliy es of Granger (1969) o confer he causaliy beween he VIX and TVIX. Firs of all if 0 and d j = 0 (d j 0 and = 0) his means ha he changes in he TVIX (VIX) will affec he changes in VIX (TVIX). Second 0 and d j 0 refers o he feedback relaionship beween he changes in VIX and TVIX. Finally if = 0 and d j = 0 his means ha here is a non-causal relaionship beween he changes in VIX and TVIX. The error erm and he jump componen are assumed o be independen ha is E( J ) = 0. The error erm 1 ( ) has a bivariae normal disribuion wih zero mean and condiional covariance marix H ~ ; besides he jump componen J 1 (J ) also has a bivariae normal disribuion wih zero mean and condiional covariance marix. In a bivariae framework he jump componen (J ) is defined as: n1 n1 Y1 i E 1( Y1 i ) i 1 i 1 J (3) n n Y1 i E 1( Y i ) i 1 i 1 n n 1 i i 1 i 1 where Y ( Y ) denoes he summaion of n i jumps or he jump inensiy for r 1 (r ) over any period. In addiion each sochasic variable Y i follows a normal disribuion wih mean for is inercep erm and variance ; in oher words he bivariae jump inensiies can be described as: Y i Y 1 N ( 1 1 ) and i N ( ). (4) In equaion (3) he variables n 1 and n boh denoe individual couning variables of jump inensiy in ha he wo variables are consruced by he independen Poisson variables. Each one of hese variables has a probabiliy densiy funcion given by: i j ( ) i * e P ni j 1. (5) j! According o Chan (003) he jump inensiy parameer is he ime-varying jump inensiy i = 1 3 and is defined as: r (6) 1 1 1 1 1 r (7) 1 r r (8) 3 3 3 1 1 4 1 where r 1-1 and r -1 denoe he changes in he US s and Taiwan s VIX a ime 1 respecively. The individual jump inensiies 1 ( ) are assumed o
Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 be relaed o marke condiions which are refleced in r ( 1 1 r 1) as an approximaion of he las period s volailiy. Similarly he covariance 3 is governed by he variaions in he las period s volailiies from boh series. By combining he GARCH model wih he CBP funcion he probabiliy densiy funcions boh for r 1 1 and r 1 are defined by: f ( X n1 j 1 N i n ) 1 H 1 ij 1 exp u H u (9) IJ ij ij r1 E 1( r1 ) i1 ( 1 3) 1 uij (10) r E 1( r ) i ( 3) where X denoes r 1 and r and u ij is he error erm. H ij is he covariance marix of r 1 and r. H ij is he summaion of he covariance marix for he normal disurbance H ~ componen and he jump ij. The covariance marix for he normal disurbance H ~ is defined as: ~ H 1 1 1 where ( 1 r ) and 1 are defined as: 1 1 1 1 1 1 1 1 1 1 1I1 1 (11) (1) (13) 1 1 1I 1 where I I 1 1 I I 1 1 1 1 1 if 0 if 1 if 0 if 1 1 1 1 0 0 1 1 0 and 0. (14) 1 1 1 Here 1 and are he error erms of he changes in he VIX and TVIX. If he error erms are greaer imply acual fear volailiy large han expeced volailiy so as his is bad news o he marke. Conversely if he he error erms are smaller his is also good news o he marke. 1 denoes he diffusion correlaion coefficien. Therefore he covariance marix for he jump componen ij can be presened as: i1 1 ij1 ij (15) 1 ij1 i where he parameer 1 denoes he jump correlaion coefficien of Y 1 and Y. The covariance marix of he CBP GARCH model is denoed by he summaion of H ~ and ij. Finally he condiional densiy funcion is defined as: P ( X ) f ( X n i n j ). (16) 1 i0 j0 1 1 The log likelihood funcion is he sum of he log condiional densiies: N ln L ln P( X ). (17) 1 1 The CBPJ model would reduce o a bivariae GARCH (BGARCH) model if one se 1 = = 1 = = 1 = 1 = = 3 = 1 = = 3 = 4 = 0.. Empirical resuls Table 1 presens he basic saisics of he changes in he VIX and TVIX (Figure 1). The means (sandard errors) of he changes in he VIX and TVIX are 0.0757 and 0.0106 (7.4756 and 5.6556). The invesor fear gauge in he US marke exhibis a larger variaion han ha in Taiwan implying ha he change in he VIX involves greaer risk. In erms of skewness and kurosis he changes in he VIX and TVIX are significanly refleced by righ skewed and lepokuric disribuions. Moreover he Jarque-Bera es resuls are 658.1634 and 381.003 for he changes in wo indices and he null hypohesis of he normal disribuion is rejeced implying ha he changes in he VIX and TVIX are non-normally disribued. The Ljung- Box Q and Q saisics for he changes in VIX and TVIX are significan a he 1% level indicaing ha he changes in VIX and TVIX exhibi auocorrelaion and linear dependence. We can see ha he VIX and TVIX move ogeher in Figure 1 and he changes in he VIX and TVIX are consisen wih he seady-sae phenomenon in Figure. Table 1. Basic saisics of he changes in VIX and TVIX Index VIX TVIX Mean 0.07567 0.0106 Sandard error 7.4756 5.6556 Skewness 0.6437** 0.688 ** Kurosis (excess) 6.9989** 1.95 ** JB 658.1634** 381.0030** Q(5) 55.9560** 45.030** Q (5) 117.6000** 95.30** Noes: ** denoes significance a he 1% level. The Jarque-Bera saisic is used o deermine wheher he daa come from a normal disribuion. Q (5) and Q (5) are Ljung-Box Q es saisics for serial correlaion in he sandardized residuals and in he squared sandardized residuals. 91
Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 100 80 60 40 0 VIX TVIX This paper uses he Augmened Dickey-Fuller (ADF) Phillips-Perron (PP) and Kwiakowski-Phillips- Schmid-Shin (KPSS) mehods of uni roo ess o deermine he series is saionary or no. In Table ADF and PP ess are unable o rejec he null hypohesis of uni roo on he VIX and TVIX and he KPSS es is significan rejeced he null hypohesis of saionary a he 5% significance level. We find he VIX and TVIX are non-saionary; herefore we ake firs differences on he series and hen repea he uni roo ess. The ADF and PP ess of he changes in VIX and TVIX are significan rejeced uni roo a 1% level bu ADF PP KPSS 9 Model 0 007 008 009 010 Fig. 1. VIX and TVIX 60 60 50 50 40 40 30 30 0 10 10 0 0 10 10 0 0 0 30 30 40 40 007 008 009 010 007 008 009 010 50 Fig.. The change in VIX (lef) and TVIX (righ) Table. Uni roo es of VIX and TVIX VIX KPSS es is unable o rejec he null hypohesis of saionary. From he above he VIX and TVIX are I(1) and saionary afer firs order difference. Hence we furher analyze long-run relaionship beween VIX and TVIX. The resuls of Johansen coinegraion es are showed in Table 3. An empirical resul of Trace and Max-Eigen saisics are 9.38476 and 3.67614 a he 5% level of significance. Hence in he remainder of he paper we furher analyze he long-run relaionship beween changes in he VIX and changes in he TVIX 1 and in equaions (1) and () above. Index level Firs differences Index level Firs differences None -0.7151 (4) -16.840** (3) -0.617 (3) -1.9004** () Inercep -.646 (4) -18.534** (3) -.187 (3) -1.888** () Trend & inercep -.1767 (4) -18.575** (3) -.68 (3) -1.910** () None -0.8038 (0) -36.368** (0) 0.6078 (37) -35.383** (35) Inercep -.6004 (14) -36.3113** (0) -.544 (6) -35.139** (35) Trend & inercep -.5603 (13) -36.3508** (0) -.5316 (6) -35.5745** (36) Inercep 0.6914* (3) 0.0771 () 0.6664* (3) 0.1504 (38) Trend & inercep 0.4930** (3) 0.037 () 0.6565** (3) 0.0333 (39) Noes: * and ** denoe significance a he 5% and 1% levels respecively. Figures in parenheses denoe he lag lengh. Table 3. Coinegraion es of VIX and TVIX Null hypohesis Trace saisic 5% criical value Null hypohesis Max-Eigen saisic 5% criical value r = 0 9.3847** 5.871 r = 0 3.6761** 19.3870 r 1 5.7086 1.5179 r = 1 5.7086 1.5179 Noe: ** denoes significance a he 1% level. TVIX
Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 Table 4 repors he resuls of a comparison beween he BGARCH and CBPJ models. The Ljung-Box Q and Q ess are no saisically significan in he BGARCH and CBPJ models implying ha sandardized residual and square residual series do no exhibi serial correlaion of linear ineremporal dependence. Therefore he wo models are good a measuring finess capaciy. Owing o he abiliy of he CBPJ model o analyze he jump relaionship beween he changes in he VIX and TVIX we apply he LR es o compare he BGARCH and CBPJ models. The resul of he LR es is significan indicaing ha he CBPJ model is beer han he BGARCH model. As a consequence we furher analyze he causaliy asymmery and jump inensiy relaionship beween he changes in he VIX and TVIX using he CBPJ model. Table 4. A comparison beween he BGARCH and CBPJ models Iems BGARCH CBPJ VIX TVIX VIX TVIX L-B Q(5) 4.17 34.376 0.641 36.44 L-B Q (5) 17.715 30.70 14.996 31.07 Log likelihood -5714.05-537.457 LR es 683.185** Noes: ** denoes significance a he 1% level. The LR es is he likelihood-raio es. Table 5 presens he esimaed resuls for he CBPJ model. In Panel A of Table 5 1 and are -0.1444 and -0.0550 and are respecively significan a he 5% level indicaing ha he lagged-one and lagged-wo changes in he VIX have a negaive influence on he curren change in he VIX. The 1 and are insignifican indicaing ha he change in he curren VIX is no affeced by he lagged changes in he TVIX. d 1 is 0.18531 and is significan a he 1% level also indicaing ha he change in he curren TVIX is posiively influenced by he lagged-one changes in he VIX. However i is no affeced by he lagged-one and lagged-wo changes in he TVIX (c ) or by he laggedwo change in he VIX (d ). Hence we furher invesigae he lead-lag relaionship beween he changes in he VIX and TVIX by performing he Granger causaliy es in he CBJ model. In Panel B of Table 5 he change in he TVIX has an insignifican impac on he change in he VIX; however he change in he VIX has a significan impac on he change in he TVIX. This implies ha he change in he VIX exers an influence on he TVIX. The coefficiens 1 and capure he speed of adjusmen back owards he long-run equilibrium. We find ha he parameers of 1 and are significan a he 5% level and ha he signs of he coefficiens are also boh negaive and posiive implying ha here will be a endency o move oward he equilibrium in he long-run relaionship 1. Then he esimaed coefficien of he error correcion erm measures he speed of adjusmen o resore equilibrium in he dynamic model. Through he error correcion erm he changes in he VIX and TVIX will finally rever back o he equilibrium. Table 5. Resuls for he models and he Granger causaliy es Panel A. Esimaion resuls for he BGARCH and CBPJ models Parameer BGARCH model CBPJ model Coefficien Sandard error Coefficien Sandard error 10 0.0686 0.31 0.1977 0.1773 1-0.1533** 0.0388-0.1444** 0.094-0.080 0.0413-0.0550* 0.080 1-0.0619 0.0489-0.0563 0.0405 0.0115 0.0456 0.005 0.0359 1-1.395 1.1576 -.1875 * 0.868 0 0.0088 0.1755-0.0957 0.1345 c1-0.0996* 0.0446-0.034 0.0304 c -0.0059 0.0417-0.0163 0.045 d1 0.1853** 0.057 0.1440 ** 0.037 d 0.0066 0.058 0.0079 0.001.0104** 0.6703 1.10 * 0.6005 1 4.3318** 1.0658 1.0071 * 0.455 4.1436** 1.101 6.6875 ** 1.636 1 0.1547** 0.0401 0.0045 0.011 0.86** 0.070 0.0176 0.015 1 0.8349** 0.0359 0.8568** 0.069 0.6957** 0.0561 0.1340 * 0.0545 1 0.1616** 0.0449 0.0810 ** 0.034 0.139** 0.0713-0.001 0.0106 1 0.071 0.0354 0.1087 * 0.0459 1 5.5976 ** 1.0133.697 * 1.1648 1 7.8967 ** 0.7764 10.5574** 1.97 1-0.535 0.093 1-0.4395** 0.0535 0.183** 0.0615 3 0.1968** 0.0535 1 0.0000 0.014 0.004 0.0107 3-0.0000 0.0110 4 0.0167 0.0178 Panel B. Granger causaliy es TVIX VIX 0.8709 0.9900 VIX TVIX 7.1869** 18.40933** Noe: * and ** denoe significance a he 5% and 1% levels. As o he parameers of he condiional variance beween he VIX and TVIX ( 1 1 and 1 ) hese are significan a he 1% or 5% levels. The 1 The esimaed coefficien of he error correcion erm will be correced by a negaive 1 = -.1875 and a posiive = 1.10 when here is a posiive deparure from he equilibrium beween he VIX and TVIX in he previous period and vice versa. 93
Invesmen Managemen and Financial Innovaions Volume 9 Issue 4 01 volailiy persisence of he changes in he VIX is 1 + 1 = 0.8613 and in he TVIX is + = 0.1516 clearly revealing ha he volailiy persisence of he changes in he VIX is greaer han ha of he changes in he TVIX. The volailiy persisence of he changes in he VIX is close o 1 indicaing he exisence of a high volailiy clusering phenomenon. On he conrary he change in he TVIX exhibis a low volailiy clusering phenomenon. Then he correlaion coefficien of volailiy ( 1 = 0.1087) shows ha here is significan posiive correlaion a he 5% level. In he case of volailiy asymmery ( 1 = 0.0810; = -0.001) only he negaive informaion regarding he changes in he VIX ( 1 ) exhibis significan evidence of enhancemen compared o he changes in curren volailiy. Therefore he US volailiy index exhibis asymmery. In erms of he jump parameers boh of he means of he jumps are 1 =5.5976 and =.697which are significan a he 1% and 5% levels respecively. Tha is when here is abnormal informaion he changes in he VIX and TVIX exhibi jump behavior. As for he variances in he jumps he parameers are 1 = 7.8967 and = 10.5574 and are significan a he 1% level implying ha he variances of he jumps will obviously be enhanced when he jump behavior occurs. Moreover he jump variance of he change in he TVIX is greaer han ha of he change in he VIX. As o he jump inensiy 1 =-0.4395 and =0.183 are significan a he 1% level indicaing ha he changes in he VIX and TVIX exhibi he jump inensiy. In he case of 1 ( ) he jump inensiy of he change in he VIX (TVIX) is insignifican a he 5% level and moreover he square of he prior reurns does no have an impac on he individual jump inensiy in he VIX (TVIX) of he change. The join jump inensiy parameer 3 = 0.1968 is significan a he 1% level indicaing ha he changes beween he VIX and TVIX exhibi join jump behavior. However he parameers 3 + 4 are insignifican a he 5% level revealing ha he join jump inensiy ( 3 ) is no affeced by he changes in he VIX and TVIX of he squared prior reurns. This finding References 94 indicaes ha here is no ime-varying join jump inensiy as he relaionship changes wih ime beween he changes in he VIX and hose in he TVIX. Conclusions This sudy invesigaes he long-run equilibrium beween volailiy indices. By comparing he BGARCH model wih he CBPJ model by performing he LR es he model is found o have a good finess capaciy. In addiion his sudy also explores he causaliy asymmery and jump inensiy relaionship using he Correlaed Bivariae Poisson Jump model of Chan (003) beween he US and Taiwan volailiy indices. The empirical resuls show ha he volailiy persisence of he change in he VIX is greaer han ha of he TVIX and he change in he VIX exhibis volailiy asymmery. Moreover he correlaion coefficien of he volailiy beween he VIX and he TVIX is found o be posiive. The changes in he VIX and TVIX exhibi an individual jump relaionship whereas he changes in he TVIX exhibi high jump risk. Alhough he changes in he wo counries exhibi join jump behavior he resuls shows ha he jump behavior does no change over ime. Finally he changes in he TVIX are deeply affeced by he pas informaion on he changes in he VIX by he Granger causaliy es in he CBJ model. Therefore he invesor fear gauge for he US will affec he invesor fear gauge for Taiwan. The resuls of his sudy also indicae ha he jump dynamics in marke volailiy indices are srong phenomenon. Clarificaion of he roles of he jump dynamics in marke volailiy indices is helpful in clarifying he risks and is helpful in improving invesmen performance. Acknowledgemens The auhors are graeful o anonymous referees whose helpful commens have led o an improvemen on he conen and exposiion of his noe. Financial suppor from he Naional Science Council (NSC 99-410-H-033-03-MY 101-410-H-033-011- and NSC 99-815-C-033-013-H) R.O.C. is graefully acknowledged. 1. Äijö J. (008). Implied Volailiy Term Srucure Linkages beween VDAX VSMI and VSTOXX Volailiy Indices Global Finance Journal 18 pp. 90-30.. Badshah I.U. (009). Asymmeric Reurn-Volailiy Relaion Volailiy Transmission and Implied Volailiy Indexes Working Paper Series SSRN. 3. Baharumshah A.Z. Sarmidi T. and Tan H.B. (003). Dynamic Linkages of Asian Sock Markes Journal of he Asia Pacific Economy 8 () pp. 180-09. 4. Becker R. Clemens A.E. and Mcclelland A. (009). The Jump Componen of S&P500 Volailiy and VIX Index Journal of Banking and Finance 33 pp. 1033-1038. 5. Cha B. and Cheung Y.L. (1998). The Impac of he US and he Japanese Equiy Markes on he Emerging Asia- Pacific Equiy Marke Asia-Pacific Financial Markes 5 pp. 191-09.
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