Quantile Regression Analysis of Asymmetric Return-Volatility Relation

Transcription

1 Regression Analysis of Asymmeric Reurn-Volailiy Relaion Ihsan Ullah Badshah Hanken School of Economics, Deparmen of Finance and Saisics, P.O. Box 287, FIN Vaasa, Finland. Phone: , Fax: , January 26, 2010 Absrac This paper uses quanile regression o invesigae he asymmeric reurn-volailiy phenomenon wih he newly adaped and robus implied volailiy indices VIX, VXN, VDAX and VSTOXX. A paricular goal is o quanify he effecs of posiive and negaive sock index reurns a various quaniles of he implied volailiy disribuion. As he level of he new volailiy index increases during marke declines, we believe ha he negaive asymmeric reurn-volailiy relaionship should be significanly more pronounced a upper quaniles of he IV disribuion han is indicaed by ordinary leas squares (OLS) regression. We find pronounced negaive and asymmeric reurn-volailiy relaionships beween each volailiy index and is corresponding sock marke index. The asymmery increases monoonically when moving from he median quanile o he uppermos quanile (i.e., 95%); OLS hereby underesimaes his relaion a upper quaniles. Addiionally, he asymmery is pronounced wih a volailiy skew-adjused new volailiy index measure in comparison o he old a-hemoney volailiy index measure. The VIX volailiy index presens he highes asymmeric reurn-volailiy relaionship, followed by he VSTOXX, VDAX and VXN volailiy indices. Our findings have implicaions for rading sraegies, hedging porfolios, pricing and hedging volailiy derivaives, and risk managemen. Keywords: Asymmeric reurn-volailiy relaion, implied volailiy, index opions, quanile regression, volailiy index. JEL Classificaions: C21, G12, G13. The Auhor would like o hank Professor Johan Knif, Professor Kenneh Högholm, and Professor Hossein Asgharian, for providing useful commens. The auhor acknowledges CEFIR (cenre for financial research) and NASDAQ OMX Nordic foundaion for providing financial suppor.

2 1. Inroducion I is widely documened ha implied volailiy (IV) is superior o hisorical volailiy (HV) when forecasing he fuure realized volailiy (RV) of he underlying asse (e.g., Day and Lewis, 1992; Chrisensen and Prabhala, 1998; Fleming, 1998; Dumas e al., 1998; Blair e al., 2001; Ederingon and Guan, 2002; Poon and Granger, 2003; Mayhew and Sivers, 2003; and Marens and Zein, 2004). IV can be recovered by invering he Black-Scholes (1973) formula. However, Brien-Jones and Neuberger (2000) and Jiang and Tian (2005) have derived a model-free implied volailiy (MFIV) under he pure diffusion assumpion and asse price processes wih jumps. They show ha he informaion conen of MFIV is superior o ha of he Black-Scholes implied volailiy (BSIV) because he MFIV measure accouns for all srikes when compuing IV a a paricular poin in ime, whereas he BSIV measure is a poin-based IV and does no accoun for all srikes in compuaion; ha is, each srike has a separae IV. Moreover, BSIV is subjec o boh model and marke efficiency, while MFIV is only subjec o marke efficiency (see Poon and Granger, 2003). The major IV indices ha used o employ a-he-money (ATM) BSIV measures in heir mehodologies have hus now adoped MFIV measures. 1 As IV is forward looking, ha is, i is implied by he marke prices of opions, and as opions represen he consensus of marke paricipans regarding expeced fuure volailiy, IV is he marke expecaion abou he fuure RV of he underlying asse over he remaining life of an opion. Similarly, he IV index capures marke expecaions. 2 Thus, IV indices are 1 The moives for adoping MFIV measures are he following. Firs, he MFIV index measure is economically appealing and robus, as i accouns for ou-of-he-money (OTM) opions (i.e., volailiy skew). Second, he previous IV index measure (now called VXO) was upward biased, he bias being induced by rading-day conversion, which is now omied from he new VIX measure. Finally, wih he new robus MFIV index measure, i is possible o replicae volailiy derivaives (e.g., variance swaps), which was no possible wih he previous measure. 2 Major opion exchanges, including he Chicago Board of Exchange (CBOE) and he Deusche Börse, have launched IV indices, robusly providing informaion on opions using MFIV measures; examples of his are he VIX index for he S&P 500 index, VXN for he NASDAQ 100 index, VDAX for he DAX 30 index and VSTOXX for he Dow Jones (DJ) EURO STOXX 50 index. 2

3 ofen referred o as he invesors fear gauge (e.g., Whaley, 2000), as he level of he IV index indicaes he consensus view abou he expeced fuure realized sock index volailiy. When he level of he IV index increases, fear increases in he marke as a resul; alernaively, when he level of he IV index decreases, run-ups are riggered in he daily sock index prices. 3 Likewise, he MFIV index measure incorporaes boh pu and call opions and herefore moves wih changes in opions prices; for example, a negaive or posiive shock o he marke induces adjusmens in hedging and rading sraegies, consequenly riggering changes in he prices of one ype (i.e., pu or call) of opion. The MFIV index measure hen moves in he direcion of he marke demand of a paricular ype of opion and he underlying asse (see Bollen and Whaley, 2004). 4 Also, Liu e al. (2005) argue ha he rare-even premia play an imporan role in generaing he volailiy skew paern observed for opions across moneyness and ha hese rare evens are embedded in he OTM opions. 5 Camara and Heson (2008) derive an opion model ha accouns for boh OTM pu and call opions. They derive he exreme negaive evens from OTM pus and exreme posiive evens from OTM calls. The MFIV index ha accouns for OTM opions hus conains a broader se of informaion and is hereby robus; he MFIV index is, as a resul, an excellen ool for examining he relaionship beween he marke percepion of volailiy and reurns. Furhermore, his relaion is asymmeric, implying ha he MFIV index reacs differenly o negaive and posiive reurns. Two main hypoheses exis in he lieraure regarding he characerizaion of his asymmeric reurn-volailiy relaionship: he leverage effec and feedback effec hypoheses. However, boh he leverage and feedback hypoheses have been unable o 3 Addiionally, he IV index level indicaes he degree of willingness of marke paricipans o pay in erms of volailiy in order o hedge he downside risk of heir porfolios wih pu opions or long posiions in call opions wih limied downside risks insead of posiions in he underlying asse (see Simon, 2003, for a deail on rading sraegies). 4 MFIV index measure corresponds o he opion raders consensus opinions opions raders are assumed o possess professional judgmen on he fuure direcion of he volailiy of he sock index for 30 calendar days. 5 Similarly, Pan (2002) showed ha volailiy skew is primarily due o invesors fear of large adverse jumps. 3

4 explain he observed srong negaive asymmeric reurn-volailiy relaion a daily frequencies (see, e.g., French e al., 1987; Breen e al., 1989; Schwer, 1989, 1990). Similarly, a recen sudy by Hibber e al. (2008) has found a very srong conemporaneous negaive asymmeric reurn-volailiy relaionship using daa of daily frequency, hereby empirically rejecing boh he leverage and volailiy feedback hypoheses. 6 Furher empirical invesigaions are imporan o characerize asymmeric volailiy using a volailiy skew-adjused robus MFIV index measure wih daily frequency. 7 Addiionally, he condiional quanile regression echniques should be preferred over OLS regression, i.e., o invesigae he asymmeric responses of volailiy a he uppermos quaniles. Few well-known sudies exis showing a significan negaive and asymmeric relaionship beween sock index reurns and BSIV index reurns using OLS (or mean) regressions (e.g., Fleming e al., 1995; Whaley, 2000; Gio, 2005; Simon, 2003; Skiadopoulos, 2004; Low, 2004; Dennis e al, 2006); as OLS ignores he responses a he ails of he IV disribuion, accouning for his is of paramoun imporance in his kind of invesigaion. Noneheless, he firs sudy on he relaion beween he old VIX (now VXO) reurns and S&P 100 index reurns was conduced by Fleming e al. (1995). They invesigaed he imeseries properies of he VXO, finding a significan negaive conemporaneous asymmeric relaionship beween VXO reurns and sock index reurns. Anoher well-known sudy is ha conduced by Whaley (2000), who examined he relaionship beween he weekly VXO reurns and S&P 100 reurns. He documened ha when he VXO falls by 100 basis poins, he S&P 100 index increases by 069%, whereas when he VXO increases by 100 basis 6 Oher sudies by Simon (2003) and Gio (2005) have also found a very srong negaive asymmeric reurnvolailiy relaionship using daa of daily frequency. Neverheless, he negaive asymmeric reurn-volailiy relaionship is oo srong a he daily level; hese hypoheses migh be ineresing o characerize an asymmeric relaion a lower frequencies, for insance, monhly or quarerly frequencies, bu no a high frequencies. 7 We also believe ha he asymmeric volailiy-reurn relaionship should be more pronounced wih he new robus MFIV index in conras o he old BSIV index measure. A possible explanaion for pronounced asymmeric volailiy is ha a pu opion is a downside-hedging insrumen and raders are always concerned abou he downward momens in he marke, so raders are always hedging heir posiions wih OTM pus. Consequenly, we find a higher volailiy for OTM pus han for calls (see, e.g., Bollen and Whaley, 2004). 4

5 poins, he S&P 100 index falls by %. He hus finds a large negaive asymmeric associaion beween VXO reurns and S&P 100 reurns, calling he VXO he invesors fear gauge. Simon (2003) sudied he NASDAQ 100 volailiy index (VXN) from January 1995 o May 2002, showing ha he VXN is inversely relaed o boh posiive and negaive index reurns. Furhermore, he found sable resuls across he bubble and pos-bubble periods. A more recen sudy by Hibber e al. (2008) used a differen approach o invesigae he negaive asymmeric reurn-volailiy relaionship using he newly developed VIX index. They found a significan negaive and asymmeric associaion beween VIX and sock index reurns when incorporaing boh daily and inraday daa, hereby confirming ha he MFIV VIX measure can beer explain he asymmeric relaionship han he BSIV VIX or he RV measures. The purpose of his paper is o invesigae he negaive asymmeric reurn-volailiy relaionship beween he sock marke reurns and he volailiy index reurns: (1) o quanify he degree o which a volailiy index is responding o he negaive and posiive reurns a differen quaniles of an IV disribuion; (2) o compare he asymmeric responses of he wo volailiy index measures, i.e., he MFIV and BSIV index measures; and (3) o rank volailiy indices according o heir asymmeries. Relaed sudies in erms of he volailiy-reurn relaionship include Simon (2003), Gio (2005) and Hibber e al. (2008). Simon (2003) sudied he relaionship beween he NASDAQ 100 index reurns and he VXN index reurns using he BSIV index measure, while Gio (2005) and Hibber e al. (2008) sudied he relaionship beween he S&P 100 and he VIX and beween he NASDAQ 100 and he VXN. Gio (2005) used he BSIV index measure, whereas Hibber e al. (2008) used he new MFIV index measure. Our sudy differs from hese hree previous sudies and herefore conribues o he lieraure in a number of ways: firs, his sudy exends heir mehodologies; for insance, hey used mean-regression models, whereas we use a robus condiional quanile 5

6 regression model o invesigae he uppermos IV quaniles responses o he negaive and posiive reurns. Second, his sudy uses a broader se of daa drawn from across he Alanic, for example, he VIX, VXN, VDAX and VSTOXX volailiy indices (using new robus MFIV measures, hereby incorporaing a broader range of informaion) and heir corresponding sock indices. 8 Finally, his sudy compares he asymmeries of he MFIV and BSIV volailiy index measures. 9 Our main findings are ha from February 2001 hrough May 2009, he MFIV indices VIX,VXN, VDAX and VSTOXX responded in a highly asymmeric fashion; i.e., negaive reurns had a much greaer impac on volailiy han posiive reurns, paricularly in he uppermos regression quaniles (e.g., q=0.95). Our quanile regression model (QRM) of he asymmeric reurn-volailiy relaion hus reveals imporan informaion ha is underesimaed by he mean-regression model (MRM). The VIX index presens he highes asymmery, followed by he VSTOXX, VDAX and VXN indices, respecively. These volailiy indices rise sharply in imes of marke urmoil and decline in marke rallies. Second, our view ha he asymmery wih he MFIV index should have pronounced responses is confirmed by comparing he asymmeric responses of VDAX (MFIV) and VDAXO (BSIV); we find ha he MFIV index responds in a pronounced fashion, in conras o he BSIV index. Third, here is a srong conemporaneous asymmery in comparison o he lags, hus rejecing he leverage hypohesis, and similar conclusions can be drawn for he feedback hypohesis. This paper is organized as follows. Secion 2 discusses he asymmeric reurn-volailiy relaion. Secion 3 discusses he daa se, he volailiy indices and heir consrucion. Secion 4 presens he condiional quanile regression model for he asymmeric reurn-volailiy relaion. Secion 5 presens he empirical resuls. Secion 6 summarizes and concludes. 8 Previously i was found ha each equiy opion marke presened somewha differen IV dynamics; herefore, his sudy is he firs o invesigae and compare he volailiy asymmeries across he Alanic. 9 We compare he new VDAX and old VDAX (denoed here VDAXO) volailiy index measures; he former is based on he MFIV measure and he laer on he BSIV measure. 6

7 2. Asymmeric Reurn-Volailiy Relaion There are wo exising hypoheses ha characerize asymmeric volailiy: he leverage and he volailiy feedback hypoheses. The leverage hypohesis proposed by Black (1976) and Chrisie (1982) aribues asymmeric volailiy o he leverage of he firm; when he financial leverage of a firm increases, he value of he firm declines, and he value of is equiy declines furher. Because he equiy of a firm has he maximum exposure o he firm s enire risk, he volailiy of he equiy should increase as a resul. On he oher hand, he volailiy feedback hypohesis proposed by French e al. (1987), Campbell and Henschel (1992) and Bakaer and Wu (2000) aribues asymmeric volailiy o he volailiy feedback effec. 10 Conrary o he leverage-based jusificaion, he volailiy feedback hypohesis saes ha increases in volailiy rigger negaive sock reurns. For insance, an increase in volailiy implies ha he required expeced fuure reurns will also increase, hereby riggering declines in curren sock prices. However, boh hypoheses empirically fail under he daily frequency daa, being unable o fully characerize he asymmeric reurn-volailiy relaionship; in ha respec, Schwer (1990) argued ha i is oo srong for he leverage hypohesis o fully characerize asymmeric volailiy. Furhermore, i is also empirically found ha he feedback hypohesis is no always consisen, and his has become a conroversial subjec; some sudies have found ha here are no always posiive correlaions beween curren volailiy and expeced fuure reurns (e.g., Breen e al., 1989), bu ohers suppor he hypohesis (e.g., French e al., 1987; Campbell and Henschel, 1992; Ghysels e al., 2005). Noneheless, he economic and accouning explanaions migh be imporan for characerizing he asymmeric reurn-volailiy relaionship a lower frequencies, for insance, monhly or quarerly daa, bu no for daily or higher frequencies. Many prior sudies have 10 Poerba and Summers (1986) characerized he volailiy feedback effec hrough economic explanaion. The main underlying facor ha induces he volailiy feedback effec is he exisence of ime-varying risk premia, which serve as he link beween flucuaions in volailiy and reurns. 7

8 documened very srong negaive asymmeric reurn-volailiy relaionships a higher frequencies, conrary o he explanaions of he wo hypoheses (see, e.g., Fleming e al., 1995; Whaley, 2000; Gio, 2005; Simon, 2003; Skiadopoulos, 2004; Low, 2004; Dennis e al., 2006; Hibber e al., 2008). However, his sudy considers new MFIV indices because we believe ha he asymmeric reurn-volailiy relaion should be more pronounced using he MFIV indices. Likewise, he imporance of he MFIV index measure increases because i accouns for volailiy skew, which may be induced by he ne buying pressure of he OTM pu opions (see Bollen and Whaley, 2004). Volailiy skew is an obvious phenomenon, previously documened by many oher researchers and imporan o capure in any volailiy measure (e.g., Alexander, 2001; Low, 2004; Goncalves and Guidolin, 2006; Badshah, 2008). Bollen and Whaley (2004) invesigaed he relaionship beween ne buying pressure and he shape of he IV funcion (IVF) for index opions. They showed ha he buying pressure of pu opions considerably affecs he changes in he IV. They assered ha when he buying pressure of index pu opions (paricularly from insiuional invesors who seek o hedge heir porfolios) increases and hus limis he abiliy of arbirageurs o bring he price back ino alignmen, his pressure permanenly drives he sloping shape of he IVF downward. Also, informaion from rading sraegies and oher shocks are well absorbed ino he MFIV index, as i accouns for boh OTM pu and call ypes of opions; herefore, when here is a shock o he marke ha leads o a change in he price of one ype of opion relaive o he oher ype, he new MFIV index adjuss and follows a similar direcion as he ne change. The MFIV index is informed by boh fear and exuberance embedded in opion prices, and he majoriy of opion markes raders are very informed and possess high skill levels (see Low, 2004; Chakravary e al., 2004). The MFIV index is a very informed measure of sock index volailiy and is herefore a good candidae for examining he asymmeric volailiy-reurn relaionship. 8

9 3. Daa Firs, he VIX, VXN, VDAX, and VSTOXX volailiy indices are inroduced, and heir consrucion is discussed. Second, he complee daa se is presened, and he descripive saisics are horoughly discussed VIX and VXN The CBOE inroduced a new VIX index in Sepember 2003 based on opions on he S&P 500 index. The VIX index is deermined from he bid and ask prices of he opions underlying he S&P 500 index. The new VIX is independen of any opion pricing model using he MFIV measure. The VIX hus provides an esimae of expeced fuure realized sock marke volailiy for he 22 subsequen rading days (over 30 calendar days). However, he old VIX index, based on opions on he S&P 100 index and inroduced in 1993, has now moved o he new icker symbol VXO. In conras o he old VIX (now VXO), which is based on near-hemoney BSIV opions on he S&P 100 index, he new VIX uses marke prices of opions on he S&P 500 index. 11 This new MF VIX mehodology accouns for boh OTM pu and call opions (i.e., volailiy skew). The new mehodology is hus more appealing and robus. The CBOE s inroducion of he new VIX was moivaed by boh heoreical and pracical deliberaions. Firs, he new VIX is economically more appealing as i is based on a porfolio of opions, whereas he old VIX was based on he ATM opion prices. Second, he new VIX makes i easy o replicae variance swap payoffs while using saic posiions in a range of opions and dynamic posiions in fuures rading. Third, he new VIX has removed he induced upward bias of he old VIX in he rading day conversion (see, e.g., Carr and Wu, 2006). Similarly, in Sepember 2003, he CBOE inroduced he VXN using he same MFIV 11 The opions on he S&P 500 index, in comparison wih he opions on he S&P 100 index, conain a much broader se of implied informaion; he new VIX is hus a more informaive measure han he old VIX (now VXO). 9

10 mehodology as ha of VIX. The CBOE has calculaed price hisories for VIX and VXN back o he years 1986 and 2001, respecively. 3. VDAX and VSTOXX The Deusche Börse and Goldman Sachs joinly developed he mehodology for he new VDAX and VSTOXX indices. The VDAX is based on opions on he DAX 30 index, whereas VSTOXX is based on opions on he Dow Jones (DJ) Euro STOXX 50 index, which consiss of he eurozone s 50 larges blue-chip socks. Opions on he DAX and he DJ Euro STOXX 50 are raded on he EUREX derivaives exchange. The VDAX measure accouns for IVs across all opions of a given ime o expiry (accouning for volailiy skew). The mehodology of he VDAX, like ha of he VIX, is based on he MFIV measure. However, he main VDAX index is furher based on eigh consiuen volailiy indices, which expire in 1, 2, 3, 6, 9, 12, 18, and 24 monhs, respecively. The main VDAX is designed as a rolling index a a fixed 30 days o expiry via a linear inerpolaion of he wo neares of he eigh available sub-indices. The VDAX and is eigh sub-indices are updaed every minue and herefore offer grea advanages in erms of rading, hedging and inroducing new derivaives on his index. The price hisories for boh VDAX and VSTOXX were calculaed back o he years 1992 and 1999, respecively Daa Se This sudy employs daa from four sources. We obained he daily ime-series price daa for he S&P 500 sock index, he NASDAQ 100 index, he DAX 30 index, and he DJ Euro STOXX 50 index from Thomson Financial DaaSream. The daa on he new VIX and VXN were obained from he CBOE, and he daa on he new VDAX and VSTOXX were obained from he Deusche Börse and STOXX, respecively. The daily frequency daa for boh sock 10

11 and volailiy indices cover a period of 8 years and 4 monhs, from February 2, 2001, o May 29, 2009, for a oal of 2172 rading days. VIX S&P VXN NASDAQ VDAX DAX VSTOXX DJESTOXX Figure 1. Sock indices versus MFIV indices from February 2, 2001, o May 29, Figure 1 shows he daily closing levels (%) of he volailiy indices, i.e., he VIX, VXN, 11

12 VDAX and VSTOXX, and he corresponding sock marke indices (levels) from February 2, 2001, o May 29, Among he four volailiy indices, he VXN index presens he highes volailiy level hroughou our sample period, whereas he VIX shows he lowes volailiy level. Similarly, he volailiy indices and sock indices move inversely o one oher. From he beginning of 2001 unil he beginning of 2003, here were considerably high volailiy levels. However, from 2004 o lae 2007, we find upward movemen in he sock marke indices, whereas he corresponding volailiy indices moved in he opposie direcion o he sock markes, showing he lowes volailiy levels. However, in he laer par of 2007, we again find somewha increasing volailiy levels, wih he sock markes again moving downward due o he beginning of he credi crunch and liquidiy crunch crises, which have caused markes o be exremely volaile and he volailiy indices o reach hisorically high levels (paricularly, in Ocober and November of 2008, he VIX level wice surpassed 80%), and he corresponding sock markes crashed aferward, herefore moving in compleely opposie direcions. This phenomenon is eviden unil he end of he sample period in May Table 1 repors he summary saisics for he daily percenage coninuously compounded reurns of four sock indices and he daily percenage reurns of five volailiy indices, as well as ess for normaliy, auocorrelaions and uni roos. The mean values for all nine sock index reurns and volailiy index reurns series are no saisically differen from zero. The ess for skewness and kurosis confirm ha he sock indices reurns are posiively skewed excep for he S&P 500 reurns, whereas all five volailiy indices reurns are posiively skewed, as hey should be. Furhermore, all nine series are highly lepokuric wih respec o he normal disribuion. Likewise, he Jarque-Bera saisics rejec normaliy for each of he sock index and volailiy index reurns series. The auocorrelaion coefficiens for he hree lags show ha he VIX, VDAX and VXN reurns series presen srong auocorrelaions, 12

13 whereas he reurns on he res of he volailiy indices presen significan auocorrelaion coefficiens a lags 2 and 3. Auocorrelaions in he S&P 500, NASDAQ 100, and DJ Euro STOXX 50 reurns series are also eviden a all hree lags, consequenly confirming he propery of mean reversion. We also invesigaed saionariy in all nine reurns series (i.e., sock and volailiy indices) by applying he augmened Dickey-Fuller (ADF) uni-roo es. The resuls in Table 1 show he rejecion of uni roos in each series a he 1% significance level. Therefore, all nine series are saionary. Table 1 Descripive saisics. S&P500 NASDAQ DAX30 STOXX50 VIX VXN VDAX VDAXO VSTOXX Mean Median Maximum Minimum Sd. Dev Skewness Kurosis JarqueBera Prob ρ *** -070*** -042* -041* *** -042* 044** ρ2-069*** -063*** ** *** *** -063*** *** *** ρ3 053*** 020*** *** 028*** 028*** *** -018*** -092*** ADF *** *** *** *** *** *** -248*** -240*** *** No. Obs This able repors he descripive saisics of he sock marke indices and volailiy indices reurns. The auocorrelaion coefficiens ρ, he Jarque-Bera and he Augmened Dickey-Fuller(ADF) (an inercep is included in he es equaion) es values are repored. ***, ** and * denoe rejecion of he null hypohesis a he 1%, 5% and 10% significance levels, respecively. 13

14 4. Regression Model for Asymmeric Reurn-Volailiy Relaion We presen a quanile regression model for assessing he negaive asymmeric relaionship beween he reurns on he sock index and reurns on he volailiy index. This model is he generalizaion of he sandard mean-regression models of Simon (2003), Gio (2005) and Hibber e al. (2008), which have empirically confirmed he asymmeric reurn-volailiy relaionship. 12 However, his paper exends hese sandard mean-regression models (MRM) by modeling he asymmeric reurn-volailiy relaionship using he condiional quanile regression model (QRM) o examine how negaive and posiive sock index reurns vary across differen quaniles of IV reurns, i.e., how much his asymmeric relaionship ends o change across differen quaniles of IV changes. Before specifying our quanile-regression model for he asymmeric reurn-volailiy relaionship, we firs specified a MRM model similar o ha of Simon (2003), Gio (2005) and Hibber e al. (2008), which is considered a sandard model in our analysis. 13 For insance, we regressed he daily volailiy index reurns (denoed ΔVI i, where i=δvix, ΔVXN, ΔVDAX, ΔVSTOX) on he daily percenage coninuously compounded reurns of he sock marke index (denoed Euro STOXX 50), where R i R i was used for posiive reurns and, where i=s&p 500, NASDAQ, DAX, DJ Ri for negaive reurns. For he posiive reurns, R i R i if R i 0, and 0 R i oherwise. On he oher hand, for he negaive reurns, R i R i if R i 0, and R i 0 oherwise. The sandard MRM for assessing he negaive asymmeric reurn-volailiy relaion hus has he form 12 They showed ha he relaionship behaves differenly for negaive and posiive sock index reurns. 13 Hibber e al.(2008) segmened negaive and posiive sock reurns ino quaniles and hen used leas squares for each quanile, which could no yield he robus resuls ha we can find using quanile regression, i.e., he effecs of negaive and posiive reurns on he upper and lower quaniles of he dependen variable would be much differen and robus using quanile regression insead of leas-squares regression (for a deailed discussion, see Heckman, 1979; Koenker and Hollack, 2001; Basse and Chen, 2001). 14

15 ΔVI i α 3 L1 β ΔVI 3 3 i L γr δr i L u (1) L0 L0 Where α is he inercep; β represens he coefficiens for he lagged IV reurns of a volailiy index i, where L 1 o 3; γ represens he coefficiens for posiive sock reurns and δ he coefficiens for he negaive reurns of a sock marke index i, where L 0 o 3 for boh ypes of reurns; and he errors u are independenly idenically disribued (iid) wih zero means. Consequenly, he sandard MRM assumes ha he effecs of boh ypes of reurns are saic across differen IV reurns (i.e., response variables); herefore, an MRM would miss imporan informaion across quaniles of IV reurns ha we could oherwise deec using a QRM, paricularly o deermine how he median or perhaps he 5 h or 95 h perceniles of he response variable IV reurns are affeced by negaive and posiive sock reurn variables (regressor variables). 14 Koenker and Basse (1978) were he firs o inroduce quanile regression ha could effecively model he uppermos quaniles. 15 QRM is a generalizaion of he MRM and is hereby a robus regression, especially in siuaions where errors are non-normally disribued, i.e., are skewed and lepokuric. Noneheless, he QRM is used for examining he asymmeric reurn-volailiy relaionship; for insance, he qh QRM, which is a generalizaion of equaion (1), has he form ΔVI i α i L γ R i L δ R i L u (2) L0 L0 q q q q L1 β ΔVI Where q α is he inercep; β (q) represens he coefficiens for he lagged IV reurns of a volailiy index i, where L 1 o 3; q γ represens he coefficiens for posiive reurns and q δ he coefficiens for negaive reurns of a sock marke index i, where L 0 o 3 for boh ype of reurns; and he errors u are assumed o be independen from an error 14 See a good discussion on his issue in Meligkosidou e al. (2009). 15 Koenker (2005) provides mahemaical deails on he differen versions of he quanile regression models. 15

16 disribuion Φ (u ) wih he qh quanile equal o zero. Equaion (2) implies ha he qh q condiional quanile of he dependen variable ΔVI i given ΔVI, ΔVI, ΔVI, R, R, R, R, R, R, R, R and denoed Q q i1 i2 i3 i i1 i2 i3 i i1 i2 i3 ΔVI i ΔVI,...ΔVI, R,.., R, R,.., R i 1 i 3 i i 3 i i 3 is equal o, α 3 q q q q L1 β ΔVI i L 3 L0 γ R i L 3 L0 δ R i L,. The main feaure of his quanile (q) regression framework is ha he effecs of he variables capured by β, γ q,and δ q vary for each qh quanile wihin he range q (0,1). Furhermore, he framework allows for heeroskedasiciy in error u, and he coefficiens are differen for differen quaniles. Consequenly, a quanile regression provides a broader se of informaion abou volailiy reurns here (i.e., he effecs on he enire disribuion of he volailiy reurns) han OLS regression would, paricularly when he error disribuion is no symmeric. 16 QRM is hus esimaed for he sample period, from February 2, 2001, hrough May 29, 2009, using he quanile regression mehod proposed by Koenker and Basse (1978), which minimizes he asymmeric sum of absolue residuals and robusly models he condiional quaniles of he response variable, i.e., in our case, changes in he volailiy index: 17 min :ΔVIi αˆ βˆ ΔVIi L γˆ Ri L δˆ Ri L :ΔVIi αˆ βˆ ΔVIi L γˆ Ri L δˆ Ri L q ΔVI αˆ βˆ i (1 q)δvi i αˆ βˆ ΔVI ΔVI γˆ γˆ R R δˆ δˆ R R 16 Because he differences beween he mean and he median produce asymmeric disribuions, see, for a more deailed explanaion, Meligkosidou e al For a discussion of quanile models and heir esimaion echniques, see Koenker (2005). 16

17 5. Empirical Resuls Figure 2 provides quanile regression resuls for S&P 500 reurns wih he VIX index, where we have 11 covariaes and an inercep. For each of he 12 coefficiens, 19 quanile regression esimaes were ploed for q ranging along q (05,0.1,...,0.9, 0.95) as he solid curve (blue) wih circles. In each plo on he x-axis, we have a quanile (or q) scale, and he y-axis indicaes he covariae effec as a percenage. For each covariae, hese esimaes could be inerpreed as he effec of a percenage-poin change of he covariae on he volailiy, holding oher covariaes unchanged. The wo red-doed lines show he convenional 95% confidence level for he quanile regression esimaes. However, more deailed resuls for he imporan upper and lower quaniles of esimaes from Figure 2 are provided in Table 2, including corresponding -saisics (in parenheses) for each of he esimaes herein. The sandard errors were obained using he boosrap mehod; herefore, robus -saisics were obained for each of he quanile esimaes. On he oher hand, for he OLS esimaes, he sandard errors were made heeroskedasiciy-consisen using Newey-Wes (1987) correcion. As he aim was o quanify he asymmeric reurn-volailiy relaionship, we limi our discussion o he posiive and negaive reurns covariaes, especially o capuring he conemporaneous effecs. When we look a he esimaed coefficiens of covariaes SP500R and SP500R in Rows 6 and 10, respecively, which represen he conemporaneous reurn-volailiy relaionship; i is apparen from he absolue difference ha here are asymmeric effecs for all quanile regression esimaes, including OLS esimaes (here, OLS esimaes are merely provided for comparaive purposes). The absolue values of SP500R are higher han he absolue values of SP 500R. Moreover, he Wald es for coefficiens was applied in order o find he saisical difference beween he coefficiens and q δ in equaion 2. The null hypohesis (i.e., he coefficiens for negaive and posiive q γ 17

18 reurns are equal) for he Wald es was significanly rejeced for each of he quanile regression esimaes. 18 These resuls imply an asymmeric reurn-volailiy relaionship, indicaing ha he negaive reurns for he sock index are linked o much higher volailiies for he VIX index han hose linked o posiive reurns. More specifically, looking a each row of Table 2 (i.e., each quanile of esimaes), he resuls indicae ha he impacs of he negaive and posiive S&P 500 index reurns on he VIX are highly asymmeric, wih boh conemporaneous coefficiens being saisically significan a he 1% significance level. The mean or OLS regression esimaes are quie similar o he q = 0.5 (median)-quanile regression esimaes; however, he changing naure of he esimaes a he oher quaniles provides an ineresing picure of how he disribuion of IV depends on he posiive and negaive reurns variables and lagged IV variables. The absolue value of SP 500R monoonically increases when moving from a median quanile o an upper quanile; i.e., he marginal effec of he negaive reurns is larger in upper quaniles (i.e., q=0.95%), and vice versa for posiive reurns. 19 As a resul, OLS underesimaes he magniude of hese effecs for he highes quaniles and overesimaes for he lowes quaniles. In deail, he coefficien esimaes wih q = 0.5 or median (and OLS) for he SP 500R variable imply ha a 1% decline in S&P 500 reurns is linked o a 140% (1.185%) increase in he VIX level, whereas he coefficien esimaes for he SP 500R variable imply ha a 1% increase in S&P 500 reurns is linked o a 0.795% (0.864%) decrease in he VIX level. 20 However, in he coefficien esimaes for quanile q=0.95, he SP500R variable implies ha a 1% decline in S&P 500 reurns is linked o a 1.646% increase in he VIX, whereas he 18 Wald ess resuls are no repored here o save space. 19 The equaliy of he coefficiens across quaniles was formally esed using he Wald es. The es resuls significanly rejeced he null hypohesis of equaliy of he coefficiens (paricularly he conemporaneous negaive and posiive reurns) across quaniles; he Wald es is repored in Table Mean-regression model (or OLS) esimaes are provided in parenheses for comparison. 18

19 coefficien esimaes for he SP 500R variable imply ha a 1% increase in he S&P 500 reurns is linked o a 001% decrease in he VIX level. Obviously, i is apparen from he quanile regression resuls ha he asymmery is much smaller in he lower and median quaniles of he disribuion and noiceably higher in he upper quaniles of he disribuion. Thus, he OLS esimae, which simply capures he mean effec, does a poor job of accouning for his asymmery in he upper quaniles. 21 Dependen Variable VIX----- Process Esimaes Inercep VIX(-1) VIX(-2) VIX(-3) SP500R+ SP500R(-1)+ SP500R(-2)+ SP500R(-3) SP500R- SP500R(-1)- SP500R(-2)- SP500R(-3) Figure 2: Regression Esimaes: Dependen Variable VIX Index 21 The sandard mean-regression models of Simon (2003), Gio (2005) and Hibber e al. (2008) for he asymmeric reurn-volailiy relaionship ignore he higher effecs of negaive and posiive reurns on he upper quaniles of he volailiy disribuion. 19

20 Table 2: Regression Resuls: Response variable VIX Index Inercep ΔVIX 1 ΔVIX 2 ΔVIX 3 SP 500 R SP 500 R 1 SP 500 R 2 SP 500 R 3 SP 500 R SP 500 R 1 SP 500 R 2 SP 500 R *** *** -079*** -042** -081*** *** 0.313*** 0.344*** (-31) (-1.39) (-0.37) (-05) (-11.65) (-3.17) (-2.51) (-3.19) (-6.72) (3.33) (2.86) (1.63) *** *** *** * -024** *** 071*** 0.314*** 0.166** (-4.19) (-0.68) (-09) (-0.11) (-13.56) (-2.94) (-1.95) (-2.82) (-12.98) (38) (37) (2.31) *** *** ** *** *** 0.186*** 060*** 0.153** (-3.32) (-1.12) (-0.52) (1.16) (-17.64) (-24) (-2.84) (-1.38) (-152) (2.98) (3.13) (2.31) *** -062* *** ** ** *** 0.119** 033*** 0.147*** (-33) (-1.81) (-0.88) (0.82) (-14.11) (-2.56) (-29) (-12) (-17.51) (28) (3.75) (2.61) *** -078** *** *** *** 097** 0.182*** 0.167*** (-34) (-2.56) (-0.17) (1.16) (-16.18) (-31) (-12) (-1.36) (-22.78) (20) (3.70) (3.53) Median ** *** -087** *** * 061 (-0.35) (-22) (0.38) (08) (-23.86) (-2.14) (0.87) (12) (-295) (13) (1.87) (14) * -054* *** 064* *** (1.70) (-1.69) (-0.61) (0.83) (-12.71) (1.72) (14) (15) (-32.60) (02) (0.16) (12) * -074** *** ** 0.111** *** (1.92) (-23) (-0.17) (09) (-9.99) (12) (21) (2.14) (-37.98) (-0.76) (-07) (0.72) *** -090** *** ** 0.105** *** (35) (-2.39) (02) (0.36) (-8.80) (0.58) (2.34) (2.17) (-243) (-16) (-03) (0.76) *** -079* *** ** *** ** (4.90) (-1.69) (0.36) (0.52) (-8.76) (01) (2.54) (0.64) (-232) (-2.11) (-0.38) (0.70) *** *** *** *** (52) (-0.39) (0.86) (1.16) (-5.67) (1.57) (2.98) (13) (-14.93) (-15) (-06) (0.12) OLS ** *** -089* *** (01) (-27) (-08) (-0.69) (-10.75) (-1.71) (-0.70) (-1.15) (-24.19) (-0.18) (0.59) (1.61) Slope Equaliy Tes Resuls: Only significan resuls of asymmery are repored wih he corresponding quaniles q 1 q. 0-0*** 0-0** 0-0*** 0-0*** 0-0* 0-0.5** 0-0.5* * ** *** ** *** *** *** *** ** The able repors resuls from he Regression and OLS Regression of he VIX index on a se of variables; specificaions 2 and 1 are esimaed. T-saisics are provided in parenheses. ***, **, and * denoe rejecion of he null hypohesis a he 1%, 5% and 10% significance levels, respecively.

21 Figure 3 presens quanile regression resuls for NASDAQ 100 reurns wih he VXN index, and he imporan full-sample daily upper and lower quaniles resuls are presened in Table The resuls are qualiaively similar o hose found for he S&P 500 and VIX asymmeric relaionship. 23 The major difference lies in he lower asymmeric responses of he covariaes (i.e., negaive and posiive reurns) across differen quaniles of he VXN disribuion in comparison wih he VIX resuls. 24 Furhermore, he significance of covariaes is lower for he VXN han for he VIX. The finding is consisen wih boh Gio (2005) and Hibber e al. (2008) in ha during volaile periods, opion raders reac less aggressively o negaive reurns. As he NASDAQ is a ech index, i inherenly presens a higher volailiy han he S&P 500; herefore, he conclusion drawn by Gio (2005) and Hibber e al. (2008) can be applied o he NASDAQ resuls. 25 On he oher hand, Figures 4 and 5 provide quanile regression resuls for DAX 30 reurns wih he VDAX and he VDAXO, respecively; he imporan daily upper and lower quanile resuls are repored in Table 4 and Table The quanile resuls for he DAX 30 reurns wih boh he VDAX and VDAXO are discussed simulaneously in order o compare he asymmeric responses of boh volailiy indices o he same negaive and posiive reurns of he DAX 30 index. 27 The coefficiens of he covariaes and are shown in Rows 6 and 10, respecively, in boh ables; he coefficiens represen conemporaneous reurn-volailiy relaionships. Based on he absolue difference in he coefficiens values, i 22 Figure 3 and Table 3 are provided in Appendix A. 23 A deailed discussion on hese resuls is avoided merely for space consideraions. 24 The Wald es for equaliy of he coefficiens across quaniles is formally esed and repored in Table 3. Here, oo, he es resuls significanly rejec he null hypohesis of equaliy of he coefficiens (paricularly he conemporaneous negaive and posiive reurns) across quaniles. 25 For more discussion on his poin, see Hibber e al. (2008). 26 Figures 4 and 5 and Tables 4 and 5 are provided in Appendix A. 27 Remember ha VDAX is he MFIV index ha incorporaes volailiy skew, whereas VDAXO is he BSIV index ha does no accoun for volailiy skew. Unforunaely, for he comparison of he wo measures we are resriced o only he DAX 30 sock index. For he oher sock indices, S&P 500, NASDAQ 100 and DJ Euro STOXX, we have no acive BSIV volailiy index; alhough he VXO, an acive BSIV volailiy index, is available, i canno be compared because i is implied from he opions on he S&P 100 index. 21

22 is clear ha here are asymmeric effecs for all quanile regression esimaes, including he OLS esimaes. The absolue values of are higher han he absolue values of. Similarly, he null hypohesis of he Wald es ha he coefficiens q γ and q δ in equaion 2 are equal is significanly rejeced for each of he quanile regression esimaes. Furhermore, looking a each row of Tables 4 and 5 (i.e., each quanile resul), he resuls indicae ha he impacs of he negaive and posiive DAX 30 index reurns on VDAX and VDAXO are asymmeric, wih boh conemporaneous coefficiens being saisically significan a he 1% significance level. The coefficien esimaes for q=0.5 or he median (and OLS) for he covariae imply ha a 1% decline in DAX 30 reurns is linked o a 0.837% (122%) increase in he VDAX level and ha a similar decline in DAX 30 reurns is linked o a 0.712% (0.778%) increase in VDAXO level. On he oher hand, he coefficien esimaes for he covariae imply ha a 1% increase in DAX 30 reurns is linked o a 0.515% (000%) decrease in he VDAX level, and a similar increase in DAX 30 reurns is linked o a 0.543% (0.370%) decrease in he VDAXO level. However, he coefficien esimaes for quanile q=0.95 of he variable imply ha a 1% decline in DAX 30 reurns is linked o a 1.389% (1.115%) increase in he VDAX (VDAXO), whereas he coefficien esimaes for he covariae imply ha a 1% increase in DAX 30 reurns is linked o a 0.130% (0.156%) decrease in he VDAX (VDAXO) level. For comparison, he coefficiens of covariaes and are lised in Tables 4 and 5. I is very clear ha he effecs of he negaive and posiive reurns are considerably differen. The VDAX (MFIV index) responds in a very asymmeric fashion in comparison o is older counerpar, he VDAXO (BSIV index), o similar negaive and posiive reurns, implying pronounced asymmeric reurn-volailiy relaionship wih he MFIV index. Furhermore, he asymmeric 22

23 responses are mos apparen in upper-quanile esimaes, where he asymmery is very pronounced; i.e., he asymmeries in he absolue differences are smaller in he lower and median quaniles of he disribuion and noiceably larger in he upper quaniles of he disribuions. Figure 6 presens quanile regression esimaes for he DJ Euro STOXX 50 reurns wih he VSTOXX index, and he imporan daily upper and lower quanile resuls are presened in Table Similarly, he resuls here are qualiaively similar o hose found for he DAX 30 and VDAX asymmeric relaionships. The major difference is he slighly more asymmeric responses of he covariaes (i.e., negaive and posiive reurns) across differen quaniles of he VSTOXX in comparison o he VDAX resuls. The main conclusion drawn from Tables 2 o 6 is ha negaive and posiive sock index reurns rigger he volailiy index o move in compleely opposie direcions and in an asymmeric fashion; i.e., negaive reurns have a much greaer impac on volailiy han do posiive reurns, paricularly a he uppermos regression quaniles (e.g., q=0.95). Our quanile regression model for he asymmeric reurn-volailiy relaion hus reveals imporan missing informaion underesimaed by he mean-regression model (OLS). Second, our argumen ha he asymmery wih skew-adjused volailiy (MFIV) should presen pronounced responses is confirmed by comparing he asymmeric responses of VDAX (MFIV) and VDAXO (BSIV), where we found ha he MFIV index responded in a pronounced fashion in comparison wih he BSIV index. 29 Third, if we look a he coefficiens of he lag covariaes of negaive and posiive reurns, hey are mosly 28 Figure 6 and Table 6 are provided in Appendix A. 29 As he MFIV volailiy indices accoun for OTM pus, he asymmery should be pronounced wih each MFIV volailiy index. Because invesors hedge heir downside risk by aking posiions in he OTM pu opions, in periods of marke urmoil here is greaer buying demand for pu opions han for call opions, which leads o higher volailiies han hose found during marke rallies. Consequenly, negaive sock index reurns induce an increase in he levels of he volailiy indices. Our resuls are also consisen wih he ne-buying-pressure hypohesis of Bollen and Whaley (2004). 23

24 insignifican; we hus asser ha a he daily level, he leverage hypohesis is unable o quanify his srong asymmeric reurn-volailiy relaion and ha similar conclusions could be drawn for he feedback hypohesis. Finally, he VIX volailiy index presens he sronges asymmeric reurn-volailiy relaionship, followed by he VSTOXX, VDAX and VXN volailiy indices, respecively. 24

25 6. Conclusion We invesigaed he asymmeric reurn-volailiy phenomenon in he newly adaped robus volailiy indices (i.e., he VIX, VXN, VDAX, VDAXO, and VSTOXX) using quanile regression. In paricular, we quanified he effecs of posiive and negaive sock index reurns a differen quaniles of IV disribuions, asking abou he degree o which he asymmeric responses a he uppermos quaniles are comparable wih he responses of median (or mean) regressions. Addiionally, as Bollen and Whaley (2004) have documened, he ne buying pressure for sock index pu opions from insiuional invesors seeking o hedge heir porfolios induces increases in IVs. Likewise, new IV indices incorporae boh OTM pu and call opions and are hus highly informed and robus measures. Accordingly, hey should presen more pronounced asymmeric reurn-volailiy relaionships in comparison o heir older counerpars. There is noiceable evidence ha he volailiy indices VIX, VXN, VDAX, VDAXO and VSTOXX from February 2001 hrough May 2009 responded in a pronounced asymmeric fashion o he negaive and posiive reurns of heir corresponding sock indices: he asymmery monoonically increases when moving from he median quanile o he uppermos quanile (i.e., 95%); herefore, OLS underesimaes his relaionship a upper quaniles. These IV indices hus sharply rise during marke declines (fear) and fall during marke rallies (exuberance). The VIX presens he highes asymmery, followed by he VSTOXX, VDAX and VXN volailiy indices, respecively. Second, our argumen ha asymmery wih he volailiy skew-adjused volailiy index measure (MFIV) should be pronounced is confirmed by comparing he asymmeric responses of VDAX (MFIV) and VDAXO (BSIV); he MFIV index responds in a pronounced fashion in comparison wih he BSIV index. Third, we also confirmed ha a significan amoun of asymmery occurs conemporaneously raher han wih a lag, hus rejecing he leverage hypohesis, and ha a similar conclusion can be drawn for 25

26 he feedback hypohesis. Our resuls have a number of implicaions. Firs, as we found ha newly adaped volailiy indices are srongly negaively correlaed wih heir corresponding sock indices, he new volailiy indices are imporan insrumens for hedging sock porfolios. Derivaives exchanges provide liquid markes for he fuures and opions underlying hese volailiy indices. Therefore, a posiion in fuures or opions on a volailiy index can more accuraely hedge a sock porfolio posiion wihou considering complicaed sock index opion rading sraegies. Second, when he sock index drops, he volailiy index rises sharply. Therefore, new volailiy indices are useful no only for assessing poenial risks, bu also for speculaive ransacions by risk-seeking invesors. Third, since he new volailiy indices are based on he robus MFIV concep and provide beer radabiliy, i is easier for issuers of derivaives o engineer srucured producs based on he volailiy indices. Fourh, rading sraegies wih regard o range could generae profis; an example of his could be a volailiy-long posiion in decreasing volailiy markes paired wih a volailiy-shor posiion in increasing volailiy markes. 26

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