Cross-sectional analysis of riskneutral

Transcription

1 Cross-secional analysis of riskneural skewness by Sephen Taylor, Pradeep Yadav, and Yuanyuan Zhang * Deparmen of Accouning and Finance, Managemen School, Lancaser Universiy, Lancaser LA1 4YX, Unied Kingdom Absrac We invesigae he imporance of various firm specific and marke wide facors in explaining he risk-neural skewness, esimaed using he prices of individual sock opions, for 149 U.S. firms. We find ha he risk-neural skewness of individual firms, which is less negaive han he marke index skewness, is negaively relaed wih he firm s opion rading volume, sock rading volume, size, sysemaic risk proporion, marke senimen raio and he firm s own volailiy, while i is posiively relaed wih he firm s leverage raio, informaion asymmery measure and he realworld volailiy asymmery. Also, he firm s risk-neural skewness ends o be more negaive during he periods when he marke index skewness is more negaive and when he sock index is more volaile. JEL classificaions: C5; G10; G14 Keywords: risk-neural disribuion, skewness; sock opions; ARCH models This version: January 008 * addresses: s.aylor@lancaser.ac.uk (S. Taylor), p.yadav@lancaser.ac.uk (P. Yadav), y.zhang18@lancaser.ac.uk (Y. Zhang). 1

2 1 Inroducion The disribuion of an asse price in he fuure is imporan in many areas of Finance. One imporan applicaion is in opion pricing. By defining he risk-neural densiy funcion of he underlying asse, we can calculae he expeced payoff of a European opion conrac. The opion price equals he discouned value of he expeced payoff in he risk-neural pricing framework. Previous research sudies have addressed differen mehods o exrac he risk-neural densiy from opion prices, especially from index opion prices, and examined is shape and properies. However, no many papers analyse he risk-neural densiy of individual socks. Bakshi, Kapadia and Madan (BKM) (003) and Dennis and Mayhew (00) find ha he risk-neural skewness of individual socks is much less negaive han he index and depends on boh marke facors and firm specific facors, wih he laer more imporan han he former. The difference beween he risk-neural skewness of individual socks and he risk-neural skewness of he sock index esablishes he need for differenial pricing of individual sock opions versus he marke index opions. This makes i relevan o find ou which, and o wha exen, firm specific facors are deermining he risk-neural skewness of individual socks. 1.1 Prior Lieraure The risk-neural momens of individual socks has drawn far less scruiny in empirical research han he corresponding momens of he sock index. One possible reason is due o daa availabiliy. Anoher possible reason is ha he opions of individual firms are far less liquid han index opions.

3 BKM (003) prove ha, if individual sock reurns are composed of a marke componen and an idiosyncraic componen, hen he skewness of sock reurns can also be decomposed ino a marke componen and an idiosyncraic componen. Their empirical sudies on 30 U.S. firms wih he highes marke capializaions and he S&P500 index show ha individual skews are nearly always negaive bu less negaive han he index, and ha here is no much informaion abou he risk-neural skewness of individual socks, which can be exraced solely from he risk-neural skewness of he index. Their resuls are confirmed again by Dennis and Mayhew (00), who direcly examine he relaion beween risk-neural skewness and six firm specific facors and wo marke-wide facors for 1,41 U.S. firms from April 1986 o December Their resuls show ha he sock rading volume, firm size and he firm s marke risk (measured by Bea) are all negaively relaed wih he risk-neural skewness of individual socks. On he conrary, hey do no find he leverage raio and he raio of pu/call opion rading volume, as a measure of marke senimen, are he driving forces behind he asymmery in he risk-neural disribuions of individual socks. The insignifican relaion beween leverage raio and risk-neural skewness in Dennis and Mayhew (00) is conrary o wha Tof and Prucyk (1997) find for 138 U.S. firms from 1993 o However, Tof and Prucyk (1997) s measure of risk-neural skewness, which is he raio of he slope of implied volailiy curve over he a-hemoney (ATM) implied volailiy, is suspicious, as i includes effecs from boh he slope and he level of he implied volailiy curve [Dennis and Mayhew (00)]. 3

4 Pena, Rubio and Serna (1999) es he deerminans of various variables on he shape of implied volailiy smiles, which is derived from he prices of Spanish index opions. They find ha he opion ransacion coss, underlying asse sandard deviaion, he long and shor erm ineres raes and he opion s ime o mauriy all influence he variaion of he implied volailiy smiles over ime. Following he work of BKM (003) in examining he marke wide effecs, Duan and Wei (006) find ha, for he same 30 U.S. firms and he S&P100 index, he sysemaic risk proporion, defined as he raio beween he sysemaic variance of a sock reurns and is oal variance, is negaively and significanly relaed wih riskneural skewness and posiively and significanly relaed wih risk-neural kurosis. Chrisoffersen, Jacobs and Vainberg (006) heoreically derive ha he risk-neural skewness of individual firms is negaively relaed wih he firms bea and posiively relaed wih skewness of marke reurns, when assuming he skewness of idiosyncraic reurn is zero. 1. Scope This paper firsly esimaes he risk-neural skewness for 149 U.S. firms, following he esimaion mehod provided by BKM (003). Secondly, we analyse he relaions beween a lis of firm specific facors and he firm s risk-neural skewness. Some of hese facors have been examined by previous lieraure, while he ohers are addressed for he firs ime o explain he deerminans of risk-neural skewness. Our 4

5 objecive is o capure more properies of he risk-neural skewness of individual firms and o find he sources of i. Our sudy is closely relaed wih Dennis and Mayhew (00) bu is differen in hree aspecs. Firsly, he calculaion of risk-neural momens derived by BKM (003) requires esimaing he inegrals funcion of opion prices. Dennis and Mayhew (00) who apply he same mehod use he marke opion prices, of which here are someimes only a few observaions for individual socks. We choose o infer a lo more opion prices from he implied volailiy curve in order o beer approximae he inegral funcions. Secondly, because days wih only a few raded opions are included, Dennis and Mayhew (00) invesigae nearly en imes he number of U.S. firms ha we use. Thirdly, he opion marke is more maure for our sample period, which is from Jan 1996 o Dec 1999, compared o he earlier period. For he firm specific variables ha have been invesigaed in previous lieraure, we find consisen resuls ha he risk-neural skewness end o be more negaive for firms wih higher sock rading volume, larger firm size, higher sysemaic risk proporion and lower leverage raio. However, differen o he resuls of Dennis and Mayhew (00), our sample resuls in negaive and significan coefficiens for he pu-o-all opion volume and he firm s implied volailiy. As for he firm specific variables ha have no been sudied in he conex of he riskneural skewness before, we find ha he opion rading volume is an imporan variable in explaining he risk-neural disribuion and is negaively relaed wih skewenss. The informaion asymmery measure, compued as he informed rader s 5

6 profis in he sock marke, increases wih he risk-neural skewness of individual firms. The real-world asymmery in volailiy is also posiively relaed wih he riskneural asymmery. However, he book-o-marke raio ha is posiively relaed wih he firms risk-neural skewness los is significan in mulivariable regressions. For he marke-wide variables, consisen wih previous lieraure, our resuls sugges ha he individual skewness moves in he same direcion as he sock index skewness and becomes more negaive when he sock index is more volaile. However, he effecs from he sock index are limied for our daa. The remainder of he paper is organized as follows. The developmens of hypohesis are described in Secion. In Secion 3, we inroduce he BKM (003) mehod o compue he risk-neural skewness. Daa sources and he consrucions of all variables are discussed in Secion 4. Secion 5 presens he regression resuls. The las secion conains conclusions. Developmen of hypohesis Mos of he hypoheses ha we make abou he risk-neural skewness of individual socks are developed according o he exising lieraure, which also sudies he individual risk-neural skewness. Some oher variables come from previous lieraure on risk-neural disribuion of sock indices, or he lieraure on he real-world disribuion of individual firm prices. 6

7 The firs explanaory variable for risk-neural skewness is he rading volume of he underlying sock. The relaion beween i and he risk-neural skewness can be moivaed from wo perspecives. Firsly, higher rading volume of socks, as a proxy for liquidiy, reduces he ransacion coss. In an opion pricing framework, lower ransacion coss make i easier o implemen dynamic arbirage sraegies and hus he arbirage bounds on opion prices are igher [Figlewski (1989), Dennis and Mayhew (00)]. Consisen wih his insigh, Dennis and Mayhew (00) find ha a higher rading volume of he underlying sock is associaed wih a less negaive riskneural skewness or a more symmeric risk-neural disribuion. On he oher hand, Hong and Sein (1999) argue ha invesor heerogeneiy is he cenral force creaing reurn asymmeries. When assuming ha here are differences of opinions among invesors abou sock values and ha some invesors face shorsale consrains, he Hong-Sein model suggess ha he negaive skewness is more pronounced in periods of heavy rading volume, where rading volume is a proxy for differences in opinions [Harris and Raviv (1993), Chen, Hong and Sein (001)]. In he empirical ess of Chen, Hong and Sein (001), he derended level of urnover over six monhs, referring o differences in opinion, has some explanaory power o predic he condiional skewness measured by daily sock reurns in he following six monhs, wih a negaive coefficien. As he relaion beween rading volume and skewness is ambiguous, we es i direcly. Following Dennis and Mayhew (00), we adop he logarihm of sock rading volume. Assuming some of he above argumens also apply o he rading volume of opions, we include he logarihm of opion rading volume as well. 7

8 Secondly, assuming he reurns of a firm s sock are relaed wih he marke index reurns. BKM (003) show ha he risk-neural skewness for he sock reurns can also be decomposed ino wo componens reflecing marke skewness and he skewness of unsysemaic risk componen, such ha: VAR β VAR =, ε, i 3 3 i m SKEW i ( 1+ ) SKEWm + (1 + ) SKEW ε, i β i VARm VARε, i where SKEW i, SKEW m and SKEW ε, i, respecively, refer o firm i s sock reurns skewness, marke reurns skewness and he skewness of he unsysemaic risk componen; VAR m and VAR ε, i are he variance of marke reurns and he variance of he unsysemaic proporion of firm i s sock reurns; β i esimaes he comovemens beween firm i s sock reurns and he marke reurns. As shown in he equaion, he individual skew is posiively linked o boh componens. When he risk-neural disribuion of he unsysemaic risk componen is symmeric or posiively skewed, he individual skew will be less negaive han he marke [BKM (003)]. Combining he heorem described by he above equaion wih he empirical findings ha risk-neural disribuion of individual socks is less negaively skewed han ha of he index, we expec ha a higher relaion beween he firms sock reurn and he index reurn is associaed wih more negaive risk-neural skewness. Dennis and Mayhew (00) find a negaive coefficien beween bea and risk-neural skewness for heir daa. However, Duan and Wei (006) sugges anoher esimae equal o βi VAR VAR ε, i m, namely sysemaic risk proporion, which is equivalen o he explanaory power, R, of he OLS regression for sock reurn: = α + +. R i, i, β i, Rm, ε i, 8

9 The sysemaic risk proporion is a beer esimae han bea. Firsly, bea measures he sysemaic marke risk bu no he sysemaic risk ha accouns for oal risk [Duan and Wei (006)]. For differen firms wih he same level of bea, he one wih lower oal risk and/or wih a higher correlaion wih he marke has a higher sysemaic risk proporion. Secondly, he sysemaic risk proporion ranges from zero o one, while bea can exceed one. Therefore, we esimae he effec of he sysemaic risk proporion on risk-neural skewness and expec ha here is a negaive relaion beween hem. Bea is used as an alernaive measure o he sysemaic risk proporion in our sudy. Moreover, according o he above equaion, he individual skewness should be posiively relaed wih he marke skewness. Dennis and Mayhew (00) find ha he risk-neural skewness of individual firms ends o move in he same direcion wih ha of he S&P 500 index over ime. We include he risk-neural skewness of he marke index ino ou analysis, expecing in he periods wih more negaive marke skewness ha he risk-neural skewness individual firm also ends o be more negaive. We adop he S&P 100 index as he marke index. Excep for marke skewness, he marke volailiy and he firm s own volailiy are also included in he above equaion. Dennis and Mayhew (00) find ha he ATM implied volailiy of he S&P500 index opion is negaively relaed wih he firm s risk-neural skewness, while he ATM implied volailiy of he firm s own sock opion is posiively relaed wih he firm s risk-neural skewness. Their findings imply ha he firm s risk-neural skewness is more negaive when he marke 9

10 volailiy is high and when he firm s own volailiy is low. We es he relaions by including boh he ATM opion implied volailiy of he S&P100 index and he ATM opion implied volailiy of he firm s sock as independen variables and expec he relaions o have he same sign. Apar from he variables connecing he firms and he marke, we also include he firm size, which helps ensure ha we do no aribue more explanaory power o oher variables han is appropriae, and he book-o-marke raio. In Chen, Hong and Sein (001) s es, book-o-marke raio is posiively relaed wih he nex period s condiional skewness. Their explanaion is he sochasic bubble model suggesed by Blanchard and Wason (198). A low book-o-marke raio implies ha he bubble has been building up for a long ime. When he bubble pops, he low-probabiliy even migh produce large negaive reurns. The leverage effec [Black (1976)] is a popular explanaion of he asymmery in sock reurn disribuions. I indicaes ha a drop in sock price raises he firm s leverage raio and, as a resul, he firm s sock becomes riskier. However, he resuled higher risk level does no necessarily relae wih a more negaive slope of he implied volailiy curve. Therefore, i is difficul o say wheher he relaion beween he riskneural skewness and he firm s leverage raio is posiive or negaive. The posiive relaion, found by Dennis and Mayhew (00) is consisen wih he empirical resuls ha he risk-neural densiy of individual socks is less negaively skewed han he index. We include he firm s leverage raio and es wheher firms wih more leverage end o have more or less risk-neural skewness. 10

11 Anoher variable ha has been examined by previous sudies is he proxy for marke senimen or rading pressure. When he marke is pessimisic, people migh expec he sock price o decline and hus he shape of he fuure reurn disribuion will appear o be lef skewed. The raio of pu-o-call rading volume is commonly believed o be a senimen index and Pan and Poeshman (006) show ha he raio consruced from reliable daa is negaively associaed wih fuure sock reurns. Dennis and Mayhew (00) do no find evidence ha he raio can explain he movemens of risk-neural skewness. We include he raio of pu-o-all rading opion volume and expec ha a higher demand for pu opions, compared o ha for calls, indicaes pessimism and implies a more negaively skewed risk-neural disribuion. The relaion beween informaion asymmery and he risk-neural disribuion has no been invesigaed in previous relaed lieraure. Invesors wih privae informaion in he sock marke can make profis by rading wih hose wihou. Easley, O Hara and Srinivas (1998) and Bardong, Barram and Yadav (006) find ha U.S. firms wih opions raded have lower informaion asymmery a he sock marke han hose wihou. The informaion asymmery should be able o influence he opion prices and hus he opion implied risk-neural disribuion. Boh French and Roll (1986) and Bardong, Barram and Yadav (006) show ha a higher informaion asymmery is relaed wih an increase in he firms risk-neural volailiy. Unforunaely, here is no heoreical research ha provides he relaion beween informaion asymmery and risk-neural skewness. We conduc he empirical ess ha may laer moivae fuure heoreical research. If a firm has more privae informaion, hen he expecaions of he fuure sock price from boh informed and 11

12 uninformed invesors will be dispersed, because hey are using differen informaion. When hese invesors paricipae in he opion marke, hey will rade opions a a variey of srike prices based on heir own expecaions. Therefore, he opion implied risk-neural densiy migh hen appear o be more volaile and more symmeric. As, a mos imes, he risk-neural skewness of individual firms is negaive, we expec a higher informaion asymmery is associaed wih less negaive risk-neural skewness. The measure of informaion asymmery documened by Naik and Yadav (003) and Bardong, Barram and Yadav (006) is adoped in our sudy. The asymmeric volailiy phenomenon (hereafer AVP) refers o he fac ha negaive reurn shocks end o imply a higher volailiy han do posiive reurn shocks of he same magniude [Nelson (1991)]. In he heory of sochasic volailiy models, he sock price S and is variance V = σ follow a pair of diffusion equaions in he real world: ds = µ S d + σ S dw and dv = α d + η dz. ρ refers o he correlaion beween he volailiy shocks and price shocks in boh he real-world and he risk-neural processes. Taylor and Xu (1994) derive ha, when ρ 0, he opion implied volailiy is approximaely a quadraic funcion of he opion s moneyness and he minimum implied volailiy does no occur a he forward price. This heoreical relaion implies ha he AVP, proxied by realisic negaive values of ρ, can influence he slope of he implied volailiy curve, which reflecs he risk-neural skewness. Based on he fuures prices of he S&P500 index. 1

13 Taylor (005) shows ha he slope of he implied volailiy curve, calculaed from he opion pricing formula of Heson (1993), is more negaive when ρ is more negaive, According o prior heoreical work, Harvey and Siddque (1999) have noed a link beween negaive condiional skewness and he AVP. Blair, Poon and Taylor (00) and Dennis, Mayhew and Sivers (006) documen ha he AVP is sronger for he sock index han for individual firms. Dennis, Mayhew and Sivers (006) furher show ha his index versus firms difference in he AVP is consisen wih he index versus firms differences of he slopes of heir implied volailiy curves. I is hus plausible o sugges ha firms wih a sronger AVP end o exhibi a more negaive slope on heir implied volailiy curves. Therefore, we use he asymmeric volailiy raio ha is defined by he GJR (1,1) model [Glosen, Jagannahan and Runkle (1993)] as he proxy for AVP. 3 Spanning and pricing risk-neural skewness The BKM (003) mehod o find risk-neural skewness and kurosis is moivaed by a heorem oulined in Bakshi and Madan (000). Le S () refer o he sock price a ime. For any claim payoff, H [ S( + )], ha is inegrable under risk-neural pricing, he risk-neural expecaion of i a ime + is: E Q 0 { H S( + )]} = H[ S( + )] q[ S( + ) [ ] ds (1), Q where E {}. refers o he risk-neural expecaion and q [ S( + )] is he risk-neural densiy of S a ime +. 13

14 Bakshi and Madan (000) show ha any payoff funcion wih bounded expecaion can be spanned by a coninuum of ou-of-he-money (OTM) European call and pu opion prices. They derive he arbirage-free price of he claim a ime as: E Q r { e H[ S] } = ( H[ S ] SH + S H SS S [ S ]) e r + H [ K] C(, ; K) dk + S 0 S H [ S ] S( ) SS [ K] P(, ; K) dk (), where r refers o he ineres rae, H S [S] and H SS [S] are he firs-order and secondorder derivaives of he payoff wih respecive o S evaluaed a any seleced number S. C(, ; K) and P(, ; K) are respecively he European call and pu opion prices a ime wih srike price K and expiry dae +. BKM (003) define he volailiy conrac, cubic conrac and quaric conrac o have he payoffs respecively equal o R (, ), 3 R (, ) and 4 R (, ), where he - period sock reurn is defined as: R(, ) log( S( + )) log( S( )). The prices of hese hree conracs a ime are expressed respecively by Q r E { e R(, ) } Q r, W (, ) E { e R(, ) } 3 Q r and X (, ) E { e R(, ) } 4 V (, ). BKM (003) derive he value of V (, ), W (, ) and X (, ) when leing H [S] in Equaions (1) and () be equal o R (, ), 3 R (, ) and R (, ) 4. For he choice S = S(), hey are: K (1 log[ ]) S( ) V (, ) = Q(, ; K) dk (3), K 0 14

15 15 = 0 ) ;, ( ]) ) ( 3(log[ ] ) ( 6log[ ), ( dk K Q K S K S K W (4), = 0 3 ) ;, ( ]) ) ( 4(log[ ]) ) ( 1(log[ ), ( dk K Q K S K S K X (5), where ) ;, ( K Q is he call opion price wih srike price K when ) S( K > and oherwise i is he pu opion price. Therefore, he value of each of hese hree conracs can be expressed by a porfolio of OTM opion prices. By Theorem (1) in BKM (003), he skewness of he risk-neural disribuion of ) log(s a ime +, which is ), ( SKEW, can be recovered from he above equaions, such ha: { } { } ] ), ( ), ( [ ), ( ), ( ), ( 3 ), ( )]), ( [ ), ( ( )]), ( [ ), ( ( ), ( µ µ µ V e V e W e R E R E R E R E SKEW r r r Q Q Q Q + = (6), wih ), ( 4 ), ( 6 ), ( 1 ) ( ) ( log ), ( µ X e W e V e e S S E r r r r Q +. This mehod shows ha he risk-neural skewness for a fuure ime can be calculaed from a coninuum of curren opion prices wih he same mauriy. The mehod has been adoped, a leas, by BKM (003), Dennis and Mayhew (00), Duan and Wei (006), and Chrisoffersen, Jacob and Vainberg (006) in measuring risk-neural momens of individual socks. 4 Daa

16 Our sample includes 149 U.S. firms 1 wih opions lised on he CBOE and ranges from Jan 1996 o Dec Daily opion daa including boh prices and rading volume for boh firms and he S&P100 index are from he IvyDB Daabase provided by Opion Merics. Opions wih less han seven days o mauriy are excluded. For mos rading days, we choose he neares-o-mauriy opions. Daily daa of he underlying socks, including rading volume, closing price and shares ousanding in he marke all comes from CRSP. Firm s financial reporing informaion used o esimae leverage and book-o-marke raio is from Compusa. The calculaion of informaion asymmery requires high-frequency sock daa, which are obained from TAQ. All variables, including risk-neural skewness, are esimaed daily and hen averaged o obain weekly measures. 4.1 Consrucion of explanaory variables Daily rading volume of underlying sock measured in shares raded is colleced direcly from CRSP. Firm size equals he firm s marke capializaion, calculaed as he daily closing sock price muliplied by he shares ousanding in he marke. In order o eliminae he effecs from exremely high or low volume and size, we use he naural log of he firm s rading volume in housands of shares and he naural log of firm size in housands of dollars. The indicaor of marke senimen is esimaed as he rading volume of pu opions divided by he rading volume of all opions, where opion rading volume is proxied by he number of raded conracs. In robusness ess, he raio of daily pu open 1 The selecion crieria of firms are same as in Taylor, Yadav and Zhang (007). We swich o he second-neares-o-mauriy opions when here are only a few observaions for neares-o-mauriy opions. 16

17 ineress o daily overall open ineress is used. Consisen wih he variable of sock rading volume, we use he naural log of daily opion rading volume as he indicaor of opion liquidiy. The firm s a-he-money implied volailiy is he average of he implied volailiies for he pu and call opions whose srike prices are closes o he sock price. As for he marke volailiy, we adop he volailiy index on he CBOE, VOX, which is he average of he volailiies implied by eigh neares-o-he-money and neares-o-mauriy opions 3. I represens he volailiy level of he S&P 100 index wih days o mauriy. The hisorical daily level of VOX index are downloaded from he CBOE s websie. The sysemaic risk proporion, defined by Duan and Wei (006), is he raio of he firm s sysemaic variance over he oal variance. For sock i, i can be viewed as he R of he OLS regression: R i = α + β R + ε (7) i i m i where R i and R m refer o he sock i s reurn and he marke reurn a ime. Following Duan and Wei (006), we run he regression in Equaion (7) for day using daily sock reurns from day 50 o day wih he S&P100 index as a proxy for he marke reurn. All reurns are compued as coninuously compounded The measure of informaion asymmery, presened by Naik and Yadav (003), calculaes, for each ransacion, he gain or loss of a rader, who deals wih a marke maker, and correcly or incorrecly anicipaes he direcion of he movemen of he 3 For consisency, we also compue he a-he-money opion implied volailiy for he S&P100 index in he same way as we calculae he implied volailiy for individual firms bu find no significan difference in regression resuls. 17

18 sock price. For a ransacion of sock i a ime ' 4, he informaion asymmery measure IA i' equals: IA i' = D i' M i( ' + ) M M i' i' (8), where D is a direcion indicaor being +1 for a buy and 1 for a sell, i' M i' is he mid-quoe corresponding o a ransacion of sock i a ime ' and M i( ' + ) is he mid-quoe minues afer he reference rade. From he definiion, a higher informaion asymmery will lead o a higher value (or more posiive value of IA ). The daily measure is calculaed as he average of all ransacions wihin he day. Bardong, Barram and Yadav (006) have esed various ime inervals,, and boh ransacion-size weighed and equally weighed when averaging o compue he daily measure and find all generae consisen resuls. We choose equal o 15 minues and daily measure equal o he equally weighed average of all ransacions wihin he day 5. To calculae he asymmeric volailiy raio, we use he GJR (1,1)-GARCH model ha incorporaes he asymmeric effec of posiive and negaive reurns in he real world. Based on 1009 daily sock reurns from Jan 1996 o Dec 1999, we esimae he parameers of he following equaions once for each firm: r = µ + ε + θε ε = h i = ϖ + αε h z, 1 z 1, ~ i. i. d. N(0,1), + α s ε βh 1 (9), 4 ' refers o a ime wihin rading day. 5 We are graeful o Florian Bardong for sharing he sofware used o calculae he informaion asymmery. 18

19 by maximizing he log-likelihood value, where s 1 is 1 if ε 1 < 0, and is 0 oherwise. From he definiion of he model, α and α + α measure he respecive effecs of posiive and negaive shocks on he nex-period condiional variance. Following Blair, Poon and Taylor (00), we define he asymmeric volailiy raio for firm i as: α A i = (10). α + α Therefore, a more pronounced AVP is consisen wih a lower value of A i. Table 1 conains he summary saisics of he explanaory variables described above. For our daa, on average here is less rading of pu opions han of call opions, as he mean pu-o-all rading volume, 0.319, is less han 50%. The average daily rading volume of underlying sock during our sample period across all firms is 94, ( e 1,000) shares and he average firm size is 5.8 billion ( e 1,000) dollars. The average level of he sysemaic risk proporion, which is 18.%, shows ha mos risk of our sample firms comes from he firm specific componen raher han he marke componen. Leverage lower han 50% means on average ha our firms are financed more by equiy han by deb. All weekly measures of informaion asymmery are posiive, indicaing ha insider profis exis in our sample firms. However, he average magniude of 1.8 basis poins is less han has appeared in Bardong, Barram and Yadav (006), where he mean of he informaion asymmery is 58 basic poins for abou 000 socks in heir sample. Since our firms on average are larger han heirs, our values of informaion asymmery are consisen wih heir findings ha larger firms have less informaion asymmery. 19

20 4. Measuring risk-neural skewness To empirically esimae he risk-neural skewness described in Equaions (6), we need o evaluae he inegrals ha appear in Equaion (3), (4) and (5). In order o reduce he errors coming from discree opion prices, we esimae implied volailiy curves from small ses of observed opion prices and hen exrac more opion prices from i. We implemen a variaion of he pracical sraegy described by Malz (1997a, 1997b), who proposed esimaing he implied volailiy curve as a quadraic funcion of he Black-Scholes opion s dela; previously a quadraic funcion of he srike price had been suggesed by Shimko (1993). As saed by Malz (1997a), making implied volailiy a funcion of dela, raher han of he srike price, has he advanage ha he away-from-he-money implied volailiies are grouped more closely ogeher han he near-he-money implied volailiies. Also, exrapolaing a funcion of dela provides sensible limis for he magniudes of he implied volailiies. The quadraic specificaion is chosen because i is he simples funcion ha capures he basic properies of he volailiy smile. Furhermore, here are insufficien sock opion prices o esimae higher-order polynomials. Dela is defined here as he firs derivaive of he Black-Scholes call opion price wih respec o he underlying forward price, wih a consan volailiy level ha permis a convenien one-o-one mapping beween dela and he srike price. Following Bliss and Panigirzoglou (00, 004), he consan volailiy level is se as he volailiy implied by he opion observaion whose srike price is neares o he forward price. 0

21 We use he implied volailiy of he observed opions provided by he IvyDB direcly. The quadraic funcion is fied by minimizing he sum of weighed squared errors beween he observed and he fied implied volailiies. The weigh of [dela * (1- dela)] ensure ha mos weigh is given o near-he-money opions. Inroducing weighs reduces he impac from any ouliers of far-from-he-money opions, which are he mos suscepible o non-synchroniciy errors. For each rading day, we exrac 1000 opion prices from he esimaed implied r e volailiy curve wih equal space in dela ranging from 1001 o r e If, following he above procedures, he lowes (highes) pu (call) opion price is sill higher han cens, we exrapolae opion prices by assuming a consan implied volailiy level and keep on reducing (increasing) moneyness, defined as he srike price divided by he forward price, by 0.01 each ime unil he minimum opion prices reach cens. However, such an exrapolaion is no ofen necessary for our daa, as almos always he exreme OTM opion prices afer inerpolaion are already oo small o have any effec on he inegral funcions. Daily risk-neural skewness for 149 firms and he S&P100 index are esimaed according o Equaions (6). Each weekly esimae is he average of daily esimaes. We also calculae he weekly esimae as he median value of daily esimaes bu he differences are small. For a few rading days, he marke opion prices imply firs order arbirage opporuniies, which means he call (or pu) opion prices are no monoonically decreasing (increasing) wih srike prices. These rading days are no included when calculaing weekly risk-neural momens. 1

22 Table provides he summary saisics of he esimaed risk-neural skewness, he auocorrelaions in skewness a lag 1 o 5 and he number of firms where Ljung-Box Q-saisics a ha specific lag are significan a he 1% level. We also sor he firms according o differen indusry secors 6 and repor he summary saisics for he firms belonging o hese secors. We find ha, consisen wih previous lieraure, our risk-neural skewness of individual firms are negaive overall, wih he mean of There is occasionally posiive skewness. Alhough we include only relaively large firms, heir risk-neural disribuion appears o be differen from ha of he sock index, which is always negaively skewed and wih a higher magniude [Dennis and Mayhew (00), BKM (003), Chrisoffersen, Jacobs and Vainberg (006)]. Secondly, he skewness shows high persisence over ime for our daa. The average auocorrelaion is a lag 1 and hen decreases monoonically from lag 1 o lag 5. These indicae ha he period of a negaive skewness ends o be followed by a period ha also has a negaive skewness. Finally, he summary saisics of risk-neural skewness for firms in differen indusry secors are close o each oher. Figure 1 plos he ime series of risk-neural skewness for boh individual firms and he S&P100 index. For each ou of 09 weeks over he four years, we calculae he median value of risk-neural skewness across 149 firms. The figure shows ha he median values of risk-neural skewness are always negaive during our sample period, ranging from 0.05 o The risk-neural skewness of he S&P100 index is 6 The definiions of indusry secors are from Professor Kenneh French s websie: hp://mba.uck.darmouh.edu/pages/faculy/ken.french/daa_library.hml, when all U.S. firms are separaed ino five main indusry secors.

23 nearly always below he median values of individual firms. The ime-series mean skewness of he S&P100 index is Table 3 presens he correlaions of all he explanaory variables used in he following regressions wih he esimaed risk-neural skewness. I appears ha opion liquidiy, underlying sock liquidiy and sysemaic risk proporion are he mos imporan variables in explaining risk-neural skewness. The marke skewness has some influences on he individual skewness, wih correlaion of 10%. The marke volailiy also has some effec on he firm s risk-neural skewness. When he marke volailiy is high, he firm s risk-neural skewness ends o be more negaive. I is no surprising o find ha he opion rading volume, sock rading volume, firm size and sysemaic risk proporion have high posiive correlaions wih each oher. Firsly, larger firms end o have more liquid opion rading and are more correlaed wih marke movemens. A he same ime, firms wih higher marke values of equiy are normally acively raded by he marke. We also find ha he informaion asymmery is negaively relaed wih firm size and posiively relaed wih book-omarke raio and he firm s volailiy. This is consisen wih he findings in Bardong, Barram and Yadav (006). 5 Regression analysis and he resuls In his secion, we show he regression resuls when using firm specific facors o explain he dynamics in risk-neural skewness. The firs subsecion inroduces wo regression specificaions ha are used for our analysis. One is he Fama-Macbeh 3

24 (1973) ype of cross-secional regressions; he oher is he ime-series cross-secional pooled regression. The resuls of univariae regressions are presened in he second subsecion. From univariae regressions, we can find he unique effec of each facor on he risk-neural skewness. The resuls of mulivariae regressions es relaive effecs form our main resuls and are presened in he subsequen subsecions. 5.1 Regression specificaions Our analysis sars from he Fama-MacBeh (1973) ype of cross-secional approach. Each week, we run he following mulivariae regression and he univariae special cases across 149 firms: SKEW i = β 0 + β 1 PUT/ALL i +β TV_OP i +β3 TV_STOCK i +β 4 SIZE i + β 5 SRP i + β 6 VOL i +β 7 LEVERAGE i +β8 B/M +β 9 i IA i 10 + β A i + ε i (11), where SKEW i is he risk-neural skewness for firm i ; PUT/ALL i is he raio of pu o all opion rading volume; TV_OP i is he opion rading volume; TV_STOCK i is he rading volume of underlying socks; SIZE i is he marke value of firm i s equiy; SRP i is he sysemaic risk proporion; VOL i is he ATM opion implied volailiy of he firm s sock; LEVERAGE i is he leverage raio; B/M i is he book-o-marke raio; IA i is he measure of informaion asymmery; A i is he real-world asymmeric volailiy raio. These regressions invesigae he cross-secional relaions beween he dependen and he independen variables and generae ime-series coefficiens hroughou 09 weeks during our sample period. For each weekly regression, he significance of he 4

25 esimaed slope coefficien is esed using he Whie (1980) -saisic aking accoun of heeroscedasiciy. The averages of hese weekly coefficiens are presened for each variable and he null hypohesis ha he mean slope coefficien over ime equals zero is esed by he -saisic adjused for he auocorrelaions in he weekly coefficiens up o he 10 h lag. Secondly, we run he pooled regressions, which es he cross-secional and he imevarying relaions simulaneously beween risk-neural skewness and various firm specific facors. The mulivariae model specificaion is as follows: SKEW = β + PUT/ALL i, 0 β 1 i, + β TV_OP i, + 3 β TV_STOCK i, β 5 SRP i, + β 6 VOL i, + β 7 LEVERAGE i, + β 8 B/M i, + β 9 IA β SIZE i, i, + β10 A i + β 11 SKEW_M + β 1 VOX_M+ ε i, (1), where i indexes for firm and indexes for week of he observaion. In hese pooled regressions, we add wo marke-wide variables, where SKEW_M refers o he riskneural skewness of he marke index a ime and VOX_M refers o he CBOE s volailiy index, VOX, on he S&P100 index a ime. The number of weekly observaions for our sample is 31,141. The hypohesis ess are based on he heeroscedasiciy and auocorrelaion consisen sandard errors presened by Newey and Wes (1987). 5. Resuls of univariae regressions Table 4 shows he univariae regression resuls of boh regression specificaions described in he las subsecion. The lef four columns are he resuls for cross- 5

26 secional univariae regressions, defined from Equaion (11). The mean coefficiens are he ime-series averages of he coefficiens obained from weekly cross-secional regressions. The null hypohesis ha he ime-series mean of weekly coefficiens equals zero is esed using he -saisics shown in he parenheses. The column labelled % -sa n/p couns he percenages of weeks when he coefficien of he explanaory variable is negaively (n)/posiively (p) significanly differen from zero a he 5% level, based on he Whie -saisics. The mean R and mean adj. R are he ime-series averages of he R values from weekly cross-secional regressions. The regressions wih opion rading volume, firm size and underlying sock rading volume generae he highes average values of R, compared wih oher regressions. Moreover, he coefficiens of opion rading volume and sock rading volume are negaively significan a he 5% level for respecively 58.0% and 60.3% ou of 09 weeks bu boh have never been posiively significan. The resuls indicae ha firms wih higher rading volume of opions, higher rading volume of underlying socks and/or higher marke value of equiy, compared o oher firms, end o have more negaive skewness. The las hree columns on he righ in Table 4 show he coefficien esimaes and explanaory powers of he univariae pooled regressions, defined in Equaion (1). The Newey-Wes -saisics, aking accoun of boh heeroscedasiciy and auocorrelaion, are presened in he parenheses. The sign and significance of he esimaed coefficien for he firm specific variables are, in general, similar o he ime-series averages of he cross-secional coefficiens repored in he lef hand side of he able. The R values are slighly lower han he mean R values in he cross- 6

27 secional regressions for mos variables. The sysemaic risk proporion has a higher explanaory power han firm size in he pooled regressions and becomes he hird mos imporan among all variables. The risk-neural skewness of he S&P100 index is posiively relaed wih ha of he individual firms, wih he coefficien esimae equal o The volailiy index of he S&P100 index, VOX, is also significanly relaed wih he firms risk-neural skewness bu wih a negaive coefficien equal o 0.9. From he cross-secional regressions and he pooled regressions, all he explanaory variables are significan in explaining he movemens in risk-neural skewness a a univariae level. However, as here is collineariy beween many pairs of explanaory variables, i is difficul o sae more conclusions abou he effecs coming from hese variables, solely based on he univariae regression resuls. 5.3 Resuls of mulivariae regressions Resuls of cross-secional regressions Table 5 presens he resuls of he mulivariae cross-secional regression defined in Equaion (11). The regression is run once a week and he means of he weekly coefficien esimaes are shown. Iniially all variables are including, labelled Model I in Table 5. From he regression resuls, firsly we find ha he coefficien on pu o all rading volume of opions, PUT/ALL, is negaive and significan. This is consisen wih our hypohesis ha when invesors are pessimisic and rade more on 7

28 pu opions relaive o he overall opion rading volume, he probabiliy of a lower price level on he risk-neural disribuion migh be driven up. However, based on he same hypohesis, Dennis and Mayhew (00) do no find any consisen evidence for heir daa. One possible reason is ha heir sample is oo large and he values of he variable, measured as he pu rading volume over call rading volume, is volaile so ha heir regression resuls migh be influenced by some exreme values. Secondly, we find a negaive and significan relaion beween he opions rading volume and he risk-neural skewness of individual firms. The index opions are much more liquid han opions on individual firms. If he index is viewed as a firm wih he highes rading volume in opions, our resuls are consisen wih he empirical findings ha he opion implied risk-neural disribuion of index reurns is much more negaively skewed han ha of individual firms. Thirdly, in univariae regressions presened in Table 4, he negaive mean coefficien on sock rading volume indicaes ha firms wih more acively raded socks end o have more negaive skewness. However, in he encompassing regression, sock rading volume loses is significance a almos all levels. I is possible ha he informaion provided by i are all subsumed by oher variables ha have high correlaions wih sock rading volume bu which remain significan in mulivariae regression. To assess he collineariy effecs, we esimae he regression again by omiing firm size. The parameer esimaes are repored in he wo middle columns of Table 5, as Model II. When firm size is ignored, here is almos no change o he coefficiens for he oher variables excep ha sock rading volume now becomes significan a very low levels ( = -3.9). 8

29 The sysemaic risk proporion, as expeced, is negaively and significanly relaed wih risk-neural skewness. Firms ha conain a higher proporion of sysemaic risk wihin heir overall risk end o exhibi more negaive risk-neural skewness. The resuls are consisen wih Duan and Wei (006) and Dennis and Mayhew (00), while he laer uses bea as he proxy of sysemaic risk. Our mean coefficien of 0.1 is much smaller han he coefficien of Duan and Wei (006) s cross-secional regressions, which is This is perhaps because hey use he S&P 500 index as he proxy for he marke porfolio and, heir sample is from Jan 1991 o Dec 1995 and conains he 30 U.S. firms wih highes marke capializaions and he S&P100 index. The risk-neural skewness of heir firms is overall more negaive han ours. Also consisen wih he finding of Dennis and Mayhew (00), we find he coefficien on deb-o-equiy raio, D/E, is posiive and significan. Therefore, he leverage effec can no be used o explain he relaion beween leverage and riskneural skewness. One possible explanaion of our posiive coefficien is ha firms wih a symmeric risk-neural disribuion or even posiive risk-neural skewness are able o ake more deb in heir capial srucure. The mean coefficien on he informaion asymmery measure, IA, is negaive and significan in he mulivariae regression. In boh he correlaion analysis in Table 3 and he univariae regression resuls in Table 4, he relaion beween i and he riskneural skewness appear o be posiive. The negaive coefficien migh be a resul of he negaive correlaions beween he informaion asymmery measure and he opion rading volume, sock rading volume, firm size and sysemaic risk proporion. All 9

30 hese values are always posiive according o heir definiions and he negaive correlaions beween hem and IA are all below 30%. To assess his negaive colineariy problem, we esimae he regression again by omiing he four covarying variables, which are opion rading volume, underlying sock rading volume, firm size and sysemaic risk proporion. The coefficien esimaes are presened in he las wo columns in Table 5, as Model III. As expeced, he mean coefficien esimae of IA is posiively significan afer he variables negaively correlaed wih i are dropped. The adjused explanaory power, 3.36%, is much lower han before, because he four imporan variables are no included in he regressions. The resuls imply ha when he firm conains more insider informaion in he underlying marke, he firm s risk-neural disribuion ends o be more symmeric or more posiively skewed. However, afer conrolling for opion and sock rading volume, firm size and sysemaic risk proporion, he coefficien becomes negaive and significan. We also find ha he coefficien esimae of booko-marke raio becomes significan when hese four variables are omied. In he allinclusive regression, informaion provided by book-o-marke raio is subsumed by he oher explanaory variables. The asymmeric volailiy raio, A, is posiively relaed wih he risk-neural skewness. A higher value of A implies a less pronounced asymmeric volailiy phenomenon. So our finding is consisen wih he hypohesis ha, when he effecs coming from negaive shocks of sock reurns are small relaive o ha from posiive shocks in he real world, he risk-neural disribuion ends o be more posiively skewed or more symmeric. 30

31 Overall he percenages of weeks when each coefficien esimae is negaively or posiively significan are low in Table 5. This migh be caused by he relaively small number of observaions each week, which is 149 firms, compared o he number of explanaory variables in he regression model. Resuls of pooled regressions The firs column of Table 6 shows he resuls of he mulivariae pooled regression I, defined by Equaion (1). The numbers below each coefficien esimae are he Newey-Wes -saisics. The F-saisic ess he null hypohesis ha he all coefficiens of he explanaory variables in he regression model are zero. In he pooled regression, we add wo marke-relaed variables, marke skewness and marke volailiy, o capure he ime-series properies of risk-neural skewness. Nearly all he coefficien esimaes of regression I have a similar magniude o he average coefficiens obained from he cross-secional approach in Table 5. The adjused R is 7.75% and he null hypohesis ha all coefficien esimaes are equal o zero is srongly rejeced according o he F-saisic. Consisen wih he crosssecional regression resuls, he coefficiens of sock rading volume, book-o-marke raio and asymmeric volailiy raio are no significan a he 5% level. For regression II in Table 6, when we omi firm size, he coefficien of sock rading volume becomes negaive and significan. The asymmeric volailiy raio is no significan probably because we fix he measure for each firm hroughou he sample period. 31

32 In regression I in Table 6, he risk-neural skewness of he S&P100 index, which is viewed as he marke skewness by us, posiively and significanly helps o explain individual firm skewness over ime, as expeced. The marke volailiy is negaively significan, indicaing he individual risk-neural skewness ends o be more negaive when he overall marke is more volaile. The firm s ATM implied volailiy is also significanly and negaively relaed wih he risk-neural skewness. This resul is differen from ha of Dennis and Mayhew (00) and hus solves he puzzle in heir paper ha he individual risk-neural skewness has a conflicing relaionship beween he marke volailiy and he firm s own volailiy. Our resuls sugges ha when he marke volailiy or/and he firm s volailiy is high, he individual risk-neural skewness ends o be more negaive. I is found again ha he informaion asymmery measure appears o be negaively relaed wih risk-neural skewness. The reason, as discussed before, lies in he srong negaive correlaions beween i and some oher explanaory variables, which are opion rading volume, sock rading volume, firm size and sysemaic risk proporion. Therefore, in regression III of Table 6, we show he resuls of he pooled regression when omiing hose four variables and find ha he coefficien esimae of he informaion asymmery measure changes sign and is significan in explaining riskneural skewness a low levels. I is also ineresing o isolae he effecs coming from he marke skewness and hose from he firms hemselves and es which is more imporan. Dennis and Mayhew (00) prove ha he risk-neural skewness of he marke index explains some of he ime-series variaion in individual skewness bu is much less imporan han he firm 3

33 specific facors. For regression IV shown in Table 6, we drop he marke variables, which are he risk-neural skewness of he S&P100 index and he volailiy index of he S&P100 index. Comparing he resuls wih hose in regression I, here is no much difference in he sign and significance of all he oher explanaory variables. The adjused R is 6.89%, which is only 0.86% lower han ha of regression I. Therefore, he risk-neural skewness and he opion implied volailiy of he S&P100 index capures only a small proporion of he ime-series variaion in he risk-neural skewness of individual firms. From he summary saisics presened in Table, we find ha he firm s risk-neural skewness has high auocorrelaions a he firs few lags. Therefore, we add he lagged risk-neural skewness as an addiional independen variable and show he regression resuls in he fifh column in Table 6. The coefficien of he lagged skewness in regression V is posiive and highly significan. The adjused R increases o 1.09%, which is abou hree imes he adjused R of regression I. This is consisen wih he resuls of Dennis and Mayhew (00) ha also find he inclusion of lagged skewness improves he explanaory power of heir regression model grealy. Their explanaion of he highly significan coefficien on lagged skewness is ha he lagged esimaes subsume he omied firm specific facors. Anoher possible reason, which maybe more credible, is he overlapping problem exising in boh heir and our samples. The coefficien on lagged skewness for our sample is 0.38 in regression V, which is abou a half of ha in Dennis and Mayhew (00). In heir sample, hey fix he risk- 33