The Factor Structure in Equity Options

Transcription

1 The Facor Srucure in Equiy Opions Peer Chrisoffersen Mahieu Fournier Kris Jacobs Universiy of Torono Universiy of Torono Universiy of Houson CBS and CREATES and Tilburg Universiy June 27, 213 Absrac Principal componen analysis of equiy opions on Dow-Jones firms reveals a srong facor srucure. The firs principal componen explains 77% of he variaion in he equiy volailiy level, 77% of he variaion in he equiy opion skew, and 6% of he implied volailiy erm srucure across equiies. Furhermore, he firs principal componen has a 92% correlaion wih S&P5 index opion volailiy, a 64% correlaion wih he index opion skew, and a 8% correlaion wih he index opion erm srucure. We develop an equiy opion valuaion model ha capures his facor srucure. The model allows for sochasic volailiy in he marke reurn and also in he idiosyncraic par of firm reurns. The model predics ha firms wih higher beas have higher implied volailiies, and seeper moneyness and erm srucure slopes. We provide a racable approach for esimaing he model on a large se of index and equiy opion daa on which he model provides a good fi. The equiy opion daa suppor he cross-secional implicaions of he esimaed model. JEL Classificaion: G1; G12; G13. Keywords: Facor models; equiy opions; implied volailiy; opion-implied bea. For helpful commens we hank Yakov Amihud, Menachem Brenner, George Consaninides, Redouane Elkamhi, Rob Engle, Bruno Feunou, Jean-Sebasien Fonaine, Jose Fajardo, Joel Hasbrouck, Jens Jackwerh, Bryan Kelly, Ralph Koijen, Markus Leippold, Dilip Madan, Mahew Richardson, Sijn Van Nieuwerburgh, Jason Wei, Alan Whie, Rober Whielaw, Dacheng Xiu, and seminar paricipans a New York Universiy (Sern, Universiy of Chicago (Booh, Universiy of Houson (Bauer, Universiy of Torono (Roman, Universiy of Zurich, as well as conference paricipans a IFM2, IFSID, NFA, OpionMerics, and SoFiE. Chrisoffersen graefully acknowledges financial suppor from he Bank of Canada and SSHRC. Correspondence o: Peer Chrisoffersen, Roman School of Managemen, 15 S. George Sree Torono, Onario, Canada M5S 3E6. peer.chrisoffersen@roman.uorono.ca. 1

2 1 Inroducion In heir pah-breaking sudy, Black and Scholes (1973 show ha when valuing equiy opions in a consan volailiy CAPM seing, he bea of he sock does no maer. Consequenly, sandard equiy opion valuaion models make no aemp a modeling a facor srucure in he underlying equiies. Typically, a sochasic process is assumed for each underlying equiy price and he opion is priced on his sochasic process, ignoring any links he underlying equiy price may have wih oher equiy prices hrough common facors. Seminal papers in his vein include Wiggins (1987, Hull and Whie (1987, and Heson (1993, Bakshi, Cao and Chen (1997, and Baes (2, 28. We show ha in a CAPM seing wih sochasic volailiy, he bea does indeed maer for equiy opion prices. We find srong suppor for his facor srucure in a large-scale empirical invesigaion using equiy opion prices. When considering a single sock opion, ignoring an underlying facor srucure may be relaively harmless. However, in porfolio applicaions i is crucial o undersand links beween he underlying socks. Risk managers need o undersand he oal exposure o he underlying risk facors in a porfolio of socks and sock opions. Equiy porfolio managers who use equiy opions o hedge large downside moves in individual socks need o know heir overall marke exposure. Dispersion raders who sell (expensive index opions and buy (cheaper equiy opions need o undersand he marke exposure of individual equiy opions. See for example Driessen, Maenhou, and Vilkov (29 for evidence on he marke exposure of equiy opions. Our empirical analysis of more han hree quarers of a million index opion prices and 11 million equiy opion prices reveals a very srong facor srucure. We sudy hree characerisics of opion prices: shor-erm implied volailiy (IV levels, he slope of IV curves across opion moneyness, and he slope of IV curves across opion mauriy. Firs, we compue he daily ime series of implied volailiy levels (IV on he socks in he Dow Jones Indusrials Average and exrac heir principal componens. The firs common componen explains 77% of he cross-secional variaion in IV levels and he common componen has an 92% correlaion wih he shor-erm implied volailiy consruced from S&P 5 index opions. Shorerm equiy opion IV appears o be characerized by a common facor. Second, a principal componen analysis of equiy opion IV moneyness, known as he opion skew, reveals a significan common componen as well. 77% of he variaion in he skew across equiies is capured by he firs principal componen. Furhermore, his common componen has a correlaion of 64% wih he skew of marke index opions. Third, 6% of he variaion in he erm srucure of equiy IV is explained by he firs principal componen. This componen has a correlaion of 8% wih he IV erm slope from S&P 5 index opions. 2

3 We use he findings from he principal componen analysis as guidance o develop a srucural model of equiy opion prices ha incorporaes a marke facor srucure. In line wih well-known empirical facs in he lieraure on index opions (see for example Bakshi, Cao and Chen, 1997; Heson and Nandi, 2; Baes, 2; and Jones, 23, he model allows for mean-revering sochasic volailiy and correlaed shocks o reurns and volailiy. Moivaed by he principal componen analysis, we allow for idiosyncraic shocks o equiy reurns which also have mean-revering sochasic volailiy and a separae leverage effec. Individual equiy reurns are linked o he marke index using a sandard linear facor model wih a consan facor loading. The model belongs o he affi ne class, which yields closed-form opion pricing formulas. I can be exended o allow for marke-wide and idiosyncraic jumps. 1 The model has hree imporan cross-secional implicaions. Firs, i predics ha firms wih higher beas have higher implied volailiies, consisen wih he empirical findings in Duan and Wei (29. Second, i predics ha firms wih higher beas have seeper moneyness slopes. Third, higher bea firms are expeced o have a greaer posiive (negaive slope when he marke variance erm-srucure is upward (downward sloping. We develop a convenien approach o esimaing he model using opion daa. When esimaing he model on he firms in he Dow-Jones index, we find ha i provides a good fi o observed equiy opion prices, and he cross-secional implicaions of he model are suppored by he daa. While i is no he main focus of his paper, our model provides opion-implied esimaes of marke beas, which is a opic of recen ineres, sudied by for example Chang, Chrisoffersen, Jacobs, and Vainberg (212, and Buss and Vilkov (212. Muliple applicaions in asse pricing and corporae finance require esimaes of bea, such as cos of capial esimaion, performance evaluaion, porfolio selecion, and abnormal reurn measuremen. Our paper is also relaed o he recen empirical lieraure on equiy opions. Dennis and Mayhew (22 invesigae he relaionship beween firm characerisics and risk-neural skewness. Bakshi and Kapadia (23 invesigae he volailiy risk premium for equiy opions. Bakshi, Kapadia, and Madan (23 derive a skew law for individual socks, decomposing individual reurn skewness ino a sysemaic and idiosyncraic componen. They find ha individual firms display much less (negaive opion-implied skewness han he marke index. Bakshi, Cao, and Zhong (212 invesigae he performance of jump models for equiy opion valuaion. Engle and Figlewski (212 develop ime series models of implied volailiies and sudy heir correlaion dynamics. Kelly, Lusig and Van Nieuwerburgh (213 use he model in our paper o sudy he pricing of implici governmen guaranees o he banking secor. Carr and Madan (212 develop a Levy-based model 1 Pan (22, Broadie, Chernov, and Johannes (27, and Baes (28 among ohers have documened he imporance of modeling jumps in index opions. 3

4 wih facor srucure bu provide lile empirical evidence. Perhaps mos relevan for our work, Duan and Wei (29 demonsrae empirically ha sysemaic risk maers for he observed prices of equiy opions on he firm s sock. 2 Our paper is also relaed o recen heoreical advances. Mo and Wu (27 develop an inernaional CAPM model which has feaures similar o our model. Elkamhi and Ornhanalai (21 develop a bivariae discree-ime GARCH model o exrac he marke jump risk premia implici in individual equiy opion prices. Finally, Serban, Lehoczky, and Seppi (28 develop a non-affi ne model o invesigae he relaive pricing of index and equiy opions. The reminder of he paper is organized as follows. In Secion 2 we describe he daa se and presen he principal componens analysis. In Secion 3 we develop he heoreical model. Secion 4 highlighs a number of cross-secional implicaions of he model. In Secion 5 we esimae he model and invesigae is fi o observed index and equiy opion prices. Secion 6 concludes. The appendix conains he proofs of he proposiions. 2 Common Facors in Equiy Opion Prices In his secion we firs inroduce he daa se used in our sudy. We hen look for evidence of commonaliy in hree crucial feaures of he cross-secion of equiy opions: Implied volailiy levels, moneyness slopes (or skews, and volailiy erm srucures. We rely on a principal componen analysis (PCA of he firm-specific levels of shor-erm a-he-money implied volailiy (IV, he slope of IV wih respec o opion moneyness, and he slope of IV wih respec o opion mauriy. The resuls from his model-free invesigaion will help idenify desirable feaures of a facor model of equiy opion prices. 2.1 Daa We rely on end-of-day implied volailiy surface daa from OpionMerics saring on January 4, 1996 and ending on Ocober 29, 21. We use he S&P 5 index o proxy for he marke facor. For our sample of individual equiies, we choose he firms in he Dow Jones Indusrial Average index a he end of he sample. Of he 3 firms in he index we excluded Kraf Foods for which daa are no available hroughou he sample. The implied volailiy surfaces conain opions wih more han 3 days and less han 365 days o mauriy (DTM. We filer ou opions ha have moneyness (spo price over srike price less han 2 See also Goyal and Sareo (29, Vasquez (211, and Jones and Wang (212 for recen empirical work on equiy opion reurns. 4

5 .7 and larger han 1.3, hose ha do no saisfy he usual arbirage condiions, hose wih implied volailiy less han 5% and greaer han 15%, and hose for which he presen value of dividends is larger han 4% of he sock price. For each opion mauriy, ineres raes are esimaed by linear inerpolaion using zero coupon Treasury yields. Dividends are obained from OpionMerics and are assumed o be known during he life of each opion. For each opion we discoun fuure dividends from he curren spo price. The S&P 5 index opions are European, bu he individual equiy opions are American syle, and heir prices may be influenced by early exercise premiums. OpionMerics herefore uses binomial rees o compue implied volailiy for equiy opions. Using hese implied volailiies, we can rea all opions as European-syle in he analysis below. Table 1 presens he number of opion conracs, he number of calls and pus, he average days-o-mauriy, and he average implied volailiy. We have a oal of 775, 67 index opions and on average 758, 976 equiy opions per firm. The average implied volailiy for he marke is 2.65% during he sample period. Cisco has he highes average implied volailiy (4.68% while Johnson & Johnson has he lowes average implied volailiy (22.79%. Table 1 also shows ha he daa se is balanced wih respec o he number of calls and pus. Table 2 repors he average, minimum, and maximum implied volailiy, as well as he average opion vega. Noe ha he index opion vega is much higher han he equiy vegas simply because he S&P5 index values are much larger han he ypical sock price. Figure 1 plos he daily average shor-erm (3 < DTM < 6 a-he-money (.95 < S/K < 1.5 implied volailiy (IV for six firms (black lines as well as for he S&P 5 index (grey lines. Figure 1 shows ha he variaion in he shor-erm a-he-money (ATM equiy volailiy for each firm is highly relaed o S&P 5 volailiy. 2.2 Mehodology We wan o assess he exen o which he ime-varying volailiies of equiies share one or more common componens. In order o gauge he degree of commonaliy in risk-neural volailiies, we need daily esimaes of he level and slope of he implied volailiy curve, and of he slope of he erm srucure of implied volailiy for all firms and he index. For each day we run he following regression for firm j, IV j,l, = a j, + b j, (S j /K j,l + cj, (DT M j,l + ɛ j,l,, (2.1 where l denoes an opion available for firm j on day. The regressors are sandardized each day by subracing he mean and dividing by he sandard deviaion. We run he same regression on 5

6 index opion IVs. We inerpre a j, as a measure of he level of implied volailiies of firm j on day. Similarly, b j, capures he slope of implied volailiy curve while c j, proxies for he slope of he erm srucure of implied volailiy. Once he regression coeffi ciens have been esimaed on each day and for each firm, we run a PCA analysis on each of he marices {a j, }, {b j, }, and {c j, }. Tables 3-5 conain he resuls from he PCA analysis and Figures 2-4 plo he firs principal componen as well as he ime series of he corresponding index opion coeffi ciens, a I,, b I,, and c I, Common Facors in he Level of Implied Equiy Volailiy Table 3 conains he resuls for implied volailiy levels. We repor he loading of each equiy IV on he firs hree componens. A he boom of he able we show he average, minimum, and maximum loading across firms for each componen. We also repor he oal variaion capured as well as he correlaion of each componen wih S&P 5 IV. The resuls in Table 3 are quie sriking. The firs componen capures 77% of he oal cross-secional variaion in he level of IV and i has a 92% correlaion wih he S&P 5 index IV. This suggess ha he equiy IVs have a very srong common componen highly correlaed wih index opion IVs. Noe ha he loadings on he firs componen are posiive for all 29 firms, illusraing he pervasive naure of he common facor. The op panel of Figure 2 shows he ime series of IV levels for index opions. The boom panel plos he ime series of he firs PCA componen of equiy IV. The srong relaionship beween he wo series is readily apparen. The second PCA componen in Table 3 explains 13% of he oal variaion and he hird componen explains 2%. The average loadings on hese wo componens are close o zero and he loadings ake on a wide range of posiive and negaive values. The sizeable second PCA componen and he wide range of he loadings sugges he need for a second, firm-specific, source of variaion in equiy volailiy Common Facors in he Moneyness Slope Table 4 conains he resuls for IV moneyness slopes. The moneyness slopes conain a significan degree of co-movemen. The firs principal componen explains 77% of cross-secional variaion in he moneyness slope. The second and hird componens explain 6% and 4% respecively. The firs componen has posiive loadings on all 29 firms where as he second and hird componens have posiive and negaive loadings across firms, and average loadings very close o zero. Table 4 also shows ha he firs principal componen has a 64% correlaion wih he moneyness 6

7 slope of S&P 5 implied volailiy. Equiy moneyness slope dynamics clearly seem driven o a non-rivial exen by he marke moneyness slope. Figure 3 plos he S&P 5 index IV moneyness slope in he op panel and he firs principal componen from he equiy moneyness slopes in he boom panel. The relaionship beween he firs principal componen and he marke moneyness slope is readily apparen, bu no as srong as for he volailiy level in Figure Common Facors in he Term Srucure Slope Table 5 conains he resuls for IV erm srucure slopes. The variaion in he erm srucure slope capured by he firs principal componen is 6%, which is lower han for spo volailiy (Table 3 and he moneyness slope (Table 4. The loadings on he firs componen are posiive for all 29 firms. The correlaion beween he firs componen and he erm slope of S&P 5 index opion IV is 8%, which is higher han for he moneyness slope in Table 4 bu lower han for he variance level in Table 3. The second and hird componens capure 14% and 5% of he variaion respecively and he wide range of loadings on his facor sugges a scope for firm-specific variaion in he IV erm srucure for equiy opions. Figure 4 plos he S&P 5 index IV erm srucure slope in he op panel and he firs principal componen from he equiy erm slopes in he boom panel. Mos of he spikes in he S&P 5 erm srucure slope are clearly eviden in he firs principal componen as well. We conclude ha he marke volailiy erm srucure capures a subsanial share of he variaion in equiy volailiy erm srucures. 2.3 Oher Sylized Facs in he Cross-Secion of Equiy Opion Prices The lieraure on equiy opions has documened a number of imporan cross-secional sylized facs. Bakshi, Kapadia, and Madan (23 derive a skew law for individual socks, decomposing individual reurn skewness ino a sysemaic and idiosyncraic componen. They heoreically invesigae and empirically documen he relaionship beween risk-neural marke and equiy skewness, which affecs he relaionship beween he moneyness slope for equiy and index opions. They find ha he volailiy smile for he marke index is on average more negaively sloped han volailiy smiles for individual firms. They also show ha he more negaively skewed he risk-neural disribuion, he seeper he volailiy smile. Finally, hey find ha he risk-neural equiy disribuions are on average less skewed o he lef han index disribuions. Oher sudies documen cross-secional relaionships beween beas, esimaed using hisorical daa, and characerisics of he equiy IVs. Dennis and Mayhew (22 find ha opion-implied 7

8 skewness ends o be more negaive for socks wih larger beas. Duan and Wei (29 find ha he level of implied equiy volailiy is relaed o he sysemaic risk of he firm and ha he slope of he implied volailiy curve is relaed o sysemaic risk as well. Finally, Driessen, Maenhou, and Vilkov (29 find a large negaive index variance risk premium, bu find no evidence of a negaive risk premium on individual variance risk. These findings are a firs blush no direcly relaed o he findings of he PCA analysis above, which merely documens a srong facor srucure of various aspecs of implied equiy volailiies. We nex ouline a srucural equiy opion modeling approach wih a facor srucure ha capures he resuls from he PCA analysis oulined above, bu is also able o mach he cross-secional relaionships beween beas and implied volailiies documened by hese sudies. 3 Equiy Opion Valuaion Using a Single-Facor Srucure We model an equiy marke consising of n firms driven by a single marke facor, I. The individual sock prices are denoed by S j, for j = 1, 2,..., n. Invesors also have access o a risk-free bond which pays a reurn of r. The marke facor evolves according o he process di I = (r + µ I d + σ I, dw (I,1, (3.1 where µ I is he insananeous marke risk premium and where volailiy is sochasic and follows he sandard square roo process dσ 2 I, = κ I (θ I σ 2 I,d + δ I σ I, dw (I,2. (3.2 As in Heson (1993, θ I denoes he long-run variance, κ I capures he speed of mean reversion of σ 2 I, o θ I, and δ I measures volailiy of volailiy. The innovaions o he marke facor reurn and volailiy are correlaed wih coeffi cien ρ I. Convenional esimaes of ρ I are negaive and large capuring he so-called leverage effec in aggregae marke reurns. Individual equiy prices are driven by he marke facor as well as an idiosyncraic erm which also has sochasic volailiy ds j S j rd = α j d + β j ( di I rd + σ j, dw (j,1 (3.3 dσ 2 j, = κ j (θ j σ 2 j,d + δ j σ j, dw (j,2, (3.4 8

9 where α j denoes he excess reurn and β j is he marke bea of firm j. The innovaions o idiosyncraic reurns and volailiy are correlaed wih coeffi cien ρ j. As suggesed by he skew laws derived in Bakshi, Kapadia, and Madan (23, asymmery of he idiosyncraic reurn componen is required o explain he differences in he price srucure of individual equiy and index opions. Noe ha his model of he equiy marke has a oal of 2(n + 1 innovaions. 3.1 The Risk Neural Disribuion In order o use our model of he equiy marke o value derivaives we need o assume a change of measure from he physical (P disribuion developed above o he risk-neural (Q disribuion. Following he lieraure, we assume a change-of-measure of he exponenial form [ where W u W u (1,1, W u (1,2 [ γ u γ (1,1 u, γ (1,2 u,.., γ (I,1 u operaor. ( dq ( = exp γ dp u dw u 1 2,.., W (I,1 u, γ (I,2 ] u, W u (I,2 ] γ ud W, W γ u u (3.5 is a 2(n vecor conaining he innovaions, is he vecor of marke prices of risk, and d.,. is he covariance In he spiri of Cox, Ingersoll, and Ross (1985 and Heson (1993 among ohers, we assume a price of marke variance risk of he form λ I σ I,. We also assume ha idiosyncraic variance risk is no priced. These assumpions yield he following resul. Proposiion 1 Given he change-of-measure in (3.5 he process governing he marke facor under he Q-measure is given by di (I,1 = rd + σ I, d W (3.6 I dσ 2 I, = κ I ( θi σ 2 (I,2 I, d + δ I σ I, d W (3.7 wih κ I = κ I + δ I λ I, and θ I = κ Iθ I κ I, (3.8 and he processes governing he individual equiies under he Q-measure are given by ds j S j = rd + β j ( di ( dσ 2 j, = κ j θj σ 2 j, d + δj σ j, d I rd (j,1 + σ j, d W (3.9 W (j,2, (3.1 9

10 where d W denoes he risk-neural counerpar of dw for which where d W = dw + d W, W γ, (3.11 Proof. See Appendix A. γ (I,1 γ (j,1 = = µ I ρ I λ I σ 2 I, σ I, (1 ρ 2 I α j σ j, (1 ρ 2 j and γ (I,2 = λ Iσ 2 I, ρ Iµ I σ I, (1 ρ 2 I and γ(j,2 = ρ jα j σ j, (1 ρ 2 j. This proposiion provides several insighs. Noe ha he marke facor srucure is preserved under Q. Consequenly, he marke bea is he same under he risk-neural and physical disribuions. This is consisen wih Serban, Lehoczky, and Seppi (28, who documen ha he risk-neural and objecive beas are economically and saisically close for mos socks. Noe ha his resul makes beas esimaed from opion daa appropriae for applicaions of he CAPM such as capial budgeing. I is also imporan o noe ha in our modeling framework, higher momens and heir premiums, as defined by he difference beween he momen under P and Q, are affeced by he drif adjusmen in he variance processes. We will discuss his furher below. 3.2 Closed-Form Opion Valuaion The model has been cas in an affi ne framework, which implies ha he characerisic funcion for he logarihm of he index level and he logarihm of he equiy price can boh be derived analyically. The characerisic funcion for he index is idenical o ha in Heson (1993. Consider now individual equiy opions. We need he following proposiion: Proposiion 2 The risk-neural condiional characerisic funcion φ j (τ, u for he equiy price, S j T, is given by φ j (τ, u E Q [ ( ( ] exp iu ln S j T = ( S j iu exp ( iurτ (A(τ, u + B(τ, u C(τ, uσ 2 I, D(τ, uσ 2 j,, (3.12 where τ = T and he expressions for A (τ, u, B (τ, u, C (τ, u, and D (τ, u are provided in Appendix B. 1

11 Proof. See Appendix B. Given he characerisic funcion for he log spo price under Q, he price of a European equiy call opion wih srike price K and mauriy τ = T is C j (S j, K, τ = S j Π j 1 Ke rτ Π j 2, (3.13 where he risk-neural probabiliies Π j 1 and Π j 2 are defined by Π j 1 = e rτ πs j Π j 2 = π Re Re [ [ ] e iu ln K φj (τ, u i du (3.14 iu ] e iu ln K φj (τ, u du. (3.15 iu While hese inegrals mus be evaluaed numerically, hey are well-behaved and can be compued quickly. 4 Model Properies In his secion we derive a number of imporan cross-secional implicaions from he model and invesigae if he model capures he sylized facs documened in Secion 2. We will also draw some key implicaions of he model for opion risk managemen and for equiy opion expeced reurns. For convenience we assume ha bea is posiive for all firms below. This is no required by he model bu i simplifies he inerpreaion of cerain expressions. 4.1 The Level of Equiy Opion Volailiy Duan and Wei (29 show empirically ha firms wih higher sysemaic risk have a higher level of risk-neural variance. We now invesigae if our model is consisen wih his empirical finding. Firs, define oal spo variance for firm j a ime V j, β 2 jσ 2 I, + σ 2 j,, and define he expecaions under P and Q of he corresponding inegraed variance by E P [V j,:t ] E P [ T ] V j,s ds and E Q [V j,:t ] E Q [ T ] V j,s ds. 11

12 By decomposing he P -expecaion ino inegraed marke variance and idiosyncraic variance, we have E P [V j,:t ] = β 2 je P [σ 2 I,:T ] + E P [σ 2 j,:t ], where σ 2 I,:T, and σ2 j,:t correspond o he inegraed variances from o T. Given our model, he expecaion of he inegraed oal variance for equiy j under Q is E Q [V j,:t ] = β 2 je Q [σ 2 I,:T ] + E Q [σ 2 j,:t ] = β 2 je Q [σ 2 I,:T ] + E P [σ 2 j,:t ]. Noe ha he second equaion holds when idiosyncraic risk is no priced so ha E P [σ 2 j,:t ] = E Q [σ 2 j,:t ]. For any wo firms having he same level of expeced oal variance under he P -measure (E P [V 1,:T ] = E P [V 2,:T ] we have E P [σ 2 1,:T ] E P [σ 2 2,:T ] = (β 2 1 β 2 2E P [σ 2 I,:T ]. Therefore ( E Q [V 1,:T ] E Q [V 2,:T ] = (β 2 1 β 2 2E Q [σ 2 I,:T ] + E Q [σ 2 1,:T ] E Q [σ 2 2,:T ] = (β 2 1 β 2 2E Q [σ 2 I,:T ] + ( E P [σ 2 1,:T ] E P [σ 2 2,:T ] ( = (β 2 1 β 2 2 E Q [σ 2 I,:T ] E P [σ 2 I,:T ]. When he marke variance premium is negaive, we have θ I > θ I which implies ha E Q [σ 2 I,:T ] > E P [σ 2 I,:T ]. We herefore have ha β 1 > β 2 E Q [V 1,:T ] > E Q [V 2,:T ]. We conclude ha our model is consisen wih he finding in Duan and Wei (29 ha firms wih high beas end o have a high level of risk-neural variance. 4.2 Equiy Opion Skews To undersand he slope of equiy opion implied volailiy moneyness curves, we need o undersand how bea influences he skewness of he risk-neural equiy reurn disribuion. The nex proposiion is key o undersanding how bea, sysemaic risk, and index skewness impac equiy skewness. Proposiion 3 The condiional oal skewness of he inegraed reurns of firm j under P, denoed 12

13 by T Sk P j, is given by T Sk P j,:t Sk P ( T dsu j Su j = SkI P (A P 3/2 j,:t + Sk P j (1 A P 3/2 j,:t. (4.1 The condiional oal skewness of he inegraed reurns of firm j under Q, denoed by T Sk Q j, is given by where T Sk Q j,:t SkQ ( T ds j u S j u = Sk Q I A P j,:t EP [β 2 jσ 2 I,:T ] E P [V j,:t ] ( 3/2 ( 3/2 A Q j,:t + Sk Q j 1 A Q j,:t, (4.2 and A Q j,:t EQ [β 2 jσ 2 I,:T ] E Q [V j,:t ] ( T are he proporion of sysemaic risk of firm j under P and Q, and where Sk I = Sk Sk j = Sk ( T σ j,sdw s (j,1 Proof. See Appendix C. is he marke and idiosyncraic skewness, respecively. This resul shows ha β j maers for deermining firm j s condiional oal skewness. Equaion (4.2 shows ha under he risk neural measure, β j affecs he slope of he equiy implied volailiy curve hrough T Sk Q j,:t by influencing he sysemaic risk proporion, AQ j,:t. A higher AQ j,:t implies a higher loading on he marke risk-neural skewness Sk Q I. Consider wo firms wih he same expeced oal variance under Q and β 1 > β 2, which implies A Q 1,:T > AQ 2,:T. Firm 1 has a greaer loading on index risk-neural skewness han firm 2. When he index Q-disribuion is more negaively skewed han he idiosyncraic equiy disribuion, as found empirically in Bakshi, Kapadia, and Madan (23, we have he following cross-secional predicion: Higher-bea firms will have more negaively skewed Q-disribuions. di s I s and Noe ha his predicion is in line wih he cross-secional empirical findings of Duan and Wei (29 and Dennis and Mayhew (22. Figure 5 plos he implied Black-Scholes volailiy from model opion prices. Each line has a differen bea bu he same amoun of uncondiional oal equiy variance defined by Ṽj β 2 j θ I +θ j =.1. We se he curren spo variance o σ 2 I, =.1 and V j, =.5, and define he idiosyncraic variance as he residual σ 2 j, = V j, β 2 jσ 2 I,. The marke index parameers are κ I = 5, θ I =.4, δ I =.5, ρ I =.8, and he individual equiy parameers are κ j = 1, δ j =.4, and ρ j =. The risk-free rae is 4% per year and opion mauriy is 3 monhs. Figure 5 shows ha bea has a subsanial impac on he moneyness slope of equiy IV even when keeping he oal variance consan: The higher he bea, he larger he moneyness slope. 13

14 The facor srucure in he model also has implicaions for he relaive imporance of sysemaic risk under he wo measures. The model implies E Q [σ 2 I,:T ] > EP [σ 2 I,:T ] AQ j,:t > AP j,:t. (4.3 A negaive marke variance premium implies a greaer imporance of sysemaic risk under he Q measure han under he P measure. This suggess ha sysemaic risk will be of even greaer imporance for pricing opions han for explaining hisorical reurns. Sysemaic risk may herefore be helpful in explaining he co-movemens in he implied volailiy moneyness slopes for equiy opions documened in Secion The Term Srucure of Equiy Volailiy Our model implies he following wo-componen erm-srucure of equiy variance E Q [V j,:t ] = (β 2j θ I + θ j + β 2 j (σ 2I, θ I e κ I(T + ( σ 2 j, θ j e κ j (T. (4.4 This expression shows how he erm srucure of marke variance affecs he erm srucure of variance for firm j. Given differen sysemaic and idiosyncraic mean revering speeds ( κ I κ j, β j has imporan implicaions for he erm-srucure of volailiies. In he empirical work below, we find ha he idiosyncraic variance process is more persisen han he marke variance. When he idiosyncraic variance process is more persisen ( κ I > κ j, higher values of bea imply a faser reversion oward he uncondiional oal variance (Ṽj = β 2 j θ I + θ j. As a resul, when he marke variance process is less persisen han he idiosyncraic variance, firms wih higher beas are likely o have seeper volailiy erm-srucures. In oher words, higher bea firms are expeced o have a greaer posiive (negaive slope when he marke variance erm-srucure is upward (downward sloping. Figure 6 plos he implied Black-Scholes volailiy from model prices agains opion mauriy. Each line has a differen bea bu he same amoun of uncondiional oal equiy variance Ṽj = β 2 j θ I + θ j =.1. We se he curren spo variance o σ 2 I, =.1 and V j, =.5, and define he idiosyncraic variance as he residual σ 2 j, = V j, β 2 jσ 2 I,. The parameer values are as in Figure 5. Figure 6 shows ha bea has a non-rivial effec on he IV erm srucure: The higher he bea, he seeper he erm srucure when he erm srucure is upward sloping. In summary, our model suggess ha ceeris paribus firms wih higher beas should have higher levels of volailiy, seeper moneyness slopes, and higher absolue mauriy slopes. 14

15 4.4 Equiy Opion Risk Managemen In classic equiy opion valuaion models, parial derivaives are used o assess he sensiiviy of he opion price o he underlying sock price (dela and equiy variance (vega. In our model he equiy opion price addiionally is exposed o changes in he marke level and marke variance. Porfolio managers wih diversified equiy opion holdings need o know he sensiiviy of he equiy opion price o hese marke level variables in order o properly manage risk. The following proposiion provides he model s implicaions for he sensiiviy o he marke level and marke variance. Proposiion 4 For a derivaive conrac f j wrien on he sock price, S j, he sensiiviy of f j wih respec o he index level, I (he marke dela, is given by f j I = f j S j S j β I j. The sensiiviy of f j wih respec o he marke variance (he marke vega is given by f j σ 2 I, = f j V j, β 2 j. Proof. See Appendix D. This proposiion shows ha he bea of he firm in a sraighforward way provides he link beween he usual sock price dela f j and he marke dela, f j S j I, as well as he link beween he f usual equiy vega, j V j,, and he marke vega f j. σ 2 I, This resul allows marke paricipans wih porfolios of equiy opions on differen firms o measure and manage heir oal exposure o he index level and o he marke variance. I also allows invesors engaged in dispersion rading, who sell index opions and buy equiy opions, o measure and manage heir overall exposure o marke risk and marke variance risk. In Figure 7 we use he parameer values from Figure 5, and addiionally se S j /I =.1. We plo he marke dela (op panel and he marke vega (boom panel agains moneyness for firms wih differen beas. The op panel of Figure 7 shows ha he differences in marke delas across firms wih differen beas can be subsanial for ATM and ITM call opions. The boom panel of Figure 7 shows ha he differences in marke vega is also subsanial paricularly for ATM calls where he opion exposure o oal variance is he larges. 15

16 4.5 Expeced Reurns on Equiy Opions So far we have focused on opion prices. In applicaions such as he managemen of opion porfolios, opion reurns are of ineres as well. The following proposiion provides an expression for he expeced (P -measure equiy opion reurn as a funcion of he expeced marke reurn. 3 Proposiion 5 For a derivaive f j wrien on he sock price, S j, he expeced excess reurn on he derivaive conrac is given by: 1 d EP [ df j f j ] rd = f j S j ( f j S j S j αj + β f j j µ I = S j f α j j + f j I I f µ I, j where f j I is given by Proposiion 4. Proof. See Appendix E. The model hus decomposes he excess reurn on he opion ino wo pars: The dela of he equiy opion and he bea of he sock. Pu differenly, equiy opions provide invesors wih wo sources of leverage: Firs, he bea wih respec o he marke, and second, he elasiciy of he opion price wih respec o changes in he sock price. In Figure 8 we use he parameer values from Figure 5 and addiionally se he equiy marke risk premium, µ I =.75. We plo he expeced excess reurn on equiy call opions (op panel and on pu opions (boom panel in percen per day agains moneyness for firms wih differen beas. The op panel of Figure 8 shows ha he differences in expeced call reurns across firms wih differen beas can be subsanial for OTM calls where opion leverage in general is highes. The boom panel of Figure 8 shows ha pu opion expeced excess reurns (which are always negaive also vary mos across firms wih differen beas, when he pu opions are OTM. In general he differences in expeced excess reurns across beas are smaller for pu opions (boom panel han for call opions (op panel. 5 Esimaion and Fi In his secion, we firs describe our esimaion mehodology. Subsequenly we repor on parameer esimaes and model fi. Finally we relae he esimaed beas o paerns in observed equiy opion IVs. 3 Recen empirical work on equiy and index opion reurns includes Broadie, Chernov, and Johannes (29, Goyal and Sareo (29, Consaninides, Czerwonko, Jackwerh, and Perrakis (211, Vasquez (211, and Jones and Wang (

17 5.1 Esimaion Mehodology Several approaches have been proposed in he lieraure for esimaing sochasic volailiy models. Jacquier, Polson, and Rossi (1994 use Markov Chain Mone Carlo o esimae a discree-ime sochasic volailiy model. Pan (22 uses GMM o esimae he objecive and risk neural parameers from reurns and opion prices. Serban, Lehoczky, and Seppi s (28 esimaion sraegy is based on simulaed maximum likelihood using he EM algorihm and a paricle filer. Anoher approach reas he laen volailiy saes as parameers o be esimaed and hus avoids filering he laen volailiy facor. This sraegy has been adoped by Baes (2 and Sana-Clara and Yan (21 among ohers. We follow his srand of lieraure. Recall ha we need o esimae wo vecors of laen variables {σ 2 I,, σ2 j,} and wo ses of srucural parameers {Θ I, Θ j }, where Θ I { κ I, θ I, δ I, ρ I } and Θ j {κ j, θ j, δ j, ρ j, β j }. Our mehodology involves wo main seps. In he firs sep, we esimae he marke index dynamic { Θ I, σi,} 2 based on S&P 5 opion prices alone. In he second sep, we use equiy opions for firm j only, we ake he marke index dynamic as given, and we esimae he firm-specific dynamics { Θ j, σj,} 2 for each firm condiional on esimaes of { Θ I, σi,} 2. This sep-wise esimaion procedure is no fully economerically effi cien bu i enables us o esimae our model for 29 equiies while ensuring ha he same dynamic is imposed for he marke-wide index for each of he 29 firms. We have confirmed ha his esimaing echnique has good finie sample properies in a Mone Carlo sudy which is available from he auhors upon reques. Each of he wo main seps conains an ieraive procedure which we now describe in deail. Sep 1: Parameer Esimaion for he Index Given a se of saring values, Θ I, for he srucural parameers characerizing he index, we firs esimae he spo marke variance each day by solving ˆσ 2 I, = arg min σ 2 I, N I, (C I,,m C m (Θ I, σ 2 I, 2 /V ega 2 I,,m, for = 1, 2,..., T, (5.1 m=1 where C I,,m is he marke price of index opion conrac m on day, C m (Θ I, σ 2 I, is he model index opion price, N I, is he number of index conracs available on day, and V ega I,,m is he Black-Scholes sensiiviy of he index opion price wih respec o volailiy evaluaed a he implied volailiy. These vega-weighed dollar price errors are a good approximaion o implied volailiy 17

18 errors and he compuaional cos involved is much lower. 4 Once he se of T marke spo variances is obained, we solve for he se of parameers characerizing he index dynamic as follows ˆΘ I = arg min Θ I N I m, (C I,,m C m (Θ I, ˆσ 2 I, 2 /V ega 2 I,,m, (5.2 where N I T N I, represens he oal number of index opion conracs available. We ierae beween (5.1 and (5.2 unil he improvemen in fi is negligible, which ypically requires 5-1 ieraions. Sep 2: Parameer Esimaion for Individual Equiies Given an iniial value Θ j and he esimaed ˆσ 2 I, and ˆΘ I we can esimae he spo equiy variance each day by solving ˆσ 2 j, = arg min σ 2 j, N j, (C j,,m C m (Θ j, ˆΘ I, ˆσ 2 I,, σ 2 j, 2 /V ega 2 j,,m, for = 1, 2,...T, (5.3 m=1 where C j,,m is he price of equiy opion m for firm j wih price, C m (Θ j, Θ I, σ 2 I,, σ2 j, is he model equiy opion price, N j, is he number of equiy conracs available on day, and V ega j,,m is he Black-Scholes Vega of he equiy opion. Once he se of T marke spo variances is obained, we solve for he se of parameers characerizing he equiy dynamic as follows ˆΘ j = arg min Θ j N j (C j,,m C m (Θ j, ˆΘ I, ˆσ 2 I,, ˆσ 2 j,/v ega 2 j,,m, (5.4 m, where N j T N j, is he oal number of conracs available for securiy j. We again ierae beween (5.3 and (5.4 unil he improvemen in fi is negligible. We repea his esimaion procedure for each of he 29 firms in our daa se. 5.2 Parameer Esimaes This secion presens esimaion resuls for he marke index and he 29 firms for he period. In order o speed up esimaion, we resric aenion o pu opions wih moneyness in he 4 This approximaion has been used in Carr and Wu (27 and Trolle and Schwarz (29 among ohers. 18

19 range.9 S/K 1.1 and mauriies of 2, 4, and 6 monhs. We esimae he srucural parameers in he model on a panel daa se consising of he collecion of he firs Wednesday of each monh. We end up using a oal of 15, 455 equiy opions and 6, 147 index opions when esimaing he srucural parameers. We esimae he spo variances on each rading day hus using more han 3.1 million equiy opions and 128, 5 index opions. Table 6 repors esimaes of he srucural parameers ha characerize he dynamics of he sysemaic variance and he idiosyncraic variance, as well as esimaes of he beas. The op row shows esimaes for he S&P 5 index. The uncondiional risk-neural marke index variance θ I =.61 corresponds o 24.7% volail- iy per year. Based on he average index spo variance pah for he sample, 1 T T =1 σ2 I,, we obain a volailiy of 22.23%. The idiosyncraic θ j esimaes range from.18 for American Express o.586 for Cisco. The esimae of he mean-reversion parameer for he marke index variance κ I is equal o 1.13, which corresponds o a daily variance persisence of /365 =.9969 which is very high, consisen wih he exising lieraure. The idiosyncraic κ j range from.15 for Bank of America o 1.29 for Merck, indicaing ha idiosyncraic volailiy is highly persisen as well. Only five firms in he sample (JP Morgan, Hewle-Packard, Inel, IBM, and Merck have an idiosyncraic variance process ha is less persisen han he marke variance. The esimae of ρ I is srongly negaive (.855, capuring he so-called leverage effec in he index. The idiosyncraic ρ j are also generally negaive, ranging from.724 for Bank of America o for Exxon Mobil. The esimaes of bea are reasonable and vary from.7 for Johnson & Johnson o 1.24 for American Express. The average bea across he 29 firms is.99. The average oal spo volailiy (ATSV for firm j is compued as ATSV = 1 T T V j, = 1 T =1 T =1 ( β 2 jσ 2 I, + σ2 j,. Comparing he bea column wih he ATSV column in Table 6 shows ha ATSV is generally high when bea is high. The final column of Table 6 repors he sysemaic risk raio (SSR for each firm. I is compued from he spo variances as follows SSR = T =1 β2 jσ 2 I, T ( =1 β 2 jσ 2 I, +. σ2 j, Table 6 shows ha he sysemaic risk raio varies from close o % for Hewle-Packard o above 19

20 7% for Exxon Mobile. The sysemaic risk raio is 46% on average, indicaing ha he esimaed facor srucure is srongly presen in he equiy opion daa. Comparison of he bea column wih he SSR column in Table 6 shows ha firms wih similar beas can have radically differen SSR and, vice versa, firms wih very differen beas can have roughly similar SSRs. This finding of course suggess a key role for he idiosyncraic variance dynamic in he model. 5.3 Model Fi We measure model fi using he roo mean squared error (RMSE based on he vegas, which is consisen wih he crierion funcion used in esimaion 1 N Vega RMSE N (C m, C m, (Θ 2 /V ega 2 m,. m, We also repor he implied volailiy RMSE defined as IVRMSE 1 N N (IV m, IV (C m, (Θ 2, m, where IV m, denoes marke IV for opion m on day and IV (C m, (Θ denoes model IV. We use Black-Scholes o compue IV for boh model and marke prices. Table 7 repors model fi for he marke index and for each of he 29 firms. We repor resuls for all conracs, as well as separae resuls for in- and ou-of-he-money pus, and for 2-monh and 6-monhs a-he-money (ATM conracs. We also repor he IVRMSE divided by he average marke IV in order o assess relaive IV fi. Several ineresing findings emerge from Table 7. Firs, he Vega RMSE approximaes he IVMRSE closely for he index and for all firms. This suggess ha using Vega RMSE in esimaion does no bias he IVRMSE resuls. Second, he average IVRMSE across firms is 1.2% and he relaive IV (IVRMSE / Average IV is 4.5% on average. The fi does no vary much across firms. Overall he fi of he model is hus quie good across firms. The bes pricing performance for equiy opions is obained for Coca Cola wih an IVRMSE of.95%. The wors fi is for General Elecric wih an IVRMSE of 1.64%. Based on he relaive IVRMSE, he bes fi is for Inel wih 2.9% and he wors is again for GE wih 5.66%. Third, he average IVRMSE fi across firms for ITM pus is 1.17% and for OTM pus i is 1.23%. Using his meric he model fis ITM and OTM pus roughly equally well. 2

21 Fourh, he average IVRMSE fi across firms for 2-monh ATM opions is 1.1% and for 6- monh ATM opions i is 1.8%. The model hus fis 2-monh and 6-monh ATM opions equally well on average. Figure 9 repors he marke IV (solid and model IV (dashed averaged over ime for differen moneyness caegories for each firm. The black lines (lef axis show he average on days wih above-average IV and he grey lines (righ axis show he average for days wih below-average IV. Moneyness is on he horizonal axis, measured by S/K, so ha and ITM pus are shown on he lef side and OTM pus are shown on he righ side. Figure 9.A repors on he firs 15 firms and Figure 9.B repors on he remaining 14 firms as well as he index. Noe ha in order o properly see he differen paerns across firms, he verical axis scale differs in each subplo, bu he range of implied volailiy values is kep fixed a 1% across firms o faciliae comparisons. Figure 9 shows ha he smiles compued using marke prices vary considerably across firms, boh in erms of level and shape. I is noeworhy ha for many of hese large firms, he smile looks more like an asymmeric smirk especially on low-volailiy days (grey lines. The smirk is of course a srong sylized fac for index opions and i is eviden in he boom-righ panel of Figure 9.B. The IV bias by moneyness are small in general across firms and no large ouliers are apparen. The model ends o slighly underprice OTM equiy pus when volailiy is high (black lines. This is no he case when volailiy is low (grey lines. The boom righ panel in Figure 9.B confirms he finding in Bakshi, Kapadia and Madan (23 ha he marke index is generally more (negaively skewed han individual firms. The boom righ panel also shows ha he model requires addiional negaive skewness o fi he relaively expensive OTM pus rading on he marke index. This can be achieved by including jumps in reurns (Baes, 2. Noe ha when allowing for a large negaive ρ I he Heson (1993 model is able o fi OTM index pu opions quie well. Figure 1 repors for each firm he average (over ime implied volailiy as a funcion of ime o mauriy (in years. We spli he daa se ino wo groups: Days wih upward-sloping IV erm srucure and days wih downward-sloping IV erm srucure. We hen compue he median slope on he upward-sloping days and he median slope on he downward-sloping days. In Figure 1 we repor he average marke IVs (solid lines as well as he average model IVs (dashed lines on he days wih higher-han-median upward-sloping erm srucure (grey lines and on he days wih lower-han-median downward-sloping erm srucure (black lines. This is done because on many days he erm srucure is roughly fla and so unineresing. The downward-sloping black lines use he lef axis and he upward-sloping grey lines use he righ axis. In order o faciliae comparison beween model and marke IVs he level of IVs differ beween he lef and righ axis and hey differ 21

22 across firms. For ease of comparison beween erm srucures he difference beween he minimum and maximum on each axis is fixed a 1% across all firms. Figure 1 shows ha he erm srucure of IV differs considerably across firms. Some firms such as Hewle-Packard ends o mean-rever raher quickly, whereas oher firms such as 3M have much more persisen erm srucures. Generally, across firms, he downward sloping black lines appear o be seeper han he upward sloping grey lines. This paern is mached by he model. I is also worh noing ha he model is able o capure he srong persisence in IV quie well: Figure 1 does no reveal any sysemaic model biases in he erm srucure of IVs. The wo-facor sochasic volailiy srucure of our equiy model is clearly helpful in his regard. We conclude from Table 7 and Figures 9 and 1 ha he model fis he observed equiy opion daa quie well. Encouraged by his finding, we nex analyze in some deail how he esimaed beas are relaed o observed paerns in equiy opion IVs. 5.4 Equiy Beas and Equiy Opion IVs The hree main cross-secional predicions of our model, as discussed in Secion 4, are as follows: 1. Firms wih higher beas have higher risk-neural variance. 2. Firms wih higher beas have seeper moneyness slopes. This is equivalen o saing ha firms wih higher beas are characerized by more negaive skewness. 3. Firms wih higher beas have seeper posiive volailiy erm srucures when he erm srucure is upward sloping, and seeper negaive volailiy erm srucures when he erm srucure is downward sloping. We now documen if hese heoreical model implicaions are suppored by he esimaes for he 29 Dow-Jones firms. Consider firs he level of opion-implied volailiy. In he op panel of Figure 11, we scaer plo he ime-averaged inerceps from he implied volailiy regression in (2.1, 1 T T =1 a j, agains he bea esimae from Table 6 for each firm j. We hen run a regression on he 29 poins in he scaer and assess he significance and fi. The slope has a -saisic of 6.81 and he regression fi (R 2 is quie high a 63%. The regression line shows he posiive relaionship beween he esimaed beas and he average implied volailiy observed in he marke prices of equiy opions. In he middle panel of Figure 11, we scaer plo he moneyness slope coeffi ciens from he IV regression in (2.1, 1 T T =1 b j, agains he bea esimae from Table 6 for each firm j. In he moneyness slope regression, he sensiiviy o bea has a -saisic of 4.66 and an R 2 of 45%. The 22

23 middle panel of Figure 11 clearly shows ha higher bea esimaes are associaed wih seeper slopes of he IV moneyness smile. Finally, in he boom panel of Figure 11 we scaer plo he absolue value of he erm srucure slope coeffi ciens from (2.1, 1 T T =1 c j, agains he bea esimae from Table 6 for each firm. In he erm slope regression, he sensiiviy o bea has a -saisic of 4.9 and he R 2 is 47%. Panel C shows ha higher beas are associaed wih higher absolue slopes of he erm srucure in equiy IVs: Firms wih high beas will end o have a erm srucure of implied volailiy curve ha decays more quickly o he uncondiional level of volailiy compared wih firms wih low beas. We conclude ha our esimaes of bea are relaed o he model-free measures of IV level, slope, and erm srucure in a way ha is consisen wih he hree main model predicions from Secion Opion-Implied and Hisorical Beas As discussed in secion 5.2, he esimaed beas seem reasonable. They vary from.7 for Johnson & Johnson o 1.24 for American Express and he average bea across he 29 firms is.99. To provide addiional perspecive we also compue hisorical beas for he same 29 firms. To be consisen wih he opion-based esimae, we esimae a consan bea using daily reurn daa for he enire sample from 1996 o 21. The hisorical bea is.97 on average across firms. Figure 12 provides a scaer plo of he opion-implied beas versus hisorical beas. I also shows he resuls of a regression of he hisorical on he opion-implied beas. A number of imporan conclusions obain. Firs, he opion-implied beas are posiively correlaed wih he hisorical beas. In fac, he relaion beween he wo bea esimaes is very srong, which is evidenced by he high R-square of he regression (84% and he fac ha Figure 12 conains very few ouliers. Second, opion-implied beas have a smaller dispersion (15% han hisorical beas (31%. This is ineresing in ligh of he well-known saisical biases in esimaing hisorical beas, and he common pracice of shrinking he beas oward one o accoun for his bias. Noe ha his larger dispersion of he hisorical beas yields a regression slope larger han one and a negaive regression inercep when regressing hisorical bea on opion implied bea. We conclude ha overall he relaionship beween hisorical and opion-implied bea is surprisingly srong. I may prove ineresing o see if his relaionship also holds for beas compued over shorer windows. We leave ha for fuure work. 23