RISK-ADJUSTED STOCK INFORMATION FROM OPTION PRICES

Transcription

1 RISK-ADJUSTED STOCK INFORMATION FROM OPTION PRICES Ren-Raw Chen, Dongcheol Kim, and Durga Panda* As opion s payoff depends upon fuure sock price, opion prices conain imporan informaion of heir underlying socks. In his paper, we price opions wih he physical measure where we can joinly esimae expeced sock reurn and implied volailiy from marke prices of opions. Using S&P 500 Index opions, we discover he following four resuls. Firs, our resul shows ha invesors have higher expecaions of sock reurns in he shor-erm, bu lower expecaions in he long-erm. Second, he erm srucure of volailiies in our model is much flaer han he erm srucure of he Black-Scholes model. Third, he empirical invesigaion shows ha a combinaion of our implied expeced reurn and implied volailiy wih Black-Scholes implied sandard deviaion provides a beer model han Black-Scholes implied sandard deviaion alone o forecas fuure volailiy of socks for any combinaion of moneyness and mauriy. Finally, he implied volailiy of our model can predic much beer fuure volailiy over he life of he opion han he implied volailiy of he Black-Scholes model, more so for shor mauriies of 90 days or less. As opion s payoff depends upon fuure sock price, opion prices conain imporan informaion of heir underlying socks. For a bullish sock, he price of he call goes up and he pu goes down. However, using he Black- Scholes model, we can only rerieve he volailiy informaion, as risk preference disappears from he pricing model. In his paper, we price opions wih he physical measure where we can joinly esimae he expeced reurn µ and implied volailiy of he underlying sock from marke prices of opions. Pricing measures are no unique. Ye he law of one price (known as no arbirage) guaranees all pricing measures lead o a unique opion price. As a resul, here exiss a pricing measure where µ is presen and he same opion price is obained. In his paper, we choose he physical measure o price opions so ha we *Ren-Raw Chen (he corresponding auhor) is a professor in he Deparmen of Finance, Graduae School of Business Adminisraion, Fordham Universiy. rchen@fordham.edu. Dongcheol Kim is a professor of finance in he Korea Universiy Business School, Korea Universiy. kimdc@korea.ac.kr. Durga Panda, is an adjunc insrucor in he Deparmen of Mahemaical Sciences, New Jersey Insiue of Technology. pandadurgap@yahoo.com. Acknowledgemens: We wish o hank Kose John, C.F. Lee, Oded Palmon, and he 009 Financial Managemen Associaion Meeings paricipans for heir helpful commens and suggesions. We hank he Whicomb Financial Cener, Rugers Business School, Rugers Universiy, for daa assisance. All errors are our own responsibiliy. Keywords: implied reurn, implied cos of equiy, risk-adjused pricing JEL Classificaion: G11, G1, G13

2 108 Review of Fuures Markes can joinly esimae he expeced reurn and implied volailiy of he underlying sock. The use of he physical measure in pricing asses has been he sandard mehodology in microeconomic heories. In fac, he earlier lieraure, such as Sprenkle (1961) and Samuelson (1965), in opion pricing used he physical measure o price opions. Our conribuion is o exend hose models and furher derive he closed form soluion o he expeced reurn of he opion as a funcion of he expeced reurn of he sock. Black and Scholes (1973) show ha if he marke is complee, 1 hen he expeced reurn of he sock should disappear from he valuaion of he opion as dynamic hedging (known as coninuous rebalancing, price by no arbirage, or risk neural pricing) should effecively remove he dependence of he opion price on he sock reurn. This is rue, however, only if he marke is ruly complee in realiy. In oher words, if he realiy were exacly described by he Black-Scholes model, i is impossible o heoreically solve for boh he expeced reurn and implied volailiy of he sock. However, i has been empirically shown ha he Black-Scholes model canno explain all opion prices (known as he volailiy smile and volailiy erm srucure). As a resul, we can solve for hese wo parameers simulaneously under our model. Excep for he expeced reurn parameer, he physical pricing measure adoped by our model makes he same assumpions of he Black-Scholes model. In paricular, we assume he same sock price process as he Black-Scholes model does. This design is o assure ha we have a closed form soluion o our model. In heory, we could relax as many assumpions by he Black-Scholes model as possible and build a model ha can explain every raded opion price in he marke place. However, in doing so, we shall lose he closed form soluion; furhermore once we have as many parameers as he number of he raded opions, he model can no longer price any opion as all opion prices are used o calculae parameers. As a resul, we need o seek balance beween over-parameerizaion (having same number of parameers as opion prices), under-parameerizaion (such as he Black-Scholes model), and compuaional feasibiliy (mainaining closed form soluion). As we shall show in our empirical sudy, wih wo parameers (expeced reurn and implied volailiy), we find ha our model can predic hisorical volailiy much beer han he Black-Scholes model. We use he erm hisorical volailiy for he ex-pos volailiy measured over he life of he opion as explained in secion III.C.1. Opion pricing models of Sprenkle (1961), Ayres (1963), Boness (1964), and Samuelson (1965) employed he physical measure and implicily or explicily assumed some form of risk-adjused model such ha he invesors buy and hold he opions unil mauriy o exrac he opion implied reurn, which hen could be linked o he sock reurn. However, none of hese models provides an adequae heoreical srucure o deermine he implied reurn values. Under he risk neural pricing 1. This is complee marke in he dynamic sense, as laer described carefully by Duffie and Huang (1985).. Galai (1978) laer showed ha he Boness model and he Black-Scholes model are consisen.

3 Risk-Adjused Sock Informaion 109 measure, Heson (1993b) shows ha, under a log-gamma dynamic assumpion for he sock price, he expeced sock reurn will show up in he pricing formula and ye he volailiy disappears. Hence, his model is no capable of joinly deermining boh he expeced reurn and volailiy of he sock price. Noneheless, Heson s paper shows he possibiliy of reaining he expeced reurn parameer in he model wih suiable adjusmens o he pricing equaion. Using he S&P 500 index call and pu opions, we esimae expeced sock reurn and implied volailiy wih our model. We use opions wih various srikes a a given day and compue expeced reurn and volailiy for each ime o mauriy. As a resul, we obain joinly he erm srucure of expeced reurn and he erm srucure of implied volailiy of he sock. We find a downward sloping erm srucure of expeced reurn ha is consisen wih exising sudies o be reviewed in deails laer in he empirical secion. We find ha implied volailiy carries more informaion in predicing hisorical volailiy of he sock han he Black-Scholes implied volailiy. The remainder of his paper is organized as follows. Secion I presens he risk-adjused discree ime model ha reains he sock expeced reurn in he opion pricing equaion. Secion II presens he daa and esimaion mehodology. Secion III discusses he empirical resuls of our esimaion. Secion IV provides he concluding remarks. I. THE MODEL I is well known ha he Black-Scholes model can be used o compue implied volailiy and no implied expeced reurn of he underlying sock, due o he fac ha no-arbirage argumen renders a preference-free model and hence conains no such parameer. In his sub-secion, we demonsrae ha such parameer can be rediscovered via an equilibrium pricing approach similar o Samuelson (1965) and Sprenkle (1961). Le he sock price follow he usual log normal process under he physical measure: ds S = µ d + σdw (1) where he annualized insananeous expeced reurn is µ and he volailiy is σ. The classical economic valuaion heory saes ha any price oday mus be a properly discouned fuure payoff: C = E [ M C ] () T, T where M,T is he pricing kernel, also known as he marginal rae of subsiuion, beween ime and ime T. The usual risk neural pricing heory developed by Cox and Ross (1976) performs he following change of measure:

4 110 Review of Fuures Markes C = E [ M C ] T, T Q T, T = E [ M ] E [ C ] rt ( ) Q e E [ CT ] if ineres rae is consan = FT ( ) PT, E [ CT] if ineres rae follows a random process (3) where Q represens he risk neural measure and F(T) represens he T -mauriy forward measure and P,T is he risk-free zero coupon bond price of $1 paid a ime T. 3 In his paper, we perform he change of measure in he oher direcion. Tha is: C = E [ M C ] T, T X T T, = EC [ ] E [ M ] kt ( ) = EC [ ] e T (4) where X represens he measure where he opion price serves as a numeraire, and k is he annualized expeced insananeous reurn on his opion in he physical world. We hen assume ha he X -measure expecaion of he pricing kernel akes a form of coninuous discouning. Now, we can derive our opion pricing formula as: kt ( ) = [max{ T,0}] kt ( ) = e STφ( ST) dst K φ( ST) dst K K ( µ k)( T ) kt ( ) = e SN( h1) e KN( h) C e E S K (5) where and T are he curren ime and mauriy ime of he opion, and K is he srike price of he opion and ( µ σ ) lns lnk + + ½ ( T ) h1 = σ T h = h 1 σ T To derive a pricing formula ha conains µ, we need he following proposiions. These proposiions describe how implied reurn and volailiy can be simulaneously esimaed from opion prices. Proposiion 1. Assume sock price S follows a geomeric Brownian moion wih an annualized expeced insananeous reurn of µ and volailiy of σ. Le a call opion on he sock a any poin in ime is given by C(S,) ha maures a ime T. Le k be he annualized expeced insananeous reurn on his opion. Then for a small inerval of ime, he relaionship beween µ and k can be given by: 3. Noe ha he las line of he equaion is a classical resul of economic valuaion.

5 Risk-Adjused Sock Informaion 111 k = r + βµ ( r) (6) where cov( rc, rs) β = (7) var( r ) and r S = S / S and r C = C / C are wo random variables represening he sock reurn and call opion reurn respecively during he period. And r is he annualized consan risk-free rae for he period of he opion. Proposiion 1 can be proved wihou assuming he CAPM. (Proof: See Appendix A.1.1.) 4 Equaion (6) holds for a small inerval of ime. We assume he disribuions of sock reurn r S and opion reurn r C are saionary over he period of he opion. This implies he annualized insananeous expeced reurn and variance over a small inerval of ime and he annualized insananeous expeced reurn and variance over he discree ime (from ime and ime T ) will be he same. This also implies β is consan over his period, which means he linear relaionship beween k and µ as in equaion (6) is valid over he life of he opion from curren ime o mauriy ime T. 5 Since our approach will be pricing he opion in a discree seing, we approximae he β over he discree ime from o T as: S β TK,, CT S cov T, C cov T, S S = = ST C var( ST) var S ( C S ) T (7a) And wih he assumpion of saionariy as above, equaion (6) holds for he life of he opion as: k = r + β ( µ r ) T, T, TK,, T, T, (6a) Equaion (6) wih equaion (7), in coninuous ime, and equaion (6a) wih equaion (7a) in discree ime can also be proved using he CAPM. However, for hese wo equaions o hold i is no necessary ha he CAPM should hold. The assumpions of he CAPM are much sronger so ha all reurn disribuions are saionary; however, here we need only he saionariy of he sock and he opion reurn o obain hese wo equaions. Hence saionariy assumpion of r S and r C is a weaker assumpion han wha is needed for CAPM. Furher, Galai (1978) shows many similariies beween he coninuous ime and discree ime properies of r C ha suppor our assumpion of saionariy of disribuion. 4. Appendix A..1 provides a similar derivaion for pu opions. 5. This can be easily seen by inegraing boh side of equaion (6) from o T.

6 11 Review of Fuures Markes Equaion (5) is obained based on he assumpion ha he expeced reurn of he opion k, expeced reurn of he sock µ, and volailiy σ are consans. We have also assumed ha he sock price follows a geomeric Brownian moion. In discree ime, equaion (5) can be wrien as: ( µ k )( T ) k ( T ) T,, K 1 T, T, T, C e S N h e KN h = ( ) ( ) (5a) where and T are he curren ime and mauriy ime of he opion, and K is he srike price of he opion and ( µ, σ, ) ln S ln K + T + ½ T ( T ) h1 = σt, T h = h1 σt, T Combining equaions (7a) and (5a), we arrive a he following proposiion. Proposiion. The β,t,k, based on he life of he opion, can be wrien as: β T,, K = ( T ) K σ µ ( T ), T, T S e N( h ) e 3 ( N( h ) N( h ) 1 ) N( h ) 1 S σt, ( T ) C e 1 (8) where (Proof: See Appendix A.1..) 6 h 3 ( µ ½σ ) lns ln K + = T, + 1 T, ( T ) σ T T, I should be noed ha we do no use he disribuional properies of he marke reurn r M o obain (8). Using (8), (5a), and (6a) we can solve for he call price C,T,K explicily in erms of he known values: sock price (S ), srike price (K), risk free rae (r), ime-o-mauriy (T ), and wo imporan unknown parameers: expeced sock reurn µ,t and volailiy σ,t. 7 If we observe he values of wo or more call opions, wih same ime-o-mauriy wih differen srike prices, we can 6. Appendix A.. provides he corresponding derivaion for pu opions. 7. µ,t and σ represen expeced sock reurn (µ) and implied volailiy of he sock (σ), respecively,,t for a specific ime period, where is he dae of observaion of opion prices, and T is he mauriy dae of he opions.

7 Risk-Adjused Sock Informaion 113 hen simulaneously solve for µ,t and σ,t. 8 In his paper we use sigma, implied volailiy, and σ,t inerchangeably. II. DATA AND ESTIMATION METHODOLOGY A. Daa To exrac implied expeced reurn from opion prices, we use he end-of-day OpionMerics daa of opions on S&P 500 (SPX) for he las business day of every monh during January 1996 April 006. This daa file conains he end-of-day sock CUSIP, srike price, offer, bid, volume, open ineres, days-o-mauriy, and Black- Scholes implied volailiy for each opion. From his daase, we exclude all pu opions and opions wih zero rading volume. We also exclude single opion records for a paricular rade dae and days-o-mauriy. 9 We obain daily levels of he index and reurns from CRSP. We need he reurns for hisorical volailiy compuaion. To mach he CRSP records wih opion records, we use he rade dae and CUSIP of he index. In our daa all S&P500 records have a common CUSIP. Merging CRSP and opion daa by rade dae and CUSIP can be used for any sock opion in general. For he ineres raes, we use he S. Louis Fed s 3-monhs, 6-monhs, 1-year, -year, 3-year, and 5-year Treasury Consan Mauriy Raes. Assuming a sepfuncion of ineres raes, we mach he days-o-mauriy in he opion record wih is corresponding consan mauriy rae. For example, if he days-o-mauriy of he opion is less han or equal o 3 monhs we use 3-monhs raes, and if he dayso-mauriy is beween 3 and 6 monhs, we use he 6-monhs rae and so on. In his paper, he resuls are based on he las business day observaions for each calendar monh. This resuls in 14 monhs, 791 differen rade dae and mauriies combinaions (on average 6.38 mauriies per monh), and a oal of 7,865 opions (9.94 differen moneyness levels per rade dae and mauriy combinaion). Taking any oher day of he monh produces similar resuls. For example, we verified our resuls by aking firs working day, second Thursday, and hird Friday of every monh. The resuls are similar. Table 1 shows he summary saisics of all moneyness S&P500 index call opion inpu daa ha are used o compue µ,t and σ,t Using opions wih wo or more srike prices, we can find values for µ,t and σ,t ha produce opion prices closes o he observed prices in he leas squares sense. A similar leas-squares mehodology was used by Melick and Thomas (1997). 9. We need a leas wo opion records for a specific rade dae and days-o-mauriy o compue µ,t and σ..,t 10. The opion daa also conain Black-Scholes implied volailiies adjused for sock dividends. Using his informaion along wih he ineres raes, we can reverse compue he corresponding European opion price. If he European opion price hus compued is higher han he bid and ask midpoin price, hen we ake he bid and ask midpoin price, else we ake he European price as he opion price o compue µ,t and σ. S&P 500 opions are European syle and he prices should,t reflec as such. However, minor differences exis beween he repored closing prices and he prices reversely compued from end of day implied volailiies.

8 114 Review of Fuures Markes Table 1. Inpu Daa Summary Saisics of S&P500 Index Opions Days-o-mauriy groups <= 90 Days > 90 Days All Mauriies Number of observaions Days-o-mauriy Mean Avg. moneyness Mean Sd. Dev Min Max Median Number of calls used Mean Sd. Dev Min Max Median Avg. spread Mean Sd. Dev Min Max Median Avg. volume Mean Sd. Dev Min Max Median Toal open ineres Mean Sd. Dev Min Max Median This able presens he summary saisics of all moneyness monh-end S&P 500 index call opions having posiive rading volume based on he monh-end observaions for he period of January April 006. Days-o-mauriy groups are formed based on opion days-omauriy. For example, if days o mauriy is less han or equal o 90 days, hen he observaion is in <=90 days-o-mauriy group. If days o mauriy is greaer han 90 days i is in > 90 dayso-mauriy group. Moneyness we define as he sock price divided by he srike price. For S&P500, sock price is he level of he index. Avg. volume is he average of volume of call opions used for a µ,t and σ pair esimae. Avg. spread is he average of spread of call,t opions used for a µ,t and σ pair esimae. Spread is defined as (offer - bid)/call price. Call,T price is he midpoin of bid and offer or he European opion price whichever is lower. European opion price is compued from Black-Scholes implied volailiy in he daa. Number of calls used is he number of opion records ha are used o compue a µ,t and σ pair.,t

9 Risk-Adjused Sock Informaion 115 B. Esimaion of Implied Expeced Sock Reurn & Implied Volailiy We joinly esimae he implied expeced sock reurn (µ,t ) and implied volailiy (σ,t ) using he risk-adjused opion pricing model described in previous secion. For a given rade dae for S&P500 index, we have many call opions wih same dayso-mauriy. We use all hese opions records o compue implied sock reurn and implied volailiy by a mehod of grid search o look for he global opima ha minimizes he square error. A square error is defined as he square of he difference beween he marke observed opion price and righ hand side of he equaion used o compue he opion price based on he observed values. 11 Since we are searching for he enire specrum for he global opima, we need o specify search inervals, wihou which we would no be able o implemen he search. 1 We use he implied expeced reurn (µ,t ) search range from 0.0% o 00.00% and implied volailiy (σ,t ) search range from 0.0% o % for he grid search. To esimae implied expeced reurns we need wo or more records wih same key value of rade dae, CUSIP, and days-o-mauriy. Thus, all he single records for a key value canno be used o compue implied reurn and are discarded. By his mehod, we exrac he marke implied reurn and marke volailiy for differen days-o-mauriy based on S&P500 index opion prices, and corresponding S&P500 index levels. III. RESULTS Using he S&P 500 monhly index opion prices from January 1996 ill April 006, we esimae expeced sock reurn (µ,t ) and volailiy (σ,t ) wih our model. We use opions wih various srikes a a given day and compue µ,t and σ for,t each ime o mauriy. As a resul, we obain joinly he erm srucure of µ,t and he erm srucure of σ,t. Using he S&P 500 index call opions of all moneyness, 13 we find he following: A downward sloping erm srucure of µ,t ha is consisen wih exising sudies o be reviewed in deails laer in he empirical secion. Much flaer erm srucure for σ,t han he Black-Scholes model. σ,t carries more informaion in predicing hisorical volailiy han he Black-Scholes implied volailiy (i.e., average implied sandard deviaion, or σ ) based on near erm opions mauring in 90-days or less. 14, T 11. The observed values used on he righ hand side of equaion (5a) are sock price, srike price, opion price, days o mauriy, and ineres rae. 1. Theoreically an inerval of - o + is he full search inerval for boh µ,t and σ. However, we,t would no be able o pracically implemen such a search for global opima given limied processing power of resources. Therefore, we choose he upper and lower bound based on he mos feasible inerval possible from prior experience. 13. We also perform combined call and pu opion esing bu he resuls, which are similar, no shown here for consideraion of space. 14. Average implied sandard deviaion is he arihmeic average of Black-Scholes implied sandard deviaion of all opions wih differen srike prices ha are used o esimae µ,t and σ.,t

10 116 Review of Fuures Markes A combinaion of our implied expeced reurn ( µ,t ) and implied volailiy (σ,t ) wih σ provides a beer model han using σ T, T, alone o forecas fuure volailiy for any mauriy and moneyness combinaion. A. The Term Srucure of µ,t Table shows he descripive saisics of implied reurn (µ,t ) and implied volailiy (σ,t ) using all moneyness S&P 500 index call opions. To analyze he resuls we classify he daa ino differen days-o-mauriy groups. Thus he opions whose days-o-mauriy is less han or equal o 90 days are classified ino <=90 group. The opions whose days-o-mauriy is greaer han 90 days are classified ino >90 group. Figure 1 shows µ,t and σ graphs for S&P500 index call opions,t of all moneyness. In hese ables and graph, we see a erm srucure of µ,t. For example in Table for <=90 days-o-mauriy µ,t is 19.5%, whereas for >90 days o-mauriy i is The erm srucure of µ,t implies he expeced reurn is impaced by he ime horizon of invesmen. McNuly e al. (00) sudy he real cos of equiy capial using opion prices. They find high expeced reurns in he shor erm and low expeced reurns in he long erm, which is consisen wih our finding. They argue ha he marginal risk of an invesmen (he addiional risk he company akes on per uni ime) declines as a funcion of square roo of ime. The falling marginal risk should be refleced in he annual discoun rae. 16 Our erm srucure of µ,t is consisen wih his explanaion. However, unlike our approach, heir approach is heurisic and lacks he heoreical foundaion. Recenly Camara e al. (009) compued he cos of equiy from opion prices using a specific uiliy funcion and arrived a downward sloping erm srucure of expeced sock reurn similar o McNuly e al. However, Camara e al. s (009) approach requires an inermediae parameer ha needs o be compued using opions of all firms before hey compue he implied expeced reurn of any individual firm. In conras o heir approach, our approach does no compue a similar parameer, and we do no assume any explici uiliy funcion. 17 The daa poins for he erm srucure graphs (Figure 1) are generaed by nonparameric spline inerpolaion using he neighborhood daa poins. Our approach can be used o esimae he cos of equiy for any ime horizon of invesmen. 18 One of he advanages of our approach is ha he expeced reurn of a sock can be compued wihou using any informaion of he marke porfolio such as he marke risk premium. This implies one does no have o define wha he marke consiss 15. We also see he erm srucure when we group he daa ino 30, 60, 90, and so on days-omauriy groups. 16. This is explained in McNuly e al. (00). 17. Noe ha our model is consisen wih he Black-Scholes and assumes normaliy of sock reurns. As a resul, our model is implicily consisen wih he quadraic uiliy funcion. 18. Our approach can be used o esimae cos of equiy for differen indusry porfolios. We do similar experimens and show he erm srucure of expeced reurn persiss for hese indusry porfolios.

11 Risk-Adjused Sock Informaion 117 Table. Implied and Hisorical Summary Saisics Using S&P500 Index Opions. Days-o-mauriy groups <= 90 Days > 90 Days All Mauriies Implied expeced reurn µ, T Mean Sd. Dev Min Max Median Implied volailiy σ,t Mean Sd. Dev Min Max Median Implied sandard deviaion ( σ T, ) Mean Sd. Dev Min Max Median Hisorical volailiy Mean Sd. Dev Min Max Median The sample consiss of all moneyness monh-end S&P 500 index call opions based on he monh-end observaions for he period of January 1996-April 006. Days o mauriy groups are formed based on opion days-o-mauriies. For example, if days o mauriy is less han or equal o 90 days hen he observaion is in <=90 days-o-mauriy group. If days o mauriy is greaer han 90 days i is in > 90 days-o-mauriy group. We use all he call opions on he same CUSIP, days-o-mauriy, and rade dae o compue he implied expeced reurn and implied volailiy by a grid search mehod ha minimizes he square of difference beween he observed and compued opion price. Hisorical volailiy is compued based on acual reurn of he index from rade dae o mauriy dae of he opion. Implied sandard deviaion ( σ, ) is he Black-Scholes implied volailiy. Resuls are in decimals. T

12 118 Review of Fuures Markes Figure 1. Term Srucures Using All Moneyness S&P500 Index Call Opions. Mu is opion implied expeced reurn, and Sigma is opion implied volailiy. Mu and Sigma are joinly esimaed using risk adjused model on opions daa.

13 Risk-Adjused Sock Informaion 119 of, and one does no have o esimae he risk premium of he marke, which is required in radiional asse pricing models, o esimae he expeced reurn. To validae he robusness of our finding, we examine he influence of marke fricion proxies such as he opion open ineres, volume, and bid-ask spread on he erm srucure of implied expeced reurn. We conrol for ime o expiraion bias, moneyness bias, and volailiy bias in his regression. 19 Our resuls show ha he marke fricion proxies do no explain his erm srucure. We also find he erm srucure of expeced reurn remains for deep-in and deep-ou of he money call opions. Furhermore, his erm srucure also persiss for combined call and pu opions (resuls available upon reques). B. Comparison of Term Srucure of σ,t and Black-Scholes Volailiy Our model also demonsraes a flaer (less variaion) erm srucure of σ,t. 0 From Figure 1, we can eyeball he wo volailiy erm srucures from σ,t of our model and σ T, of he Black-Scholes model ha he erm srucure of σ,t is much flaer han he erm srucure of σ T,. While i is no easy o compare he wo erm srucures saisically, we can compue he relaive variaion of he wo erm srucures from Table. For all mauriies, he mean and sandard deviaion of σ,t are and , respecively; and of σ T, are and , respecively. Hence, he relaive variaion, defined as sandard deviaion divided by he mean, is for our model and for he Black-Scholes model. 1 This demonsraes ha he σ,t of our model presens a flaer erm srucure han he σ T, of he Black-Scholes model. When we divide he sample ino shor erm (<=90 days) and long erm (>90 days), we find ha our model performs beer han he Black-Scholes model for he shor erm opions o versus , ye worse for he long erm opions o versus This demonsraes ha he erm srucure of he Black- σ, T Scholes dissipaes off, for higher days o mauriy opions. To have a deailed comparison of he characerisics of σ,t of our model and he implied volailiy (i.e., implied sandard deviaion, or σ ) of he Black-Scholes T, model, we esimae various aribues of comparison as shown in Table 3. σ,t is joinly esimaed wih µ,t using muliple opion records as described in secion II.A and II.B. To compue he values in his able, firs, we esimae he mean and sandard deviaion of σ,t and Black-Scholes implied volailiy ( T ) for each year and days-o-mauriy based on our enire daase. Then we compue he difference of hese means and sandard deviaions of σ,t and σ T, for each year and days-omauriy. Panel A of Table 3 provides he summary saisics of he difference of 19. Papers by Chiras and Manaser (1978), Macbeh and Merville (1980), Rubensein (1985), and Canina and Figlewski (1993) find hese biases. Longsaff (1995) has similar conrols for hese biases. 0. By erm srucure of σ,t, we mean he value of σ,t for differen days o mauriy of T, for same observaion dae,. 1. Table 3 provides a deailed comparison of σ,t and σ. T,. Difference of he means is compued as he mean of σ,t minus he mean of σ. Similarly we T, compue difference of sandard deviaion and difference of coefficien of variaion. σ,

14 10 Review of Fuures Markes Table 3. Comparison of Sigma and Black-Scholes Implied Volailiy. Days-o-mauriy groups <= 90 Days > 90 Days All Mauriies B S Panel A: Tes of difference beween level of Sigma and σ, T Difference: Mean Sandard Deviaion saisics ** **.88** B S Panel B: Tes of difference beween sandard deviaion of Sigma and σ, T Difference: Mean Sandard Deviaion saisics -.813* ** Panel C: Tes of difference beween coefficien of variaion of Sigma and σ, T Difference: Mean Sandard Deviaion saisics -4.1** * ** This able presens he summary saisics of comparison of our sigma (σ,t ) esimaes and Black-Scholes implied volailiy ( σ T, ) for differen days-o-mauriy groups based on all moneyness S&P500 Index call opions for he period of January 1996-April 006. Days o mauriy groups are formed based on opion days-o-mauriies. For example, if days o mauriy is less han or equal o 90 days hen he observaion is in <=90 days-o-mauriy group. If days o mauriy is greaer han 90 days i is in > 90 days-o-mauriy group. For his able, firs, we compue mean and sandard deviaion of sigma and σ T, for each year and days-o-mauriy. Panel A presens he es of difference beween mean level of sigma and for differen mauriy groups. Panel B presens he es of difference beween sandard deviaion level of sigma and σ T, for differen mauriy groups. For Panel C, we compue he coefficien of variaion (CV) of sigma and σ T, as corresponding sandard deviaion divided by he mean for each year and days-o-mauriy. Then we ake he difference of CV of sigma and σ T, for each year and days-o-mauriy. The -saisics shows wheher hese differences are significan for differen days-o-mauriy groups. ** and * represen he p-values of less han 0.01, and beween 0.01 and 0.05 respecively. he means for differen days-o-mauriy groups. Panel B provides he summary saisics of he difference of he sandard deviaions for differen days-o-mauriy groups. In Panel A, he -saisics are significan for all mauriy groups. Similarly in Panel B he -saisics is significan for boh <=90 days-o-mauriy and all mauriies groups and hey are negaive, showing he sandard deviaion is lower for sigma han σ,. Panel C shows he summary saisics of he difference of coefficien of variaion (CV) of σ,t and σ T, for differen days-o-mauriy groups. We see he CV of σ,t and are saisically differen. Similar o Table, we see CV of σ,t σ, T T

15 Risk-Adjused Sock Informaion 11 are lower compared o CV of T and hus σ,t is flaer han T. Overall, Table 3 shows ha σ,t has lower sandard deviaion, lower CV, and higher mean compared o σ T,. This implies ha σ,t of our risk-adjused model migh have addiional informaion beyond σ T, ha migh be valuable o esimae he characerisics of he underlying sock. C. Volailiy Forecas σ, In his secion we analyze wheher he µ,t and σ pair of our model carries,t more informaion han σ T, of he Black-Scholes model in forecasing hisorical volailiy. We use he erm hisorical volailiy for he ex-pos volailiy measured over he life of he opion as explained in secion III.C.1. We find ha σ,t alone can predic fuure hisorical volailiy significanly beer han Black-Scholes σ when T, we use opions of all moneyness. More ineresingly, we find ha when µ,t σ,t and σ T, are all used in he predicion, he resul is significanly beer han eiher σ,t or σ alone. These resuls are sronger for near erm opions. Firs, when we use all T, moneyness, σ,t and is second order erm do beer han σ T, and is second order erm for boh he days-o-mauriy groups, namely <=90 and >90 based on adjused R-square. Second, for near erm opions, he coefficiens of σ,t, and he secondorder erm are significan even in he presence of σ T,. Furhermore, a likelihood raio es rejecs he null hypohesis ha resriced model wih σ T, and is secondorder erm is beer han he unresriced model wih all he hree variables and heir second-order erms for all near and far mauriy groups, and for any moneyness level. 3 A vas body of lieraure exiss on he volailiy forecasing fron ha invesigaes he forecasing capabiliy of implied volailiy from opion prices. 4 In a recen comparison sudy, Granger and Poon (005) find ha he Black-Scholes (1973) implied volailiy provides a more accurae forecas of volailiies. In heir paper, hey show he oucomes of 66 previous sudies in his area ha uses differen mehods o forecas volailiy. These mehods are hisorical volailiy, ARCH, GARCH, Black-Scholes (1973) implied volailiy, and sochasic volailiy (SV). 5 Based on heir ranking, hey sugges ha Black-Scholes (1973) implied volailiy provides he bes forecas of fuure volailiy. Despie he added flexibiliy of SV models, he auhors find no clear evidence ha hey provide superior volailiy σ, 3. As we show in Table 5, we ake all moneyness or near-he-money opions; we ake <=90 days and >90 days-o-mauriy groups. In all hese cases we rejec he resriced model ha uses only Black-Scholes implied volailiy and is second order erm o predic he realized volailiy. 4. Papers are by Laane and Rendleman (1976), Chiras and Manaser (1978), Beckers (1981), Day and Lewis (199), Canina and Figlewski (1993), Chrisensen and Prabhala (1998), Lamoureux and Lasrapes (1993), and Blair e al. (001). Granger and Poon (005) provide a comparison of differen mehods of forecasing volailiy. 5. Opion pricing models by Meron (1976), Cox and Ross (1976), Hull and Whie (1987), Sco (1987), and Heson (1993a) exend basic Black-Scholes (1973) model o incorporae sochasic volailiy and jumps.

16 1 Review of Fuures Markes forecass. Furhermore, hey find Black-Scholes (1973) implied volailiy dominaes over ime-series models because he marke opion prices fully incorporae curren informaion and fuure volailiy expecaions. Therefore, we choose Black-Scholes implied volailiy ( σ T, ) as he benchmark and compare he informaion conen of implied expeced reurn (µ,t ) and implied volailiy (σ,t ) wih σ T,. To undersand he forecasabiliy of hisorical volailiy using µ,t, σ,t and σ T,, we plo hese ime series values in Figures and Figure 3 for <=90 days-o-mauriy and >90 days-omauriy groups, respecively, for S&P500 index opions using all moneyness. 1. Informaion Conen of he Nesed Model σ, T σ, T The comparison of informaion conen of over a model of µ,t, σ,t, and can be evaluaed using he following regressions: HV = α + α + α + T, T, 1 T, 1 T, σ σ σ ω (R1) HV T, = σt, + σt, + T, σ α α α ω (R) HV α α α α α α T, T, 4 T, 43, T 44 T, 45, T 46 T, 4 T, σ = + σ + σ + µ + α µ + σ + σ + ω (R3) Pas lieraure ypically uses equaion (R1) wihou he second-order erm. In our invesigaion we include he second-order erms 6 o capure he higher order HV effecs o explain he annualized hisorical volailiy ( σ T, ), where is he dae of observaion of opion prices for a given sock, and T is he mauriy dae. To compue he σ T,, we use he dividend adjused Black-Scholes implied volailiies given in he OpionMerics daa file. σ T, is he average of hese implied volailiies of all opions ha are used o esimae he µ,t and σ,t pair. 7 HV To compue σ T, firs we compue daily hisorical volailiy based on ex-pos daily reurns of he underlying asse for he remaining life of he opion and hen muliply by 5 : σ 5 τ HV T, = i i τ 1 i= 1 ( u u ) where τ is he remaining life (in working days) of he opion; u i = 1n(1 + r i ); r i is he daily reurn of he underlying asse for day i in CRSP daabase; u i is he mean of he u i series. 8 Table shows he summary saisics of hisorical volailiies HV ( ) of S&P500 index opions for differen day-o-mauriy groups of opions. σ T, 6. We es he validiy of he resriced model wihou he square erm. Based on he likelihood raio es, our resuls in mos cases rejec he resriced model. Therefore, we ake he variables ( µ,t, σ,t, or Black-Scholes implied sandard deviaion) wih he square erms. 7. µ,t, σ,t represen µ and σ respecively for a specific ime period, where is he dae of observaion of opion prices, and T is he mauriy dae of he opions. 8. Hull (00) uses a similar procedure o compue hisorical volailiies.

17 Risk-Adjused Sock Informaion 13 Anderson e al. (001) show ha he convenional squared reurns produce inaccurae forecas if daily reurns are used. The inaccuracy is a resul of noise in hese reurns. They furher show ha impac of noise componen is reduced if high-frequency reurns are used (e.g., 5-minue reurns). However, a relaively recen sudy by Aï-Sahalia, Mykland, and Zhang (005) demonsraes ha more daa does no necessarily lead o a beer esimae of realized volailiy in he presence of marke microsrucure noise. They show ha he opimal sampling frequency is joinly deermined by he magniude of marke microsrucure noise and he horizon of realized volailiy. For a given level of noise, he realized volailiy for a longer horizon (e.g., one monh or more) should be esimaed wih less frequen sampling han he realized volailiy for a shorer horizon (e.g., one day). Since our experimens are mosly for more han one monh ime horizon, he opimum daa frequency should neiher be 5-minues nor he daily reurns. In he absence of high-frequency daa, o he exen he opimum frequency is closer o one day, our measure (of hisorical volailiy) based on his frequency should closely represen he realized volailiy. 9 Using he above regression models, (R1) ~ (R3), we can es hree hypoheses. Firs, we can es if σ,t predics beer han. Second, we can verify if he coefficiens of µ,t and σ,t are significan even in he presence of σ. Third, we T, can es he hypohesis H 0 : a 43 = a 44 = a 45 = a 46 = 0. If we rejec his null hypohesis, hen we can argue ha µ,t and σ,t have significan conribuion in forecasing he fuure volailiy using he model as given in equaion (R3). The regression resuls are shown in Table 4. We have separae regressions for differen mauriy groups. As before, if days-o-mauriy is less han or equal o 90 days, hen he observaions are in <=90 days-o-mauriy group. If days-o-mauriy is greaer han 90 days, hen he observaions are in >90 days-o-mauriy group. We esimae hese regressions using he generalized mehod of momens. Using OLS may no be appropriae for our daa in he presence of nonspherical disurbances. Panel A of Table 4 shows he regression resuls using all moneyness of S&P500 index call opions. 30 As shown in his panel, he coefficiens of T, σ,t and are significan using models (R1) and (R) respecively. However, he adjused R-square is higher for he equaion conaining σ,t and σ,t for every mauriy group. This shows, when we ake all opions, σ,t provides a beer forecas of hisorical volailiy of he sock han he σ T,. To invesigae he performance of σ,t furher we have similar regressions in Panel B and Panel C of Table 4. As we see in Panel B, for sock price/srike price beween 0.95 and 1.05, he adjused R-squares are no higher for he equaions conaining σ,t and σ,t. However, he adjused R-squares are higher for he equaions conaining σ,t and using far-he-money opions Noneheless, o differeniae our measure of ex-pos volailiy from he Andersen e al. (001) measure, we use he erm hisorical volailiy for our measure. The way we compue his hisorical volailiy is explained in secion III.C In all our samples, we do no include opions ha have zero rading volume. 31. Opions are defined o be far-he-money if he sock price divided by srike price is eiher higher han 1.05 or lower han σ, T σ,t σ, σ,t

18 Mu is opion implied expeced reurn, and Sigma is opion implied volailiy. Mu and Sigma are joinly esimaed using risk adjused model on opions daa. 14 Review of Fuures Markes Figure. 90 Days or less Predicabiliy of Hisorical Volailiy by Sigma, Mu, and B-S Implied Volailiy of S&P500 Index.

19 Risk-Adjused Sock Informaion 15 Figure 3. More han 90 Days Predicabiliy of Hisorical Volailiy by Sigma, Mu, and B-S Implied Volailiy of S&P500 Index. Mu is opion implied expeced reurn, and Sima is opion implied volailiy. Mu and Sigma are joinly esimaed using risk adjused model on all moneyness call opions daa. DTM is days-o-mauriy.

20 16 Review of Fuures Markes Table 4. Informaion Conen of he Nesed Model Using Mu, Sigma, and Black-Scholes Implied Volailiy. Panel A. SPX Call Opions Using All Moneyness Inercep Adj. RP P LR Days-o Mauriy of Less han or Equal o 90 Days (-1.17) 1.336** (4.8) * (-.8) (-0.79) ** (3.68) (0.68) ** (4.67) -.98** (-3.77) Days-o Mauriy of Greaer han 90 Days **(-4.45) ** (1.17) ** (-8.89) (0.36) (-0.95) * (-.49) **(-6.84) ** (10.11) **(-4.4) (0.81) All Mauriies **(-.89) 1.468** (9.5) (-0.35) ** (-5.37) 0.67* (.08) (-1.65).485** (3.5) **(-.6) ** (5.96) (-0.8) 1.809** (6.83) ** (-6.38) (-0.11) (-0.3) (-1.85) (-1.18) ** (3.3) ** (-7.67) ** (-3.19) -.575** (-3.05).585* (.49) ** ** ** ** ** **

21 Risk-Adjused Sock Informaion 17 Table 4, coninued. Informaion Conen of he Nesed Model Using Mu, Sigma, and Black-Scholes Implied Volailiy. Panel B. SPX Call Opions Using SockPrice/SrikePrice beween 0.95 and 1.05 (near he money) Inercep Adj. RP P LR Days-o Mauriy of Less han or Equal o 90 Days 0(0) ** (4.55) (-1.48) (-0.43) ** (3.64) (1.89).1375** (3.76) ** (-3.46) Days-o Mauriy of Greaer han 90 Days (0.38) ** ** (-7.7) (-1.7) (1.6) * (-.14) (1.63) **(-5.73).931** (8.03) **(-5.78) * ** (-0.69) (0.7) (.11) (-1.61) (6.76) All Mauriies (-0.63) ** -0.89** (-9.58) (16.1) (-1.94) 1.447** (5.04) * ** * (-1.49) (.55) (-.6) (-0.57) (1.4) (.15) (-0.73) ** (.77) ** (-5.88) ** (-6.07) * (-.08) (-1.45) ** ** ** * ** **

22 18 Review of Fuures Markes Table 4, coninued. Informaion Conen of he Nesed Model Using Mu, Sigma, and Black-Scholes Implied Volailiy. Panel C. SPX Call Opions Using SockPrice/SrikePrice of Less han 0.95 or Greaer han 1.05 (far from he money) Inercep Adj. RP P LR Days-o Mauriy of Less han or Equal o 90 Days (-0.08) ** (10.64) ** (-8.19) (-0.5) 1.185** (4.17) 0.046(0.91) 1.148** (6.7) ** (-6.76) Days-o Mauriy of Greaer han 90 Days (-1.1) ** (10.01) ** (-6.63) (-0.75) (-1.1) (-1.16) **(-5.71).6113** (10.18) **(-3.7) (0.9) All Mauriies (0.5) 1.088** (10.17) (-0.1) (1.58) (-1.16).044** (.95) (-1.96) (1.3) ** (-7.61) -4.54** (-.86) * ** * ** **(-5.84) * (-1.96) **(6.68) (0.17) ** (6.79) **(-5.44) (-0.76) (-0.97) 0.004(0.0 6) -.06** (-3.69) (0.07) ** This able presens he generalized mehod of momens regression for forecas of hisorical volailiy using µ,t, σ, T, and Black-Scholes implied volailiy ( ) for differen mauriy groups for he period of January 1996-April 006. Days o mauriy groups are formed based on opion days-o-mauriies. For example, if days o mauriy is less han or equal o 90 days hen he observaion is in '<=90' days-o-mauriy group. If days o mauriy is greaer han 90 days i is in '> 90' days-o-mauriy group.values in parenhesis are -saisics. Dependen variable is hisorical volailiy of he index for he period of he opion. is Black-Scholes implied volailiy, µ,t and σ, T are he esimaed values from our model. For S&P500 index (SPX), sock price is he level of he index. LR is he likelihood raio o es wheher he resriced regressions are valid. ** and * represen he p-values of less han 0.01, and beween 0.01 and 0.05, respecively.

23 Risk-Adjused Sock Informaion 19 This shows σ,t provides a beer represenaion of ex-ane volailiy han using he informaion in far-from-he-money opions. Even hough σ T, does beer when we ake only near-he-money opions, i is unable o provide a single implied volailiy measure ha we can use for opions of all moneyness. On he oher hand, σ,t provides a beer measure of ex-ane volailiy ha can be used for opions of all moneyness. How does equaion (R1) compare wih equaion (R3) in explaining he hisorical volailiy? To address his quesion firs we see for all panels using near-he-money, all moneyness, and far-he-money opions, he adjused R-square is higher for he unresriced regression (R3) as shown in Table 4. For example, in Panel A for <=90 days-o-mauriy group, he adjused R-square for he unresriced model (R3) is 46.9%, and for he resriced model (R1) i is 41.87%. This shows ha equaion (R3) provides a beer model, such ha i has a higher adjused R-square for nearhe-money, far-he-money, and opions of all moneyness. Second, for all mauriies σ,t he coefficiens of σ,t and are significan for all panels of Table 4 in he unresriced equaion (R3). However ha is no he case wih σ T,. For example, in Panel A and Panel C he coefficiens of σ T, are no significan. Finally, we use he likelihood raio o es he hypohesis H 0 : a 43 = a 44 = a 45 = a 46 = 0. The likelihood raios are significan in our experimen for all panels of Table 4. Therefore, we rejec he resriced model as given in equaion (R1) for all mauriy groups shown in his able. This resul indicaes ha he inclusion of µ,t and σ,t and heir second-order erms provides a beer model han simply using Black- Scholes implied volailiy o forecas he hisorical volailiy for all near and far mauriy groups, and for any moneyness level.. Informaion Conen of Non-Nesed Models In his subsecion we compare he non-nesed models ha have only he riskadjused variables (µ,t, σ,t, and he square erms) or he σ T, variable (and is square erm) o forecas hisorical volailiy. We use wo differen variaions of J- es ha are popularly used in he lieraure. The non-nesed models ha we use o forecas hisorical volailiy can be given by he following regressions: σ, T HV T, T, 1 T, 1, T σ = α + α σ + α σ + ω (R4) σ = α + α σ + α σ + ω HV T, 0 1, T T,, T σ = α + α σ + α σ + α µ + α µ + ω (R6) HV T, T, 3 T, 33, T 34 T, 3 T, To compare (R5) or (R6) wih (R4) we ake he fied values of equaions and use he following J-es regressions: HV σ T, (R5) from hese

24 130 Review of Fuures Markes σ = φ[ α + α σ + α σ ] + (1 φ)[ σ ω ] + e HV HV, T T, T, 1 T, 1 T, T, (R7) HV HV σt, = φ1 [α30 + α31 σt, + α3 σ T, + α 33 µ, T + α 34 µ T, ] + (1 φ1 )[ σt, ω1 T, ] + et, (R8) HV HV T, = [ T, + 1 T, ] + (1 )[ T, T, ] + et, σ φ α α σ α σ φ σ ω (R9) Using (R7) and (R9), we can es wheher he Black-Scholes implied sandard deviaion offers any incremenal informaion over risk-adjused implied volailiy. If he Black-Scholes model does no have any incremenal informaion, hen φ 1 should be close o 1 and significan, and φ should be insignifican. 3 To find wheher φ 1 is in fac 1, we es he null hypohesis of H 0 : φ 1 = 1. Since our null hypohesis is he resul inended, in his es, o minimize he Type II error he criical p-value should be higher. 33 The lef side of Table 5 shows he resuls of his comparison. As we see from lef side of Panel A using all moneyness, φ is insignifican for <=90 days-o-mauriy group. Also, φ 1 is significan and we fail o rejec he null hypohesis ha φ 1 = 1 for his mauriy group. This shows ha for <=90 days-o-mauriy group Black-Scholes implied sandard deviaion provides no incremenal informaion over our implied volailiy. However, for >90 days-o-mauriy group we canno say ha he Black-Scholes implied sandard deviaion provides no incremenal informaion over he risk-adjused σ,t. Resuls are similar when we ake boh µ,t and σ,t o compare wih he Black-Scholes implied sandard deviaion. Even for he near-hemoney opions (Panel B) for <=90 days-o-mauriy group, φ is insignifican, and we fail o rejec he null hypohesis ha φ 1 = 1. This indicaes ha even when we do no have a volailiy smile he risk-adjused σ,t performs marginally beer han σ T,. Furhermore, as we see from Panel C of Table 5, consisen wih he prior lieraure, when we have many far-from-he-money opions, σ T, does no provide any incremenal informaion. These resuls sugges, o forecas volailiy for shorer mauriy of 90-days or less, he risk-adjused σ,t provides a beer alernaive over he σ T, for any moneyness level. Furhermore, if we have many far-from-hemoney opions, hen σ,t is a beer choice irrespecive of days-o-mauriy. We also es anoher variaion 34 of he above J-es using he following regressions: σ = (1 ψ )[ α + α σ + α σ ] + ψ [ σ ω ] + e (R10) HV HV T, 0 1 T, T, T, 1 T, T, 3. Our discussions compare (R5) wih (R4). However, we can also compare (R6) wih (R4) o find if Black-Scholes implied sandard deviaion offers any incremenal informaion over risk-adjused σ,t and µ,t. In ha case we use (R8) insead of (R7) and (R9) is given by: HV HV T, [ T, 1 T, ] (1 )[ T, 3 T, ] et, σ = φ α + α σ + α σ + φ σ ω + (R9) We show he resuls for boh (R5), (R4) comparison, and (R6), (R4) comparison in Table We ake 5% significance level as he cuoff poin, approximaely in he middle of 10% and 1%. 34. Davidson and MacKinnon (1981).