Capital Structure Effects on Prices of Firm Stock Options: Tests Using Implied Market Values of Corporate Debt

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1 Capial Srucure Effecs on Prices of Firm Sock Opions: Tess Using Implied Marke Values of Corporae Deb By Rober Geske and Yi Zhou* Noember, 2007 This reision January, 2009 JEL Classificaion: G12 Keywords: Deriaies, Opions, Leerage, Sochasic Volailiy We appreciae he commens and suppor from Waler Torous, Richard Roll, Mark Grinbla, Michael Brennan, Mark Garmaise, Liu Yang, Geoffrey Tae, Charles Cao, Mark Rubinsein, Hayne Leland, Jurij Albero Plazzi, Bernd Brommund, Xiaolong Cheng and Michael Nowony and all seminar paricipans where his paper was presened. All remaining errors are ours. Rober Geske is he corresponding auhor a The Anderson School a UCLA, 110 Weswood Plaza, Los Angeles, California 90095, USA, , *Yi Zhou: The Michael F. Price College of Business a Uniersiy of Oklahoma, Diision of Finance, 307 W. Brooks, Norman, Oklahoma 73019, , Elecronic copy aailable a: hp://ssrn.com/absrac=

2 Absrac This paper inroduces a new mehodology for measuring and analyzing capial srucure effecs on opion prices of indiidual firms in he economy. By focusing on indiidual firms we examine he cross secional effecs of leerage on opion prices. Our mehodology allows he marke alue of each firm s deb o be implied direcly from wo conemporaneous, liquid, a-he-money opion prices wihou he use of any hisorical price daa. We compare Geske s parsimonious model o he alernaie models of Black Scholes (BS) (1973), Bakshi, Cao, and Chen (BCC, 1997) (sochasic olailiy (SV), sochasic olailiy and sochasic ineres raes (SVSI), and sochasic olailiy and jumps (SVJ)), and Pan (2002) (no-risk premia (SV0), olailiy-risk premia(sv), jump-risk premia (SVJ0), olailiy and jump risk premia (SVJ)) which allows sae-dependen jump inensiy and adops implied sae-gmm economerics. These alernaie models do no direcly incorporae leerage effecs ino opion pricing, and excep for Black-Scholes hese model calibraions require he use of hisorical prices, and many more parameers which require complex esimaion procedures. The comparison demonsraes ha firm leerage has significan saisical and economic cross secional effecs on he prices of indiidual sock opions. The paper confirms ha by incorporaing capial srucure effecs using our mehodology o imply he marke alue of each firm s deb, Geske s model reduces he errors pricing opions on indiidual firms by 60% on aerage, relaie o he models compared herein (BS, BCC, Pan) which omi leerage as a ariable. Howeer, we would be remiss in no noing ha afer including leerage here is sill room for improemen, and perhaps by also incorporaing jumps or sochasic olailiy a he firm leel would resul in an een beer model. Elecronic copy aailable a: hp://ssrn.com/absrac=

3 1. Inroducion Ross (1976) demonsraed ha almos all securiies and porfolios of securiies can be considered as opions. Black and Scholes (BS) showed ha all opions are acually leered inesmens in he underlying opioned securiy or asse. I is well known ha mos corporaions hae some form of direc or indirec leerage. 1 Thus, i seems puzzling ha in he asse pricing lieraure, here hae been few deailed examinaions or ess for leerage effecs using a model which direcly incorporaes leerage, based on economic principles. In hree recen papers, Geske and Zhou (2006, 2007a,b) hae demonsraed ha by including a new measure of implied marke leerage in a parsimonious mehodology using conemporaneous equiy and equiy opion prices, hey can significanly improe on he pricing of indiidual sock opions and index opions. Furhermore, hey also show (2007) ha heir mehodology allows an implied equiy olailiy measure ha dominaes he CBOE s VIX, and seeral GARCH echniques for forecasing fuure olailiy. Empirically, researchers hae documened a negaie correlaion beween sock price moemens and sock olailiy, which was firs idenified by Black (1976) as he leerage" effec. A few papers hae confirmed ha deb is relaed o he obsered negaie correlaion (Chrisie (1993), and Tof and Pruyck (1998)). Tof and Prucyk (TP) (1997) adap a ersion of Leland and Tof (1996) o indiidual sock opions, and using ordinary regression in cross-secional ess hey demonsrae significan 1 A firm no direcly issuing bonds has many indirec promised payous (loans, receiables, axes, ec). Elecronic copy aailable a: hp://ssrn.com/absrac=

4 correlaions beween heir model s deb ariables and he olailiy skew for a 13 week period in 1994 for 138 firms in heir final sample. Howeer, TP do no inesigae he exen of opion pricing improemen aribuable o leerage by comparison o more complex models which omi leerage. Insead hey examine he cross-secional correlaions beween olailiy skew for indiidual socks and heir model deb ariables which are: (i) LEV, he raio of book alue (no implied marke alue) of deb and preferred sock o deb plus all equiy, and (ii) CVNT, raio of shor mauriy deb (less han 1 year) o oal deb, as a proxy for a proecie coenan. Some opion pricing papers hae modeled and esed his negaie correlaion beween a sock s reurn and is sochasic olailiy. Among hese papers are he sochasic olailiy models of Heson (1993), BCC (1997) and Pan (2002), which is a more complex exension of Baes (2000). Howeer, hese papers all assume arbirary funcional forms for he correlaion beween a sock s reurn and changes in he sock s olailiy. None of hem proides he economic moiaion of leerage for his correlaion. If his negaie correlaion is parially caused by deb as idenified firs by Black, hen he ariaions in acual marke leerage should be boh saisically and economically imporan o pricing equiy opions. Thus, i is imporan o isolae and analyze he magniude of he leerage effec independen of oher assumed possible complexiies such as sochasic olailiy, sochasic ineres raes, and sochasic jumps. Oherwise, hese addiional assumed sochasic parameers may be esimaed wih error because of a relean omied ariable. In order o incorporae deb ino asse pricing, we adop Geske s (1979) no arbirage, parial equilibrium, compound opion model. Geske s model proides a unique mehod o imply he marke alue of deb. His leerage based sochasic equiy olailiy model does no assume any arbirary

5 funcional form, and i proides he economic reason for he negaie correlaion beween olailiy and sock reurns. The sock reurn olailiy is no a consan as assumed in he Black and Scholes heory, bu is a funcion of he leel of he sock price, which also depends on he alue of he firm. As a firm s sock alue declines, he firm s leerage raio increases. Hence he equiy becomes more risky and is olailiy increases. This model can explain he negaie correlaion beween changes in a sock s reurn and changes in he sock s olailiy. Geske s opion model also resuls in he obsered faer (hinner) lef (righ) ail of he sock reurn disribuion. By incorporaing he implied marke alue of each firm s deb direcly and modeling is economic impac, Geske s opion model uses Modigliani and Miller (M&M) o ake he opion pricing heory deeper ino he heory of he firm. 2 His model incorporaes he differenial implied marke alue of sochasic deb, differenial defaul risk, and differenial bankrupcy. Thus, he Geske approach gies rise o sochasic equiy olailiy naurally, and his has he adanage of a direc economic inerpreaion for he sochasic olailiy. This paper demonsraes he parsimonious Geske model performs much beer wih far fewer parameers and less difficul esimaion han he more complex parameerized models of BCC and Pan which omi deb bu include parameers for sochasic equiy olailiy, sochasic ineres raes, and sochasic equiy jumps. Geske also is shown o dominae Black-Scholes. Boh he size of he implied marke alue of deb and he duraion of deb effec he sochasically changing shape of each firm s sock reurn disribuion. I is he shape of he condiional equiy reurn disribuion a any poin in ime ha deermines he model alues for opions wih differen srike prices and differen imes o expiraion. Thus, 2 Since he sock price is known inpu, gien M&M, he soluion is acually for he marke alue firm deb.

6 he omission of an imporan and measurable economic ariable, deb, causes he reurn disribuion o be mis-specified. This paper shows ha he omission of deb is parially responsible for opions alued wih eiher BS or he more complex models of BCC and Pan o exhibi greaer errors. Howeer, afer including leerage here is sill room for improemen, and perhaps by also incorporaing jumps or sochasic olailiy a he firm leel would resul in an een beer model. This is he firs paper in he exising lieraure o empirically examine capial srucure effecs on he pricing of indiidual sock opions by using Geske s closed-form compound opion model. In a relaed paper, Geske and Zhou (2007) presen he firs eidence of he ime series effecs of deb on prices of S&P 500 index pu opions. Since an index has no cross secional ariaion in leerage, he paper examines he changes in aggregae index deb wih ime. The index paper shows ha by including he ime ariaions in leerage as a ariable, Geske s model is superior for pricing index opions o he models of Black-Scholes (BS) and Bakshi, Cao and Chen (1997) which omi leerage. Furhermore, he adanage of including deb is monoonic in he changing amoun of leerage oer ime, and in ime o opion expiraion. This paper is relaed o many papers in he opion pricing lieraure. For example, Rubinsein (1994) (and ohers) deelops a laice approach o bes fi he cross-secional srucure of opion prices wherein he olailiy can depend on he asse price and ime. Dumas, Fleming and Whaley (1998) describe he approach of Rubinsein and ohers as a deerminisic olailiy funcion (DTV) and find ha hese implied ree approaches work no beer han an ad hoc ersion of Black-Scholes where he implied olailiy is modified for srike price and ime.

7 The negaie correlaion beween equiy reurn and olailiy has been modeled by Heson (1993) and ohers. 3 Heson deelops a closed-form sochasic olailiy model wih arbirary correlaion beween olailiy and asse reurns and demonsraes ha his model has he abiliy o improe on he Black-Scholes biases when he correlaion is negaie. Heson and Nandi (2000) deelop a closed-form GARCH opion aluaion model which exhibis he required negaie skew and conains Heson s (1993) sochasic olailiy model as a coninuous ime limi. They demonsrae ha heir ou of sample aluaion errors are lower han he ad hoc modified ersion of Black-Scholes which Dumas, Fleming and Whaley (1998) deeloped. Liu, Pan and Wang (2005) aemp o furher disenangle he rare-een premia by separaing he premia ino diffusie and jump premia, drien by risk aersion, and hen adding an inuiie componen drien by imprecise modeling and subsequen uncerainy aersion. All of he laer hree papers es heir models on S&P 500 index opions. In all cases, hese more generally specified models wih many more inpu parameers ouperform he (ad hoc) Black-Scholes soluions. Howeer, in his paper we focus primarily focus on he following hree papers: Black-Scholes, Bakshi, Cao and Chen (1997), and Pan (2002). 4 Bakshi, Cao and Chen (1997) formulae a series of all-encompassing models which include sochasic olailiy, ineres raes, and jumps wih consan jump inensiy, and which hey es by comparing he implied saisical parameers o hose of he underlying processes, as well examining ou-of-sample pricing and hedging performance for S&P 500 index opions. Pan (2002) examines he join ime series of he S&P 500 index and 3 See Sco (1987), Sein and Sein (1991) and Wiggins (1987). Wih respec o Heson (1993), Pan (2002) says Our firs se of diagnosic ess indicaes ha he sochasic olailiy model of Heson (1993) is no rich enough o capure he erm srucure of olailiy implied by he daa." 4 Eraker, Johannes and Polson (2003) exend Pan(2002) by assuming uncerainies in boh he jump iming and jump size in boh he olailiy and he reurns, wih eiher simulaneous arrials wih

8 near-he-money shor-daed opion prices wih a no-arbirage model o capure boh sochasic olailiy and jumps. She inroduces a parameric pricing kernel o analyze he hree major risk facors which she assumes effec he S&P 500 index reurns: he reurn risk, he sochasic olailiy risk and he jump risk. Pan (2002) exends Baes (2000) by allowing he jump premium o depend on he marke olailiy by assuming ha he jump inensiy is an affine funcion of he olailiy for a sae-dependen jump-risk premium so ha he jump risk premium is larger during olaile periods. She also shows ha his jump risk premium dominaes he olailiy risk premium. By omiing deb as a ariable and insead assuming arbirary funcional forms for olailiy, correlaion and jump processes, he exising lieraure fails o address direcly he imporance of capial srucure in asse pricing. This paper direcly ess he exen of a leerage effec in indiidual sock opions by measuring and using he acual daily implied marke deb for each indiidual firm. The Geske model requires he curren oal marke alue of he firm s deb plus equiy, and he insananeous olailiy of he rae of growh of his oal marke alue, neiher of which are direcly obserable. This problem is parsimoniously circumened by obsering wo conemporaneous, liquid marke prices, one for he indiidual sock price and he second for he price of a call opion on he indiidual sock. Then soling hree simulaneous equaions for he oal marke alues, V = S + B, marke reurn olailiy, σ and he criical oal marke alue, V*, for he opion exercise boundary. V We firs show ha Geske s model improes he ne opion aluaion of oer 2.5 million lised in-he-money and ou-of-he-money indiidual sock call opions on oer 11,500 firms by on aerage by abou 60% compared o oher models. Furhermore, we show for each firm s opions his improemen is direcly and monoonically relaed o boh correlaed jump sizes or independen arrials wih independen jump sizes.

9 he firm s deb and he ime o expiraion of he opion. The pricing improemen is monoonic wih respec o ime o expiraion because leerage has a longer ime effec. I may no be compleely surprising ha Geske dominaes simple Black-Scholes when pricing equiy opions if he daa qualiy for measuring leerage is good. Howeer, when we compare Geske s model wih more complex compeing models which require many more parameers (Bakshi, Cao and Chen (1997) and Pan (2002)), we find ha Geske s model produces he bes performance in boh absolue and relaie pricing error measures. The res of he paper proceeds as follows. Secion 2 describes he Geske model and is relaiely parsimonious implemenaion. Secion 3 describes he daa and explains in deail how he necessary daa inpus are calculaed. Secion 4 compares he Geske resuls wih he BS model and repors boh saisical and economic significance. Secion 5 describes and compares he hree BCC model ersions, SV, SVSI and SVJ wih Geske. Secion 6 describes and compares Pan s SV0, SV, SVJ0 and SVJ models wih Geske. Secion 7 concludes he paper. 2. Compound Opion Model In his secion, we briefly reiew he model of Geske (1979), and in laer secions we reiew BS, BCC, and Pan. Recall ha Geske s opion model, when applied o lised indiidual equiy opions, ransforms he sae ariable underlying he opion from he sock o he oal marke alue of he firm, V, which is he sum of marke equiy and marke deb. In his case he olailiy of he equiy of he indiidual sock will be random and inersely relaed o he alue of he indiidual sock equiy. This

10 inerpreaion of he Geske s model inroduces a new mehod by which o measure indiidual firm s implied marke deb alue and a new measure of indiidual firm s credi risk. Geske s model is consisen wih Modigliani and Miller, and allows for defaul on he deb and bankrupcy. The Black-Scholes model is a special case of Geske s model which will reduce o his equaion when eiher he dollar amoun of leerage is zero or when he leerage is perpeuiy. The boundary condiion for he exercise of an opion is also ransformed from depending on he srike price and sock price o depending on he alue of he firm, V, and on a criical oal marke alue, V*. This resuls in he following equaion for pricing indiidual sock call opions: ) ( ) ;, ( ) ;, ( 1 1 ) ( ) ( h N Ke h h N Me T h T h VN C T r T r F F + + = ρ ρ σ σ (1) Where T T r V V h F + = ) )( 2 1/ ( *) / ln( σ σ T T r M V h F + = ) )( 2 1/ ( ) / ln( σ σ ρ = ( )/( ) T T 1 2. Here V* a opion expiraion dae =T 1 is he criical oal marke alue a which he equiy index leel, S T1 = K, and S T1 is deduced from Meron s applicaion of he Black-Scholes equaion which reas sock as an opion: ) ( ) ( 2 1 ) ( h N M e T h V N S T r F + = σ (2) and hus a =T 1 where S T1 = K, K h N M e T T h N V S T T r T T F = + = ) ( ) ( 2 1 ) ( * σ (3) where h 2 is gien aboe. The face alue of a firm s deb ousanding is M and T 2 is he

11 duraion of his deb. The eens of exercising he call opion and he firm defauling are correlaed. If a firm is more likely o defaul a T 2, i.e., V is less han M a T 2, V will also be more likely o be less han V* a T 1, hus he call opions are less likely o be exercised. For Geske s compound opion here are wo correlaed exercise opporuniies a T 1 for he call opion and a T 2 for he deb duraion. The correlaion is measured by ρ = ( T1 )/( T2 ) where indiidual sock opion expiraion T 1 is less han or equal o marke deb duraion, T 2. When he firm has no deb or when he deb is perpeuiy, V = S and σ = σ, and equaion (1) reduces o he well-known Black-Scholes equaion: s rf 1( T1 ) C = SN h + σ T ) Ke N ( h ) (4) 1( The noaion for hese models can be summarized as follows: C = curren marke alue of an indiidual sock call opion, S = curren marke alue of he indiidual sock, V = curren marke alue of he firm s securiies (deb B + equiy S), V* = criical oal marke alue of he firm where V V* implies S K, M = face alue of marke deb (deb ousanding for he firm), K = srike price of he opion, r F = he risk-free rae of ineres o dae, σ = he insananeous olailiy of he marke firm alue reurn, σ s = he insananeous olailiy of he equiy reurn, = curren ime, T 1 = expiraion dae of he opion, T 2 = duraion of he marke deb,

12 N 1 (.) = uniariae cumulaie normal disribuion funcion, N 2 ( ) = biariae cumulaie normal disribuion funcion, ρ = correlaion beween he wo opion exercise opporuniies a and T 1 and T 2. Because of leerage, he olailiy of an opion is always greaer han or equal o he olailiy of he underlying sae ariable, and from Io s Lemma, he exac relaion beween he olailiy of he indiidual sock and he olailiy of he firm alue conains he marke alue of he deb/equiy raio, and is expressed as follows: 5 S V σ s = σ = N1( h2 +σ T2 ) ( 1+ B / S ) σ (5) V S The parial deriaie of he olailiy of he equiy reurn wih respec o he firm is σ s V V S V ( ) σ = N 2 1( h2 +σ T2 ) σ < 0 (6) S V S = 2 Thus, while Black-Scholes assume he equiy s reurn olailiy is no dependen on he equiy leel, Geske s model implies ha he olailiy of he equiy s reurn depends direcly on leerage, and is inersely relaed o he indiidual sock leel. When he firm alue and hus indiidual sock leel drops (rises), assuming he firm does no reac insananeously o sabilize he leerage, hen firm leerage rises (falls), and he indiidual sock olailiy also rises (falls). In he nex secion we describe he daa necessary o es for he presence of any leerage effecs in indiidual sock call opion prices. 5 See Geske (1979) for deails. This equaion arises direcly from Io s lemma.

13 3. Daa Collecion and Variable Consrucion 3.1. Opion Daa The Iy DB OpionMerics has he Securiy file, he Securiy_Price file and he Opion_Price file. The OpionMerics daa was colleced in June I conains opion daa from January, 1996 hrough December, This 120-monh sample period coering 10 years has abou 2500 obseraion days. From he Securiy file, we obain Securiy ID (The Securiy ID for he underlying securiies. Securiy ID s are unique oer he securiy s lifeime and are no recycled. The Securiy ID is he primary key for all daa conained in Iy DB.), CUSIP (The securiy s curren CUSIP number), Index Flag (A flag indicaing wheher he securiy is an index. Equal o 0 if he securiy is an indiidual sock, and 1 if he securiy is an index.), Exchange Designaor (A field indicaing he curren primary exchange for he securiy: Currenly delised, NYSE, AMEX, NASDAQ Naional Markes Sysem, NASDAQ Small Cap, OTC Bullein Board, The securiy is an index.). We choose all he securiies ha are equiies and we exclude all indices. An exchange-raded sock opion in he Unied Saes is an American-syle opion. We furher selec he securiies ha are aciely raded on he major exchanges. Now we hae a sample of 11,539 securiies whose sock opions are American-syle opions. From he Securiy_Price file, we obain Securiy ID, Dae (The dae for his price record) and Close Price (If his field is posiie, hen i is he closing price for he securiy on his dae. If i is negaie, hen here was no rading on his dae, i is he aerage of he closing bid and ask prices for he securiy on his dae.). We selec he securiy price

14 records when here are definiely rades on he daes. From he Opion_Price file, we obain Securiy ID, Dae (The dae of his price), Srike Price (The srike price of he opion imes 1000), Expiraion Dae (The expiraion dae of he opion), Call/Pu Flag (C-Call, P-Pu), Bes Bid (The bes, or highes, closing bid price across all exchanges on which he opion rades.), Bes Offer (The bes, or lowes, closing ask price across all exchanges on which he opion rades.), Las Trade Dae (The dae on which he opion las raded), Volume (The oal olume for he opion), and Open Ineres (The open ineres for he opion). We merge he seleced daases from he Opion_Price file and he Securiy_Price file, and we furher merge he newly generaed daase wih he seleced daase from he Securiy File. We keep all he opions on he securiies ha are presen in boh files. In order o minimize non-synchronous problems, we keep he opions whose las rade dae is he same as he record dae and whose opion price dae is he same as he securiy price dae. Nex we check o see if arbirage bounds are iolaed ( C S K e r T T ) and eliminae hese opion prices. If non-synchroniciy occurred because he sock price moed up afer he less liquid in or ou of he money opion las raded, hen opion under-pricing would be obsered, and some of hese opions would be remoed by he aboe arbirage check. If non-synchroniciy occurred because he sock price moed down afer he less liquid in or ou of he money opion las raded, hen opion oer-pricing would be obsered. Because we canno perfecly eliminae non-synchronous pricing for he in and ou of he money opions wih his daa base we keep rack of he amoun of under and oer-pricing in order o relae his miss-pricing o he resulan under (oer) pricing of in (ou of) he money indiidual sock call opions.

15 3.2. Diidends The diidend informaion is obained from CRSP. From CRSP, we collec he following diidend informaion: CUSIP, Closing Price (o cross check wih he securiy price from OpionMerics), Declaraion Dae (he dae on which he board of direcors declares a disribuion), Record Dae (on which he sockholder mus be regisered as holder of record on he sock ransfer records of he company in order o receie a paricular disribuion direcly from he company) and Paymen Dae (he dae upon which diidend checks are mailed or oher disribuions are made). A diidend paid during he opion s life reduces he sock prices a he ex-diidend insan and reduces he probabiliy ha he sock price will exceed he exercise price a he opion s expiraion. Because of he insurance reason and ime alue of he money, i is neer opimal o exercise an American call opion on a non-diidend-paying sock before he expiraion dae. Therefore, we use he colleced diidend informaion o resric my sample o be all he eligible call opions on socks wih no diidend prior o he opion expiraion. Thus, all he socks in my sample can be separaed ino wo groups: he firs group of sock neer pays any diidend beween January 4, 1996 and December 30, 2005; he second group of sock pays diidends in ha period a leas once. For he firs group of sock, we use all he opions wrien on hese socks in he whole sample period; for he second group of socks, we use all he opions whose expiraion daes are before he firs ex-diidend dae and all he opions whose expiraion daes are afer he preious ex-diidend daes and before he nex ex-diidend daes. There are ypically four days beween he ex-diidend day he record dae for he indiidual socks in U.S. As we canno obain he ex-diidend daes direcly from CRSP bu we can obain he record

16 dae from CRSP, we assume ha he ex-diidend dae occurs 4 rading days prior o he record dae o ge he ex-diidend daes. For he opions on he second group of socks, he opions seleced are no subjec o diidend paymen and can be aken as he American call opion on non-diidend-paying socks; he underlying securiy prices are he daily closing prices of he securiies and we do no need o ake ino accoun of diidends Balance Shee Informaion From he COMPUSTAT Annual daabase (colleced as of June 10, 2007), from year 1996 o 2005, by CNUM (CUSIP Issuer Code), here are 95,769 single firm-year obseraions and 293 duplicae firm-year obseraions due o mergers. These duplicae firm-year obseraions hae differen alues for each daa iem because hey are differen firms before he merger and acquisiion. CNUM (CUSIP) is he only way o merge he COMPUSTAT daabase wih IVY OpionMerics. If firms are duplicaes on CNUM, we canno differeniae wo (or more) firms by CNUM, we am no able o know which opions belong o which firms. Therefore, we excluded hose 293 duplicae records from he COMPUSTAT sample and he opions wrien on hese firms from he IVY OpionMerics daa sample. The 95,769 single firm-year obseraions from COMPUSTAT is composed of he following records: 1996: 10,604; 1997: 10,328; 1998: 10,654, 1999: 10,685, 2000: 10,221, 2001: 9,645, 2002: 9,192, 2003: 8,899, 2004: 8,411, 2005:7130. The balance shee informaion we collec from COMPUSTAT is he book deb ousanding. The deb o be maured in one year is defined as he sum of deb due in one year (Daa 44: no included in curren liabiliies Daa 5), he curren liabiliies (Daa 5), he accrued expense (Daa 153), he deferred charges (Daa 152), he deferred federal

17 ax (Daa 269), he deferred foreign ax (Daa 270), he deferred sae ax (Daa 271) and he noes payable (Daa 206). The deb of mauriy of he 2nd years is Daa 91. The deb mauring in he 3rd year is he oal of he repored deb mauring in he 3rd year (Daa 92) and he capialized lease obligaion (Daa 84). The deb of mauriy of he 4h years is Daa 93. The deb o be maured in he 5h year is he oal sum of he repored deb mauring in he 5h year (Daa 92), he consolidaed subsidiary (Daa 329), he deb of finance subsidiary (Daa 328), he morgage deb and oher secured deb (Daa 241), he noes deb (Daa 81), he oher liabiliies (Daa 75) and he minoriy ineres (Daa 38). The deb caegorized o be due in he 7h is eiher zero or he oal of debenures (Daa 82), he coningen liabiliies (Daa 327), he amoun of long-erm deb on which he ineres rae flucuaes wih he prime ineres rae a year end (Daa 148), and all he repored deb wih mauriy longer han 5 years (Daa 9 - Daa 91 - Daa 92 - Daa 93 - Daa 94). 6 In addiion, we delee firms whose conerible deb is (Daa 79) more han 3% of oal asses (Daa 6) and/or finance subsidiary (Daa 328) is 5% of oal asses. Among all hese annual daa iems, Daa 5, 75 and 9 are updaed quarerly from he COMPUSTAT quarerly daa file as Daa 49 (Q), 54 (Q) and 51 (Q). This srucure of deb ousanding permis he compuaion of he daily duraion of he corporae deb and he daily amoun due a he duraion dae. In order o make sure ha he key deb informaion is no missing from he COMPUSTAT daa, we check Daa 44, Daa 9, Daa 91 o Daa 94. If all of he six daa iems are missing, hen we do no include his company s record. If only some of he daa iems are missing while ohers hae posiie alues, hen we se he missing iems as zero and keep his company s record. For he oher daa iems besides he aboe six ones, if hey are missing, we se hem as zero. We also need o make sure ha Daa 25 6 The mean duraion of issued US corporae deb was 7 years ( ). See Guedes and Opler (1996).

18 (Common Shares Ousanding) is no missing, as he marke leerage will be calculaed on a per share basis. We exclude all uiliy firms (DNUM=49), financial and non-profi firms (DNUM60) Ineres Rae and Discoun Rae Esimaing he presen alue of deb and duraion requires esimaes of he riskless ineres raes and he discoun raes. The riskless rae and discoun rae appropriae o each opion were esimaed by inerpolaing he effecie marke yields of he wo Treasury Bills of U.S. Treasury securiies a 6-monh, 1-, 2-, 3-, 5-, 7- and 10-year consan mauriy from he Federal Resere for goernmen securiies. The ineres rae for a paricular mauriy is compued by linearly inerpolaing beween he wo coninuous raes whose mauriies sraddle Characerisics of he Final Sample We diide he opion daa ino seeral caegories according o eiher erm o opion expiraion or moneyness. Fie ranges of ime o expiraion are classified: 1. Very near erm (21 o 40 days) 2. Near erm (41 o 60 days) 3. Middle erm (61 o 110 days) 4. Far erm (111 o 170 days) 5. Very far erm (171 o 365 days) Opions wih less han 21 days o expiraion and more han 365 days o expiraion were omied. 7 The fie ranges of opion mauriy classificaion are se such ha he numbers of each caegory are relaiely een. 7 Rubinsein (1985) also used his pracice.

19 The raio of he srike price o he curren sock price is defined as he moneyness measure. The opion conrac can hen be classified ino seen moneyness ranges: 1. Very deep in-he-money (0.40 o 0.75) 2. Deep in-he-money (0.75 o 0.85) 3. In-he-money (ITM) (0.85 o 0.95) 4. A-he-money (ATM) (0.95 o 1.05) 5. Ou-of-he-money (OTM) (1.05 o 1.15) 6. Deep ou-of-he-money (1.15 o 1.25) 7. Very deep ou-of-he-money (1.25 o 2.50) We omi opions wih a raio less han 0.40 or larger han 2.5 because heir ligh rading frequency and hus possible non-synchroniciy of rading. The coerage of my erm o expiraion and moneyness is he larges in all he lieraure on indiidual sock opions. Afer he diidend resricions, he final sample is composed of nearly 3.5 million eligible indiidual sock call opions on 1,683 firms. Table 1 describes he sample properies of he eligible indiidual sock call opion prices. we repor summary saisics for he aerage bid-ask mid-poin price, he aerage effecie bid-ask spread (i.e., he ask price minus he bid-ask midpoin), he aerage rading olume and he oal number of opions, for all caegories pariioned by moneyness and erm of expiraion. Noe ha here are a oal of 3,487,894 call opion obseraions. ITM consiss 26.5% of he sample; ATM akes up 27.8% of he oal sample and OTM consiss 45.7% of he sample. There are almos wice as many OTM as ITM or ATM indiidual sock call opions. The ery near erm ATM has he larges number per caegory (272,856). Wih he longer erm o expiraion, he aerage call opion prices in all moneyness

20 caegories increase monoonically. Wih he larger raio of K/S, he aerage call opion prices in all erms of expiraion caegories decrease monoonically. The mos expensie aerage opion price is in he caegory of he ery deep in-he-money and he ery far expiraion erm opions. The leas expensie aerage opion prices are from he deep and ery deep ou-of-he-money opions and of he ery near erms of expiraion. Very deep in-he-money opions (0.40 <= K/S<0.75) are he mos expensie wih he aerage price across all erms o expiraion around $17.11 while ery deep ou-of-he-money (1.25 <= K/S<2.50) are he leas expensie wih he aerage price across all erms o expiraion around $0.25. The aerage price of ATM opions is $3.45. The aerage effecie bid-ask spreads also decrease monoonically wih he increase of from $0.22 o $0.08. The aerage effecie bid-ask spreads are abou $0.12 for all he erms of expiraion. In fac, hey do no ary oo much across erms o expiraion gien any leel of moneyness. The ery near erm ATM opions hae he highes aerage rading olume in conracs (on 100 shares). Across all erms o expiraion, he ATM opions hae he aerage rading olume ITM opions aerage rading olumes are from o and OTM opions aerage rading olumes are from o The deeper he moneyness and he furher he expiraion erms are, he less he aerage rading olumes of he opions are, which has been repored by he preious papers. Table 2 describes he disribuion of opions in each moneyness and erm o expiraion caegory for each year coered by he sample. From 1996 o 2003, he aerage number of opions is around 320,000 per year. In 2004 and 2005, he aerage number is 450,000 per year. A he money opions conain almos 30% he oal opions. The numbers of

21 opions decrease wih respec o ime o expiraion and moneyness. This able also shows ha in each caegory, we hae sufficien amoun for daa o draw saisical conclusions Combined Final Inpus Gien he daa defined in Secion 2 as C, S, M, K, r FT,, T 1, T 2, D, and ρ, and, i is possible o compue V, V*, and σ. In order o compue V, V *, and σ, we simulaneously sole equaions (1), (2), and (3), gien marke alues for C, S and he conraced srike price K. In his paper we choose o es hese models using he mehodology of mos professionals. Thus, we allow a erm srucure of olailiy, possibly differen for differen opion expiraions bu he same for all srikes of he same expiraion, and compue his erm srucure of olailiy daily. All he ess use ou of sample daa and forward looking implied olailiies for boh models. Usually, he markes for he mos a-he-money opions are he deepes and mos liquid as shown from he open ineres and olume daa, we base he olailiy erm srucure on he mos a-he-money and mos liquid (MATM) opions. If he mos a he money opions are no he mos liquid opions, hen we choose he mos liquid opions. This se-up is also based on he fac ha hese opions conain mos of he informaion. Thus, daily we compue he V for he mos a-he-money and he shores-mauriy opion, gien marke alues for C, S, and he conraced srike price K. Then we keep he V he same for all he opions in ha day, and we compue he implied olailiies for he indiidual sock opion from Black-Scholes and for he indiidual firm s marke alue from Geske for each ime o expiraion for he mos a-he-money opion, gien he sock price, opion price, and srike price. For differen imes o expiraion we hold he sock price, S, and he marke alue, V, consan and allow he implied olailiy for

22 he mos a-he-money opion o produce his opion s marke price. This is he mehodology, which we undersand mos professionals using Black-Scholes follow. Gien he obsered marke prices of indiidual sock call opions, his mehodology produces he well-documened Black-Scholes pricing biases obsered for indiidual sock call opions. The Black-Scholes model underprices he as majoriy of in he money call opions and oerprices he as majoriy of ou of he money call opions. As is he case wih many of he more recen models discussed in Secion 1, he hree ersions of BCC models, SV, SVSI, and SVJ, and he four ersions of Pan models, SV, SV0, SVJ0 and SVJ, hae many addiional parameers o be esimaed for he sochasic processes assumed. To esimae hese addiional parameers i is necessary for BCC o use mos of he opions presen on each day in order o find olailiy ha day ha minimizes he sum of squared errors across all hose opions. Thus, in order for BCC s parameer esimaes o remain ou of sample", he researchers ypically esimae he required parameers from prices lagged one day, and hen use he parameer esimaes o price opions he nex day. To esimae all he parameers for Pan s model, one opion per day is chosen for all he days in he sample and all opions are pooled as one single se. The opion series is combined wih a daily sock reurn se o se up he opimal momen condiions of reurn and olailiy. The daily olailiies are implied from he daily opions chosen. Pan specifically menioned ha by using her mehod, he complexiy of a ime dependency in he opion-implied olailiy due o moneyness and expiraion is compromised. To compare Geske s model wih BCC and Pan s models, we implemen Geske s model using he MATM erm srucure of olailiy, we follow he BCC s esimaion echnique by minimizing he sum of squared errors and we follow Pan s esimaion echnique by using implied sae GMM. Gien he daa and esimaes described, we can now examine wha improemen, if any, Geske s leerage based opion model may proide.

23 4. Comparison wih he Black-Scholes Models In his secion, we sar wih Black-Scholes and presen more deails abou he model comparison mehodology, graphs of he model errors wih respec o he opion s ime o expiraion and moneyness. Also presened are ables illusraing boh he saisical and economic significance of he Black-Scholes errors and Geske s improemens wih respec o moneyness and ime o expiraion by calendar year and by leerage Model Pricing Error Comparison Figure 1 presens a graph of indiidual sock call opion marke prices, Black-Scholes model alues, and moneyness, K/S, which is represenaie of mos research findings for he indiidual sock call opions. Black-Scholes model underprices mos in he money call opions (low K) and oerprices mos ou of he money call opions (high K) on he indiidual sock. Since he indiidual sock leel, S, is he same for all a any poin in ime during or a he end of any day, as aries in Figure 1, ITM indiidual sock call opions (low K) are shown o be under alued and OTM indiidual sock call opions (high K) are shown o be oer alued by he Black-Scholes model relaie o he marke prices. 8 Figure 1 shows ha Geske s compound opion model has he poenial o improe or een eliminae hese Black-Scholes aluaion errors because of he leerage effec. Leerage creaes a negaie correlaion beween he indiidual sock leel and he indiidual sock olailiy. This ineracion beween he indiidual sock leel and 8 Figure 1 presens he mos ubiquious resul. There are 15 differen model disance comparisons: boh oer marke, boh under, one oer while he oher is under, one equal o he marke while he oher is eiher oer or under, boh equal o each oher bu eiher oer or under, boh equal o each oher and equal o he marke, and here are muliple cases for each siuaion when he models are no equal o each oher.

24 indiidual sock olailiy implies ha he indiidual sock olailiy is boh sochasic and inersely relaed o he leel of he indiidual sock, and ha he resulan implied indiidual sock reurn disribuion will hae a faer lef ail and a hinner righ ail han he Black-Scholes assumpion of a normal reurn disribuion. Thus, Geske s compound opion model produces opion alues ha are greaer (less) han he Black-Scholes s alues for in (ou of) he money European indiidual sock call opions, and could poenially eliminae he known Black-Scholes bias. Figure 1 presens how we measure he amoun of improemen Geske s model proides for sock indiidual sock call opions during his sample period. For each opion, we calculae he compound model alue and he Black-Scholes model alue. The improemen of Geske s compound opion model compared o he Black-Scholes is calculaed wih he following formula: 9 BS error CO error = (Marke - BS) - (Marke - CO) (7) BS error (Marke - BS) We presen his analysis for all mached pairs of opions for a ariey of caegories wih differen imes o expiraion, differen moneyness, and for he differen marke leerage exhibied during my sample ime period. This is he firs paper o repor on Geske s compound opion model and is poenial o correc hese errors when used o price indiidual sock call opions Error Significance by Year, Leerage, Expiraion and Moneyness In he following ables, we presen a more deailed analysis of he aboe resuls relaing hese ITM and OTM Black-Scholes pricing errors and Geske s improemens o he opion s ime o expiraion by calendar year and by leerage. We also presen he 9 Care mus be aken wih he sign of he ariey of mached pair errors, especially if one model alue disance is aboe and he oher disance is below he marke, when compuing he aerage error across all mached pairs. Howeer, he resul depiced in Figures 1 is found for he as number of all opions.

25 number of opions aailable in hese caegories during his ime period, and examine boh he saisical and economic significance of Geske s model relaie o Black-Scholes. The ATM opion region is considered o be wihin 5% of he indiidual sock price. Consider he number of mached pairs of raded ITM call opions presened in Table 3 Panel A. Year 1999, 2000, 2004 and 2005 conain 451,100 ou of 923,353 oal opions, which is abou 50%. As expeced, he able shows ha ITM ery near erm o expiraion caegory is raded more heaily han he far expiraion ones in eery year. The ery near erm o expiraion caegory (21-40 days) conains 223,509 of he 923,353 oal opions, abou 24%. Table 3 Panel B presens he ne pricing error improemen of Geske s model relaie o Black-Scholes by calendar year for he arious imes o expiraion for all ITM indiidual sock opion mached pairs. The improemen of Geske s model wih respec o ime o expiraion aries from 14% for shores expiraions o 47% for longes expiraions, and is sricly monoonic across all years. Nex, consider he number of mached pairs of raded ITM call opions presened in Table 4. Panel A presens he ITM indiidual sock call opions by ime o expiraion and by deb/equiy (D/E) raio. The D/E raio during his ime period ranges beween 0% and 200%. Panel A shows ha abou 50% of his sample of ITM opions raded when he D/E raio ranged from 30% o 200%. Each opion expiraion caegory has a leas 20% of he oal opions. Panel B presens he ne pricing error improemen of Geske s model relaie o

26 Black-Scholes by D/E raio for he arious imes o expiraion for all ITM call indiidual sock opion mached pairs during his sample period. As in Table 3 Panel B, he improemen of Geske s model wih respec o ime o expiraion aries from 14% for shores expiraions o 47% for longes expiraions, and is sricly monoonic across all ranges of leerage. Relaie o Black-Scholes, he improemen of Geske s model s increases wih he D/E raio almos monoonically for eery ime o expiraion. From he lowes D/E caegory o he highes D/E caegory, he improemen increase from 11% o 64%. Table 5 presens similar daa o Table 3 for ou of he money (OTM) indiidual sock call opions. Firs consider he number of raded indiidual sock calls presened in Table V Panel A for OTM opions. Panel A shows he mos acie rading years for OTM indiidual sock opions during my sample period are 2000, 2001, 2004 and Each opion expiraion caegory has abou 20% of he oal opions. Table 5 Panel B demonsraes ha Geske s compound opion model s pricing error improemen for each year. Almos monoonically for eery ime o expiraion, he improemen of Geske s model wih respec o ime o expiraion aries from 49% for near erm expiraions o 65% for longes expiraions, and is sricly monoonic across all years and ranges of leerage. Year 1996, 1997, 1999, 2000 and 2005 exhibi more han 70% pricing error improemen and he smalles ye subsanial improemen around 30% happen in he year 2002 and Similar paerns also can be found in he Table 3 and 4 s Panel Bs for ITM indiidual sock opions. Table 6 presens similar daa o Table 4 for OTM indiidual sock call opions. Firs consider he number of raded indiidual sock calls presened in Table 6 Panel A for

27 OTM opions. Panel A shows ha abou 50% of his sample of OTM opions raded when he D/E raio ranged from 30% o 200%. 22% of opions hae D/E raios from 30% o 60%, and 20% of opions hae D/E raios higher han 60%. Each opion expiraion caegory has abou 20% of he oal opions. Table 6 Panel B demonsraes ha Geske s compound opion model s improemen also increases wih he D/E raio, almos monoonically for eery ime o expiraion, he improemen of Geske s model wih respec o ime o expiraion aries from 49% for shores expiraions o 65% for longes expiraions, and is sricly monoonic across all years and ranges of leerage. Relaie o Black-Scholes he improemen of Geske s model s increases wih he D/E raio almos monoonically for eery ime o expiraion from 20% o 83% Alernaie Tesing We also ried a differen olailiy mehodology of basing he aggregae ne pricing errors and improemen of Geske s model compared o Black-Scholes on he olailiy ha minimizes he sum of squared errors. We find ha his does no change he characerisics of my resuls, and his is eiden regardless of wheher we allow or do no allow a erm srucure of olailiy. This resul is no surprising because moing he pricing olailiy ha minimizes he sum of squared errors away from ATM oward eiher he ITM or OTM will exhibi a more han off-seing effec Saisical Significance Here we use non-parameric saisics o es he significance of he differences beween

28 Black-Scholes and Geske s model. As can be seen in Table 3 and 4 Panel C for ITM opions and Table 5 and 6 Panel C for OTM opions, we find Geske s model improemens are all significan a -alue smaller han he 0.001% by rank-sum es. The rank-sum es (also called Wilcoxon es or Mann-Whiney es) is a nonparameric or disribuion-free es which does no require any specific disribuional assumpions. We firs lis all obseraions from boh samples in a increasing order, label hem wih he group number, creae a new ariable called rank". For ies, we gie hem he same rank. Then we sum up he ranks for each group. The sum of he ranks is called T. The es saisics is Z saisic = [T Mean(T)]/SD(T), Where T : he sum of he ranks, Mean (T ): n imes he mean of he whole (combined) sample, SD(T): he sandard deiaion of Mean (T ). A p-alue is he proporion of alues from a sandard normal disribuion ha are more exreme han he obsered Z -saisic. My p-alues which are all 0 lead us o conclude ha here is significan difference beween Black-Scholes and Geske s model. We also did oher non-parameric ess: signed rank es, sign es and Kruskal-Wallis es (for wo independen samples, i.e. Mann-Whiney U Tes). All of hem yield he same resuls ha Geske s model improemens are all significan a p -alue smaller han he 0.001% for all erms o expiraion and calendar years and leerage raios. 5. Comparison wih Bakshi, Cao and Chen (1997) In his secion, we presen more deails abou he model comparison mehodology, graphs of he model errors wih respec o he opion s ime o expiraion and moneyness. Also included are ables of he saisical and economic significance of he Bakshi, Cao and Chen s SV, SVSI and SVJ errors and Geske s improemens wih

29 respec o moneyness and ime o expiraion by calendar year and by leerage BCC Descripion and Srucural Parameer Characerisics To conduc a comprehensie empirical sudy on he relaie adanages of compeing opion pricing models, we furher compare Geske s model wih he hree compeing BCC models: he sochasic-olailiy (SV) model, he sochasic-olailiy and sochasic-ineres-rae (SVSI) model, and he sochasic-olailiy random-jump (SVJ) models (Bakshi, Cao and Chen (1997)). These models relax he log-normal sock reurn disribuional assumpions and do correc some of he bias of he Black-Scholes model. The implici sock reurn disribuion is negaiely skewed and lepokuric. To derie a close-form jump diffusion opion pricing model, BCC specify a sochasic srucure under a risk-neural probabiliy measure. Under his measure, he dynamics of sock reurn process, he olailiy process and he ineres rae process are: ds( ) S( ) = [ R( ) λμ ] d + V ( ) dw ( ) J ( ) dq( ) (8) ( J S + dv ( ) = [ θ κ V ( )] d + σ V ( ) dw ( ) (9) ln[ 1+ J ( )] ~ N(ln[1 + μ J ]) σ J, σ J ) (10) 2 dr( ) = [ θ κ R( )] d + σ R( ) dw ( ) (11) R R R R whereas R () is he insananeous spo ineres rae; λ is he jump frequency per year; μ J is he mean relaie jump size; V() is he diffusion componen of reurn ariance (condiional on no jump occurring); ω ( ), ω ( ) is sandard Browning moion wih correlaion ρ ; q () is a Poisson jump couner wih inensiy λ ; κ mean-reersion rae of he process; S θ / κ is he long-run mean of he V() process; is he σ

30 is he ariaion coefficien of he diffusion olailiy V(); J () is he percenage jump size (condiional on a jump occurring) ha is he iid disribued wih mean μ J and 2 ariance σ J ; σ is he sandard deiaion of ln[ 1+ J ( )] ; κ R is he mean-reersion J rae of he R() process; θ R / κ R is he long-run mean of he R() process; σ R is he ariaion coefficien of he R() process. Under he risk-neural measure, he opion price is a funcion of he risk-neural probabiliies recoered from inering he respecie characerisic funcions. For deailed expression, please refer o Bakshi, Cao and Chen (1997). The SV model is by seing λ =0 and θ R = κ R = σ =0. The SVSI model is by seing λ =0. The SVJ model is by seing θ R = κ R = σ =0. The SV model assumes ha here exiss a negaie correlaion beween olailiy and spo asse reurns and he olailiy follows a sochasic diffusion process. The negaie correlaion produces he skewness and he ariaion coefficien of he diffusion olailiy conrols he ariance of he olailiy kurosis. The SVJ model assumes ha he disconinuous jumps causes negaie skewness and high kurosis. SVSI model assumes ha he ineres-rae erm srucure is sochasic o reduce he pricing error across opion mauriy. This is no relaed o he implici sock reurn disribuion, bu o improe he aluaion of fuure payoffs. All hree models are implemened by backing ou daily, he spo olailiy and he srucural parameers from he obsered marke opion prices of each day. R R In order o measure he laen srucural parameers of he SV, he SVSI and he SVJ models, we adop he Bakshi, Cao and Chen (1997) s approach mehod of minimizing he sum of squared dollar pricing errors. We collec all he opions for a firm in one day,

31 for any opion number greaer or equal o one plus he number of parameers o be esimaed. For each opion wih a erm o expiraion and srike price, we calculae he model price. The difference beween he model price and he marke price is he dollar pricing error. Then we sum all he squared dollar pricing errors as he objecie funcion o minimize o imply he laen srucural parameers and he olailiy. In implemening he aboe procedure, we firs use all indiidual sock call opions aailable for each firm on each gien day, proided ha he opion number is greaer or equal o he one plus he number of parameers o be esimaed, regardless of mauriy and moneyness, as inpus o esimae he laen srucural parameers and he olailiy. Table 7 repors ha daily aerage and he sandard error of each laen parameer and olailiy, respeciely for he BS, and BCC s SV, SVSI and SVJ models. The firs obseraion is ha he implied spo olailiy is quie differen among he four models. The BS model has he highes implied olailiy (55%), which is no so differen from he second highes SV and SVSI implied olailiies (52%), while SVJ model has he lowes implied olailiy (49%). The second obseraion is ha he esimaed srucural parameers for he spo olailiy process differ across he SV, he SVSI, and he SVJ models. Noe ha all he hree models hae he similar esimaedκ, he implied speed-of-adjusmen θ, which is around The SV, SVSI and SVJ models hae esimaes ha are no significan, indicaing he long-run mean of he diffusion olailiy is ignorable. Recall ha in he SV model, he skewness and kurosis leels of sock reurns are conrolled by he correlaion ρ and olailiy ariaion coefficien σ. The ariaion coefficien σ is significan for all hree models and is he highes for SV model, followed by SVSI model and he lowes for SVJ model. The magniudes of correlaion ρ are similar for

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