CONSISTENCY OF (INTERTEMPORAL) BETA ASSET PRICING AND BLACK-SCHOLES OPTION VALUATION

Transcription

1 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 55 CONSISTENCY OF INTERTEPORAL BETA ASSET PRICING AND BLAC-SCHOLES OPTION VALUATION Anj Hnn, P Rihling Absa I is wll-known ha h CAP valuaion omula suls om a quadai uiliy o h psnaiv invso In his pap w show ha h CAP valuaion ul mains valid i h psnaiv invso xhibis an onnial uiliy and ass and mak uns a bivaia nomally disibud In onas o quadai uiliy, onnial uiliy implis a posiiv sohasi disoun ao ha guaans posiiv opion pis In paiula, wihin ou dis-im amwok, opions a pid aoding o h Blak-Shols omula In addiion, ou appoah allows h valuaion o singl asss i hi uns ollow an inmpoal mak modl wih sohasi ba Th suling valuaion omula dis om h sandad CAP only in ha h d ba plas h dminisi on I uns ou ha h d ba an asily b simad om h un im sis y wods: Ass piing, opion valuaion, inmpoal CAP JEL Classiiaion: G, G3 Inoduion Th apial ass piing modl CAP is sill gadd as a paadigm o apial mak hoy Pimaily, his may b du o is simpl suu Bsids, duing h las dads, sah was mo onnad on h piing o divaiv insumns ah han on h valuaion o undlying asss In is sandad vsion h CAP laims ha h d un o a isky ass onsiss o wo pas Th is on is a liquidiy pmium a h lvl o h isk- ins a Th sond on is a isk pmium ha quals h mak isk pmium adjusd o h sysmai isk o h ass Thus, h aaivnss o h modl oms om wo as On on hand, h linaiy o h valuaion omula is onsisn wih an abiag- apial mak baus a poolio's ba quals h wighd avag o h bas o h soks ha onsiu h poolio On h oh hand, h modl in is sandad vsion has h onomially omphnsibl inpaion ha only h sysmai isk ao is valud baus unsysmai isks an b divsiid Alhough h is a lag numb o spial vsions and vsions wih wak assumpions 3 h modl is osd o qun iiqu Fama and Fnh 99 ound ou ha ba oiins vn in long im sis do no hav any lanaoy pow o h oss-sion o uns a h sok mak Insad, dins wihin his oss-sion a laind by mioonomi aos lik h mak valu and h book-o-mak aio o quiy 4 Fuhmo, Roll and Ross 994 plid on h basis o Roll's 977 iiqu Thy showd ha a small dg o ini- Oo-von-Guik-Univsiy agdbug, Gmany Oo-von-Guik-Univsiy agdbug, Gmany Two main appliaions o h CAP an b ound in poman masumn and ompany valuaion Fo xampl, audios suggs o ompu h os o quiy via h so alld ax CAP o Bnnan 97 Only w o h vaiy o xampls a h ax CAP o Bnnan 97 mniond abov, h inmpoal CAP o on 973a, h muli-ba CAP o Losq and Chaau 98, h onsumpion-basd CAP and h lognomal CAP o Rubinsin 976, and h CAP wih sohasi inlaion o Roll On xampl is Blak's 97 zo-ba vsion wihou a isk- ass Tunbull 977 analyzd spial vsions wih sp o mak impions Linn 969 showd ha hognous aions do no lad o sv dviaions in ass valuaion 4 Fom sudis ould ah onim h CAP S Blak, Jnsn and Shols, 97; and Fama and abh, 973 Howv, a h las sin h Fama and Fnh, 99 sudy, h sandad CAP is gadd as mpiially alsiid Anj Hnn, P Rihling, 6

2 56 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 iny o h mak indx, ha is usd o ompu h ba oiins, is suiin o obsv a ovaian o zo bwn ba oiin and man un Sin h sandad CAP is gadd as mpiially alsiid, w un o h ollowing hoial poin o iiqu Dybvig and Ingsoll 98 showd ha h valuaion omula o h CAP suls om a lina sohasi disoun ao usd by h psnaiv invso o valu isky uu ash lows Th sohasi disoun ao psns h maginal uiliy o a psnaiv invso A lina maginal uiliy suls om a quadai uiliy union Hn, h quadai uiliy union implis h CAP valuaion ul Bu a quadai uiliy union xhibis a ngaiv maginal uiliy in h nd and inasing laiv and absolu isk avsion Fuhmo, ngaiv opion pis may ou i h psnaiv invso xhibis a quadai uiliy In his pap w show ha h valuaion omula o h CAP likwis suls om a psnaiv invso wih an onnial uiliy union in as o bivaia nomally disibud ass and mak uns In onas o a quadai uiliy, an onnial uiliy union always xhibis a posiiv maginal uiliy and onsan absolu isk avsion Fuhmo, man and vaian omplly haaiz nomally disibud andom vaiabls Wih nomally disibud ass uns, in mak quilibium h CAP valuaion ul an b divd om man-vaian poolio slion Fuhmo, i uns ou ha in ou dis-im modl wih onnial uiliy and nomally disibud ass uns, opions an b valud by h Blak and Shols 973 omula In onas o a quadai uiliy union, his xluds ngaiv opion pis Addiionally, h valuaion amwok wih onnial uiliy and nomally disibud ass uns has h ollowing advanag Using h osponding sohasi disoun ao, h valuaion o singl inanial asss is possibl vn in as o sohasi ba oiins Th assumpion o h mak modl wih a sohasi ba as un gnaing poss is moivad by on's 973a inmpoal CAP In as h ba oiin is sohasi, i is unknown a ah poin in im In his siuaion h las squas gssion analysis is no hlpul baus i yilds jus on singl simao Th valuaion by a psnaiv invso wih onnial uiliy suls in a valuaion omula ha is lina in h d ba oiin i ass bas and h mak un a bivaia nomally disibud Founaly, i is possibl o sima h d ba om h un im sis wihou h nd o obsving alizd ba oiins On his no, w psn a sabl vsion o on's 973a inmpoal CAP Th pap is oganizd as ollows In sion h valuaion o opions and undlying asss by a psnaiv invso is psnd Th psnaiv invso xhibis a quadai uiliy union a is and an onnial uiliy union awads Sion 3 dals wih sohasi ba oiins Th valuaion omula wih d ba is divd Sion 4 onluds wih a bi summay Valuaion Famwok wih a Rpsnaiv Invso L P dno h un pi and P h andom uu pi o an invsmn a h nd o h piod in ou on-piod modl Th un R on a poolio onsising o his invsmn wih aion x and h isk- ass wih ins a wih aion x ads as ollows: P P R x RP x wh RP P Th maximizaion o d uiliy wih sp o h aion x givs h ollowing is od ondiion: I h suiy mak lin is usd in poman masumn, naly vy anking an b ad by an appopia indx hoi S Dybvig and Ross, 985

3 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 57 E u R! E u R R P x x R x P P This lads o h ollowing valuaion P o h isky uu ash low P: u R R E P P u R R x Th isky uu ash low is ansomd in wo sps A is, i is ansomd ino a osponding isk- ash low ha is disound by h isk- ins a subsqunly Thus, h sohasi disoun ao 3 u R R 4 u R R x givs a haaizaion o h maginal a o subsiuion bwn a isky and a isk- ash low Th valuaion aoding o quaion 3 osponds o h isk-nual valuaion hniqu Wih h sa-pi dnsiy, h man P is disound by h isk- ins a In ou dis-im singl-piod modl, h inomplnss o h mak wih oninuous sas is ssd by h dpndny o h valuaion on isk pns Th sohasi disoun ao psns h nomalizd maginal uiliy o h psnaiv invso Tho, is man quals uniy: 5 By aanging quaion, i an b sn ha h isk-nual valuaion o h isky un R P quals h isk- ins a: R 6 Rpsnaiv Invso wih a Quadai Uiliy Funion Dybvig and Ingsoll 98 showd ha h CAP valuaion omula holds i h psnaiv invso xhibis a quadai uiliy union Howv, in his as ngaiv opion pis may ou in an inompl mak wih dis-im ading A divaion o h Dybvig and Ingsoll, 98 sul is givn in h ollowing L R dno h xss un Fo iiny asons w analyz poolios whih onsis o h mak poolio wih aion x and h isk- ass wih aion x oov, w us h ollowing quadai uiliy union: u R R b R wh R x R x x 7 Th maximizaion o d uiliy wih sp o h aion o h mak poolio givs h ollowing is od ondiion: E u R! R b x b 8 x x x Raanging yilds h opimal aion o h mak poolio: P S also Glosn and Jagannahan 994

4 58 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 b x 9 To ompu h sohasi disoun ao o a quadai uiliy union, w pu his aion ino h divaiv o h uiliy union ur wih sp o R: u R R x Taking h man yilds: R u R E R x x b b b E Finally, using mak volailiy, h sohasi disoun ao qu o a quadai uiliy union ads as ollows: qu To show ha h valuaion ul wih h quadai sohasi disoun ao lads o h CAP quaion, w us quaion 6 and g: qu R E RP R P R R RP 3 Raanging lads o h CAP valuaion ul: RP Cov RP, R R Equaion shows ha h quadai sohasi disoun ao is lina in h un o h mak poolio wih a ngaiv slop I h nd-o-piod pi and h un o h mak poolio, spivly, a unboundd, h quadai sohasi disoun ao an ak ngaiv valus Aoding o quaion, his ous i P 4 5 A ngaiv sohasi disoun ao may lad o ngaiv ass pis Exampls a all opions on h mak indx ha a dp ou-o-h-mony I h payo o suh a all is posiiv only i ondiion 5 is ulilld, is payo is posiiv only i h valuaion ao aks ngaiv valus Tho, all opions on h mak indx ha a suiinly dp ou-o-h-mony g a ngaiv pi O ous, his dos no onadi h law o on pi Howv, an abiag mak asks o a posiiv sohasi disoun ao Jaow and adan 997 onludd h linaiy o h sohasi disoun ao om h man-vaian iion Howv, his impliaion only holds o abiaily disibud uns ha a oninuously ompud Ohwis, i wihou any assumpion onning h un disibuion, only h quadai uiliy union lads o a man-vaian iion Convsly, h ollowing sion shows ha wih bivaia nomally disibud ass and mak uns an onnial sohasi disoun ao implis h valuaion omula o h CAP, oo

5 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 59 Rpsnaiv Invso wih an Exponnial Uiliy Funion To show how h CAP valuaion ul is lad o a psnaiv invso wih an onnial uiliy union, w assum: A Th psnaiv invso xhibis onsan absolu isk avsion, i an onnial uiliy union A Runs o singl asss and h mak un a bivaia nomally disibud Using hs assumpions w obain h ollowing poposiion: Poposiion : Und h assumpions A and A, h CAP valuaion omula holds Poo: Wih onsan absolu isk avsion a, i onnial uiliy union u R { a R} wh R x, 6 and wih nomally disibud un R, i is suiin o maximiz h ainy quivaln a a C R x R x Va R x x x 7 Th maximizaion o h ainy quivaln CRx wih sp o h aion o h mak poolio x yilds: x 8 a Wih h opimal aion x o h mak poolio, w a abl o div h onnial sohasi disoun ao Th divaion o h uiliy union ur wih sp o R ads: u R a { a } 9 R x Taking h man yilds: u R E E a { a } R x Finally, h sohasi disoun ao o an onnial uiliy union ads as ollows: To show ha h valuaion ul wih h onnial sohasi disoun ao lads o h CAP quaion, w again us quaion 6 and g: E RP R P I ass and mak uns hav a bivaia nomal disibuion h Sin-Rubinsin ovaian omula yilds : H, w uiliz ha o a nomally disibud andom vaiabl X wih man and sandad dviaion i holds ha { X} I andom vaiabls X and Y hav a bivaia nomal disibuion, g: is an a las on oninuously diniabl union, ' Y g xiss, and lim y g y y, wh y is h dnsiy o Y, hn

6 6 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 R Cov, RP w RP Raanging and plaing by R lads o h CAP valuaion omula: 3 RP P R 4 Hn, wih a bivaia nomal disibuion o ass and mak uns, h CAP valuaion ul an b divd und h assumpion ha h psnaiv invso xhibis an onnial uiliy Tho, h CAP dos no nssaily imply h quadai uiliy union o h psnaiv invso To valu opions wih h onnial sohasi disoun ao, w onsid h payo C o a Euopan indx all opion wih xis pi : wh dnos h indx pi Fuhmo, w assum: C max{ ;}, 5 A3 Th indx pi ollows a gomi Bownian moion: d d dw 6 Using assumpion A3 a oninuous-im sohasi indx pi poss is supposd Howv, w a only insd in h nd-o-piod pi and h un ov h ni piod, spivly Tha mans ha w assum a oninuous sa spa wih dis ading poins in im No, ha in onas o h Blak and Shols 973 and on 973b opion valuaion amwok, in ou dis-im modl, i is no possibl o duplia h opion's payo Nvhlss, using assumpion A3 w omula h ollowing poposiion ha givs h laionship bwn a psnaiv invso wih onnial uiliy and h Blak-Shols opion piing omula: Poposiion : Und h assumpions A and A3, opions a valud aoding o h Blak-Shols omula Poo: Th soluion o h sohasi dinial quaion 6 in assumpion A3 ads as ollows: W 7 Raanging lads o: W 8 I w wi quaion 6 as a sohasi ingal quaion, w g: d s s W 9 Now, w din h indx un as R d s 3 s Using quaions 8 and 9 yilds: Cov X, g Y d g Y / d y Cov X, Y ound in Rubinsin 976 A poo o h Sin-Rubinsin ovaian omula an b

7 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 6 R 3 Hn, h d indx un quals h di om h gomi Bownian moion 6 ims im In ou on-piod modl, w simpliy h noaion o : R 3 L dno h oninuously ompoundd isk- ins a ha osponds o h disly ompoundd isk- ins a in quaion 3 Thn, h valuaion o h indx all opion using h onnial sohasi disoun ao aoding o assumpion A ads as ollows: dy y dy y d d d d µ µ C C N N 33 wh N dnos h umulaiv sandad nomal disibuion union I h uiliy union o h psnaiv invso is onnial, his maginal uiliy is always posiiv Thus, h sohasi disoun ao is always posiiv In onas o h quadai sohasi disoun ao qu, his implis posiiv opion pis I holds ha R S also on and on,

8 6 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 Th valuaion o indx opions by a psnaiv invso wih an onnial uiliy union lads o h Blak-Shols omula i h un o h undlying ass is nomally disibud Tho, wih lognomally disibud pis o h undlying ass, h valuaion o opions by h Blak-Shols omula is adqua dspi dis-im ading i h psnaiv invso xhibis onsan absolu isk avsion This ompls h suls o Rubinsin 976 and Bnnan Valuaion wih Sohasi Ba Coiins Basd on on's 973a inmpoal CAP, w assum a un gnaing poss ha dis om h sandad mak modl by im-dpndn ba oiins: A Ass uns ollow h inmpoal mak modl: and wh Cov, Cov, P P P P P P, P o all poolios P, o any wo din poolios o all poolios P P and P, Addiionally, w assum: A Ba oiins and h mak un a bivaia nomally disibud Using hs assumpions and assumpion A, w obain h ollowing poposiion ha givs h laionship bwn a psnaiv invso wih an onnial uiliy union and h CAP valuaion ul in a siuaion wih sohasi ba oiins: Poposiion 3: Wih h inmpoal mak modl aoding o assumpion A4 in onjunion wih assumpions A and A5, h CAP valuaion ul ads: 34 RP P R 35 Poo: Th isk-nual valuaion o a isky un R P using h onnial sohasi disoun ao lads o h ollowing quaion: RP Cov, RP RP Baus o Cov P, i holds: 36 Cov, RP Cov, P P P Cov, P P P 37 Wih h sohasi disoun ao, h mak un gs a valuaion amouning o h isk- ins a Tho, h valu o h xss un in omula 37 is zo: 38 Th Sin-Rubinsin ovaian omula givs: Cov, E P Cov, P Cov, P Cov P, 39

9 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 63 Puing h suls o quaions 38 and 39 ino quaion 37 and h sul o quaion 37 ino quaion 36, w g h ollowing valuaion omula: This yilds: Cov P, P RP 4 RP P R 4 Hn, i h psnaiv invso xhibis an onnial uiliy union and ba oiins and h mak un a bivaia nomally disibud, h CAP valuaion ul wih sohasi ba oiin dis om h sandad CAP omula only in ha h om vals a lina dpndny on h d ba oiin Howv, h ba oiins a no known a ah poin in im Tho, i is no possibl o ompu hi avag o sima h d ba oiin Nvhlss, h ollowing poposiion sas ha h man ba an b simad om h un im sis und h sam assumpions whih w usd o div h CAP valuaion ul in as o sohasi ba oiins: Poposiion 4: Und h assumpions A, A4, and A5, d ba oiins an b simad om h un im sis Poo: Fom P in quaion 34, i ollows ha: ha: P P Cov P, P 4 Fuhmo, om P and Cov P, in quaion 34, i ollows P P Cov, P P Cov P, Puing h las sul ino quaion 4 and aanging yild: 43 P P P 44 All d valus on h igh-hand sid o his quaion an b simad om h un im sis 4 Summay Th apial ass piing modl CAP is gadd as mpiially alsiid Fuhmo, ss o h CAP only hk h iiny o h mak indx usd o ompu h ba oiins Som hoial iiqu applis o h a ha h valuaion omula o h CAP suls om a quadai uiliy union o h psnaiv invso A quadai uiliy union implis an in-h-nd-ngaiv maginal uiliy and inasing isk avsion In addiion, h quadai uiliy union lads o a valuaion ao o isky payos ha is lina in h mak un wih a ngaiv slop Wih his valuaion ao, all opions on h mak indx ha a vy dp ouo-h-mony g ngaiv pis Ou amwok wih an onnial uiliy union o h psnaiv invso und h addiional assumpion o nomally disibud uns ovoms hs waknsss A is, h maximizaion o d uiliy by h psnaiv invso lads o h CAP valuaion omula, oo Howv, in onas o h quadai sohasi disoun ao, h onnial sohasi disoun ao mains posiiv Tho, ou appoah is onsisn wih an abiag-

10 64 Invsmn anagmn and Finanial Innovaions, Volum 3, Issu 4, 6 apial mak In paiula, in ou onsidd dis-im modl, opions a pid aoding o h Blak-Shols omula Addiionally, h valuaion amwok wih an onnial uiliy union o h psnaiv invso allows h valuaion o singl asss i hi uns ollow an inmpoal mak modl wih sohasi ba oiins Th suling valuaion omula dis om h sandad CAP ul only in ha now h d ba oiin masus sysmai isk Wih h onnial sohasi disoun ao, d ba oiins an b simad om h un im sis wihou h nd o obsving alizd ba oiins Rns Blak, F Capial ak Equilibium wih Rsid Boowing // Jounal o Businss, pp Blak, F, C Jnsn & Shols Th Capial Ass Piing odl: Som Empiial Tss // in C Jnsn Edio Sudis in h Thoy o Capial aks Nw Yok, 97 pp 79-3 Blak, F, Shols Th Piing o Opions and Copoa Liabiliis // Jounal o Poliial Eonomy, pp Bnnan, J Taxs, ak Valuaion, and Copoa Finanial Poliy // Naional Tax Jounal, 97 3 pp Bnnan, J Th Piing o Coningn Claims in Dis Tim odls // Jounal o Finan, pp Dybvig, PH, JE Ingsoll an-vaian Thoy in Compl aks // Jounal o Businss, pp Dybvig, P H, SA Ross Th Analyis o Poman asumn Using a Suiy ak Lin // Jounal o Finan, pp Fama, EF, R Fnh Th Coss-Sion o Expd Sok Runs // Jounal o Finan, pp Fama, EF, JD abh Risk, Run, and Equilibium: Empiial Tss // Jounal o Poliial Eonomy, pp Glosn, LR, RA Jagannahan A Coningn Claim Appoah o Poman Evaluaion // Jounal o Empiial Finan, 994 pp 33-6 Jaow, RA, DB adan Is an-vaian Analysis Vauous: O was Ba Sill Bon? // Euopan Finan Rviw, 997 pp 5-3 on, R, E on Opion Piing and Poolio Opimizaion odn hods o Finanial ahmais // Povidn, 3 Linn, J Th Agggaion o Invso's Divs Judgmns and Pns in Puly Compiiv Suiy aks // Jounal o Finanial and Quaniaiv Analysis, pp Losq, E; JPD Chaau A Gnalizaion o h CAP Basd on a Popy o Covaian Opao // Jounal o Finanial and Quaniaiv Analysis, 98 7 pp on, RC An Inmpoal Capial Ass Piing odl // Eonomia, 973a 4 pp on, RC Thoy o Raional Opion Piing // Bll Jounal o Eonomis and anagmn Sin, 973b 4 pp Roll, R Ass, ony, and Commodiy Pi Inlaion und Unainy // Jounal o ony, Cdi, and Banking, pp Roll, R A Ciiqu o h Ass Piing Thoy's Tss Pa I: On Pas and Ponial Tsabiliy o h Thoy // Jounal o Finanial Eonomis, pp Roll, R, SA Ross On h Coss-Sional Rlaion Bwn Expd Runs and Bas // Jounal o Finan pp - Rubinsin, Th Valuaion o Unain Inom Sams and h Piing o Opions // Bll Jounal o Eonomis, pp Tunbull, S ak Impions and h Capial Ass Piing odl // Jounal o Businss Finan & Aouning, pp