Abstract. Keywords: Warrants, Executive Stock Options, Value of the Firm, Risk-neutral, Distribution. JEL Classification: G12, G13.

Transcription

1 Absac This pape ses ou o poie a isk-managemen ool namely he isibuion o he sock pice o a waan-issuing im an a he same ime esoles an ousaning issue beween he heoy an he empiical eience o he waan picing lieaue. In hei seminal aicle on waan picing Galai an chnelle 978 make he ollowing saemen: i he isibuion o he im s liquiaion alue is lognomal he alue o is shae pice is no lognomally isibue. On he ohe han ecen empiical suies sugges ha assuming lognomaliy o he sock pice isibuion o a waan-issuing im gies a ey goo appoximaion o he alue o a waan his is he so-calle opion-like waan aluaion appoximaion. In his pape we eie he heoeical isibuion o he sock pice o a waan-issuing im an show ha iluion is elece an incopoae in he unelying sock pice pio o expiaion. e also show ha espie o he ac ha he iskneual isibuion o a waan-issuing im an a non-waan issuing im is ieen aluaion by aking expecaions o he iscoune payo o he waan oe he wo ieen isk-neual isibuions pouces waan pices ey close o each ohe o a lage numbe o cases een when he log-sock pice isibuion o he waan-issuing im exhibis make skewness an kuosis. Excepions occu o eep-ou-o-he-money an close o mauiy ou-o-he-money waans in geneal. In such cases he opion-like appoximaion will signiicanly oepice waans. The isincion we make in his pape beween waans an execuie sock opions is simply a mae o whehe he conac is ae o no. e use he em waan o coe boh cases. eywos: aans Execuie ock Opions alue o he Fim Risk-neual Disibuion. JEL lassiicaion: G G3.

2 THE IMPLIED DITRIBUTION FOR TO OF OMPANIE ITH ARRANT ANDOR EXEUTIE TO OPTION THEOFANI DARINO AND TEPHEN E. ATHELL * Faculy o Economics an Poliics Uniesiy o ambige May. INTRODUTION A naual applicaion o opion picing moels is he aluaion o waans. Howee unlike call opions waans an execuie sock opions ae wien by companies on hei own sock an om pa o hei equiy. hen execise hey incease he numbe o ousaning shaes in he im an hus hae a iluion eec. The exbook eamen o waan aluaion heeoe manaes ha insea o aking he sock pice o he im as he unelying asse one shoul use he oal equiy alue o he im as he sae aiable. To illusae; in he classical Black-choles amewok o opion aluaion i is assume ha he sock pice pocess is goene by a Geomeic Bownian Moion o lognomal iusion. In a simila spii a Black-choles amewok o waan aluaion assumes ha Geomeic Bownian Moion is he pocess goening he oal equiy alue o he im insea o he sock pocess; we illusae why his is he case in secion below. Bu hen i lognomal iusion is he pocess goening he equiy alue o he im wha is he pocess goening is sock pice? An een moe impoanly wha is he isibuion o he sock pice o his waan-issuing im? Answeing his quesion is he pupose o his pape. The is insigh is gien by Galai an chnelle 978: " i he isibuion o he im's liquiaion alue oal equiy is lognomal he alue o is shae pice is no ". 3 nowlege o he isibuion o he sock pice o a waan issuing im is ey impoan o a numbe o easons such as isk managemen puposes e.g. alue-a-risk calculaions cei managemen puposes e.g. esimaing he pobabiliy o eaul o a im 4 no o menion he ac ha as noe by many auhos Galai an chnelle 978 Galai 989 ienius * Theo Dasinos gaeully acknowleges inancial suppo om he A.G. Leenis Founaion he enbuy cholaship Fun a he Uniesiy o ambige an he Economic an ocial Reseach ouncil ER. e hank hiaki Haa an Geo Meeks o useul suggesions an Alan cowco Alexane Ineichen ephen oope an Laun Mileon o poiing us wih inomaion on waanexecuie sock opion issuing ims. T. Dasinos achell ee.achell@econ.cam.ac.uk. Ohe mino ieences beween call opions an waans coul ega ajusmen o iiens o he possibiliy ha expiaion aes o ceain waans may be change. In ohe wos a waan is o be egae as a call opion on a shae o he oal equiy o he im whee equiy is eine as he sum o he alue o is shaes an he alue o is waans oiginally suggese by Black an choles The same obseaion is mae by ienius In his case he eb o he im shoul be also incopoae in is capial sucue.

3 996 he alue o a waan is equal o he alue o a call opion on he sock o he waanissuing im. In ac his lack o knowlege o he isibuion o he sock pice was one o he moiaions ha le Galai an chnelle o pice a waan by eeence o he alue o a call opion on he sock o an ienical im wihou waans ajuse by a iluion aco. O puing his las saemen in anohe way whee no eeence o "ienical" ims is mae he sae aiable is nohing else han he oal equiy alue o he waan-issuing im which o simpliciy an acabiliy can be assume o be lognomally isibue see o example chulz an Taumann 994. Oe he pas ecae empiical lieaue on waan picing see o example Bensoussan ouhy an Galai 99 chulz an Taumann 994 ienius 996 has suggese ha hee is no nee o ollow he exbook eamen o alue waans i.e. he equiy alue appoach. Insea base on hei empiical esuls an simulaions hese suies ecommen opion-like waan aluaion. This simply means aluing he waan as i i was ienical o a call opion wihou inoling in he aluaion pocess he ypically unobseable alue o he im an ignoing any iluion. upisingly such an appoximaion woks ey well o a lage numbe o cases see ecion. o moe eails. Inee ox an Rubinsein 985 an Ingesoll 987 hae long ago ecognise ha when iluion is suicienly small hen i may me ignoe. Moeoe in hamony wih he inings o chulz an Taumann 994 we illusae in his pape ha iluion can oen also be ignoe when he iluion poenial is high. Noe howee ha his esul is no uniesal an we pesen a ew cases whee he pacice o opion-like aluaion will lea o oepicing. Regaing now he size o he iluion aco hee appea o be no speciic limis place upon he oal numbe o opions gane by copoaions. In ac oe ime he oal gane can ise o a consieable pecenage oen well in excess o % o 5% o he issue capial. In elaion o he aboe paagaphs i is impoan o sess hee ha waan aluaion heoy oes no sae ha opion-like aluaion is wong. On he conay i saes ha such an appoach is equialen o he equiy alue appoach. Howee in oe o his o be exac one shoul use he pocess o he sock pice o he waan-issuing im. This pocess is no he same as o example he pocess o he sock pice o an ohewise ienical non-waanissuing-im. Thus he naue o he so-calle opion-like waan aluaion appoximaion is ha i akes he isibuion o he sock pice o a non-waan-issuing-im as an appoximaion o he isibuion o he sock pice o a waan-issuing im. Moeoe he pocesses ollowe by he sock pice an he oal equiy o he waanissuing im canno in geneal be o he same om. Inee as pu by ienius 996 once he oal equiy pocess is gien he sock pocess is compleely ixe an ice-esa an one canno inepenenly speciy he om o boh pocesses. In paicula o assume ha hey ae boh Geomeic Bownian Moion is inconsisen excep in special cases e.g. no ousaning waans. In his pape we show ha he iniial assumpion ha he equiy alue o he im is lognomal leas o a pocess o he sock pice wih sochasic i an aiance paamees. Moeoe he olailiy o he sock pocess epens on he moneyness an ime o mauiy o he waans an exhibis a smile o smik simila o he one obsee in opions makes. This agees wih chulz an Taumann 994 whee hey sugges ha een i he olailiy

4 o he equiy alue is consan he sock olailiy will be non-saionay. Noe howee an excepion: Fo he special case whee he waan is a-he-money as is someime he case wih execuie sock opions 5 an ineesing esul luks. e show ha boh he equiy alue an he sock pice ae ien by Geomeic Bownian Moions an ae hus lognomally isibue. Alhough conenien he esul is noneheless quie esicie since i is only locally insananeously applicable o equies he waan o be pemanenly a-he-money. Depaues om ha assumpion imply ha he lognomal isibuion will be jus an appoximaion o he ue isibuion o he sock pice an in ac an inceasingly ba one as he waan moes pogessiely away om he money. e heeoe go on nex o eie he "ue" isibuion o he sock pice o a waan-issuing im. This isibuion mus aleay elec he poenial iluion o equiy hus making he waan ienical o a call opion on he sock pice o he waan-issuing im. To sum up we ae ace wih wo phenomena. On he one han we hae Galai an chnele 978 inicaing ha he isibuion o he sock pice o a waan issuing im is no lognomal. On he ohe han we hae empiical suies suggesing ha assuming lognomaliy o he sock pice isibuion gies a ey goo appoximaion o he alue o a waan. uely somehing ineesing is going on an equies inesigaion. This is he gap in he lieaue ha his pape ses ou o ill. The sucue o he pape is as ollows. In secion we se up a amewok o waan picing. e sa wih a iscussion o he heoeical appoaches ha exis o aluing waans. In paicula we show ha he alue o a waan can be wien eihe as an opion on a shae on he oal equiy alue o he im o as an opion on is sock pice an we elaboae on some ineesing insighs ha aise om his elaionship. e also pesen some appoximaions ha hae been suggese o he aluaion o waans. e hen go on o eie he pocess ha goens he sock pice o a waan-issuing im base on he ac ha we hae peiously assume ha he alue pocess is a Geomeic Bownian Moion. Following is a subsecion ha illusaes how he unobsee alue an olailiy an i o an execuie sock opion issuing im can be inee by soling a sysem o simulaneous equaions. e en he secion wih a noe on he non-aabiliy o he alue o an execuiesock-opion-issuing im an he implicaions o his on no-abiage picing an heging agumens. In secion 3 we go on o eie he "ue" isibuion o he sock pice o ou waan-issuing im. e o his no by soling he sochasic pocess ha goens he im s sock pice bu by exploiing he monooniciy o he sock pice wih he equiy alue o he im an applying a non-linea ansomaion. In secion 4 we elax an assumpion ha we mae in secion 3. The assumpion was ha he olailiy o he equiy alue o he im is obseableineable. e elax his assumpion by uilising a Bayesian appoach whee he iusion paamees o he alue pocess ae eae as anom aiables an can heeoe be inegae ou o eliminae ia non-linea ansomaions. ecion 5 illusaes empiically 5 ee o example Noeen an olson 98 whee hey sae ha execuie sock opions ae ypically issue wih he execise pice equal o he sock pice. p. 385 Foonoe. Fo he economic aionale behin his pacice see Hall an Muphy whee he auhos show ha by seing he execise pice a o nea he gan-ae make pice copoaions maximise he incenies o hei employees an execuies. 3

5 wha has been sai in he peious secions. oncluing emaks an issues o uhe eseach ollow in secion 6.. A FRAMEOR FOR ARRANT PRIING. The Texbook Teamen o he aluaion o aans aans an execuie sock opions ae wien by companies on hei own sock. hen hey ae execise he company issues moe sock an sells hem o he opion hole o he sike pice. This subsequenly ilues he equiy o exising shaeholes. A way ou o his illusae below an oiginally suggese by Black an choles 973 an also iscusse in Lauebach an chulz 99 chulz an Taumann 994 is o ega a waan as a call opion on a shae o he oal equiy o he im whee equiy is eine as he sum o he alue o is shaes an he alue o is waans. In his case he unelying sae aiable is he oal equiy alue o he im an in he absence o abiage he alue o a waan is shown o be equal o he alue o a call opion on he equiy o he im muliplie by a iluion aco equal o 6. One iiculy ha aises wih his appoach is ha he alue an olailiy o he im s equiy is no iecly obseable. I is possible howee as we illusae lae o ine hese alues using numeical ouines. An alenaie way is o pice a waan as a call opion on he sock o he waan-issuing im. This o couse equies knowlege o he isibuion o he sock-pice o he waanissuing im. In his case he poenial iluion shoul aleay be elece in he sock pice an hus hee is no nee o a speciic iluion ajusmen. Hence he waan is ienical o a call opion on he sock pice o he im. Alhough his appoach has nee been use in empiical wok o aluing waans ue o he lack o knowlege o he sock pice isibuion i aises impoan issues in opion picing an suggess ha using o example he Black-choles moel o pice exchange-ae sock opions on ims ha happen o issue waans may esul in mispicing since he wong pocess i.e. he lognomal is assume o he sae aiable i.e. he sock pice. O in ohe wos wo sock opions one on a im wih waans an he ohe on an ienical im wihou waans shoul nomally sell o ieen pices since hei pice pocesses will ypically be ieen. 7 Looking a he bigge picue his implies ha when picing a sock opion he capial sucue o he im shoul be moelle an elece in he sock pice pocess. e leae an inesigaion on he impac o he capial sucue o a im on is sock pice isibuion o a ohcoming pape. The isincion we make in his pape beween waans an execuie shae opions is simply a mae o whehe he conac is ae o no. Moeoe we ake he em execuie sock opions o conain he popula subse o employee shae opion schemes which hae been successully inouce amongs copoaions uing he pas ew yeas. Issues o non- 6 epesens he aio o he oal numbe o new shaes issue upon execise o he waans o he oal numbe o exising shaes. 7 Noe hough ha i is possible o ieen pocesses o he sae aiable o hae he same isibuion. 4

6 anseabiliy elaye esing an execise policies o Execuie ock Opions EO s ae no iscusse in his pape. Fo hese an a ew ohe ieences beween exchange-ae opions an EO s we ee he eae o Rubinsein 995. Hee i suices o say ha he eucion in alue o an execuie opion ue o he non-anseabiliy consain is usually hanle my muliplying he alue ha woul ohewise be obaine i.e. he alue o a ae waan by one minus he pobabiliy ha he execuieemployee will leae he im beoe execise is possible. The eaul in his suy will be ha he pobabiliy o pemaue leae is zeo i is saighowa howee o moiy his assumpion. Regaing now he elaye esing consain o execise policy o EO s in geneal i is inee he case ha mos opion plans o no pemi employees o execise hei gane opions unil ae a peeine peio o ime has elapse o example an execuie opion has ypically a mauiy o yeas; howee hough elaye esing execise is usually no pemie o a peio ae gan ypically 3 yeas. In ohe wos EO s ae neihe Euopean no Ameican. In is Exposue Da Accouning o ock-base ompensaion FAB allows aluing EO s as Euopean bu equies using he so calle expece lie o he opion insea o he acual ime o expiaion. apene 998 eines he cos o an EO as he make alue o he opion o an unesice ousie ineso an examines he execise policies o manages who ae subjec o anseabiliy an heging consains. he shows ha ealy execise o an EO is no consisen wih execise paens obsee in he aa. Execuies hol opions long enough an eep enough ino he money beoe execising o capue a signiican amoun o hei poenial alue. In anohe noe now EO s ae paiculaly popula amongs high-ech ims. uch copoaions pomise api gowh bu also pay lile o no iiens. see amongs many ohes Micoso opoaion Oacle isco ysems AoL Time-ane ec. This has he aanage ha Ameican EO s can be alue as Euopean. Geneally in his suy we shall use he em waan o coe all cases i.e. waans execuie opions an employee opions. e sa is wih a simpliying assumpion: AUMPTION : The waan-issuing im is an all equiy im wih no ousaning eb. 8 Assumpion implies ha socks an waans ae he only souces o inancing ha he company is using. Hence he company has a cuen oal equiy alue o: N n.. whee N: The numbe o ousaning shaes. : The pice pe shae a ime. n: The numbe o ousaning waans : The pice o a waan on a shae a ime. Occasionally we will jus wie o 8 e make his assumpion o simpliciy. Noe howee ha his appoach can be exene o he moe ealisic case whee he im also has eb in is capial sucue. 5

7 enoe he alue o he waan. imilaly he alue o he im pe shae oal equiy pe shae N is:.. whee n N is he iluion aco. uppose now ha each waan eniles he hole o puchase one shae o he im a ime T o a sike pice o pe shae. 9 I he waans ae execise he company eceies a cash inlow o n an he oal equiy alue inceases o T n. This alue is hen isibue among N n shaes so ha he pice pe shae immeiaely ae execise becomes n N n..3 T T Hence i he waan is execise he payo o he waan hole is T n N n N T T T N n N N..4 O couse he waan will be execise only i hole a mauiy is T T >. Hence he payo o he waan..5 e hae heeoe shown ha he alue o he waan is equal o he alue o call opions on a shae on he oal equiy alue o he im :..6 whee... enoes a call opion aluaion uncion an epesen he nonsochasic ime o mauiy equiy olailiy sike pice an isk-ee ae especiely. LEMMA : As noe in Galai an chnelle 978 he alue o a waan is also equal o he alue o a call opion on he sock o he waan-issuing im. The execise alue o such a call a mauiy is T..7 9 e make his assumpion o simpliciy. I is saighowa o amen i an assume ha each waan eniles he hole o puchase p numbes o shaes. Fo claiy we shoul noe hee ha when we ee o equiy olailiy o sock olailiy we on eally mean he absolue olailiy o he equiy o he sock bu ahe he olailiy o he ae o eun on he equiy o sock especiely. 6

8 which is he same as he mauiy alue o he waan. Howee in his case he aluaion poceue equies knowlege o he isibuion o he sock pice o he waan-issuing im. Moeoe since he pocess o he sock pice is no in geneal lognomally isibue he Black-choles aluaion omula will no longe be ali. Assuming no abiage he alue o he waan is gien in his case by Q T e E \ I e \ T RN T T..8 o D D..9 whee D... enoes a call opion aluaion uncion no necessaily he Black-choles. enoes he sock eun olailiy. Q is he isk-neual maingale measue o he iscoune sock pice i i exiss; we show lae ha i oes. I is he ilaion o he sock pice a ime. RN T \ is he isk-neual coniional isibuion o he sock pice T gien he secuiy pice a ime. Noe ha we hae no ye mae any isibuional o pocess assumpions o o o. e hae shown howee ha he alue o a waan can be wien eihe as an opion on a shae on he alue o he im : o as an opion on is sock pice : D I is woh noing ha... an D... epesen wo ieen opion picing uncions:... is eie base on he pocess ollowe by while D... epens on he pocess ollowe by. O couse as aleay menione an as i will be shown below once one o he pocesses is speciie he ohe will be eie as a by-pouc an hus be compleely ixe as well... ome uggese Appoximaions o he aluaion o aans To alue waans ia he omula one nees o esimae he alue an olailiy o he im. Fo he case o ae waans his is a saighowa ask. Fo he case o execuie sock opions howee his can be icky since is unobseable an equies numeical ineence. e iscuss his issue in ecion.6 7

9 below. imilaly aluing waans ia he omula D equies knowlege o he pocess o he isibuion o he sock pice. The lae has no so a been aailable. As a esul acaemics an paciiones hae popose appoximaion echniques no all necessaily coec. Fo example as noe by Galai 989 an ouhy an Galai 99 a common misake ha is oen mae in pacice is ha waan pices ae calculae as.. This poceue which is base upon a misinepeaion o he Galai an chnelle 978 esul is incoec an will lea o ownwa biase waan alues. Galai an chnelle 978 suggese ha a waan can be pice by eeence o he alue o a call opion on he sock o an "ienical" im wihou waans ajuse by a iluion aco. ince howee "ienical" ims haly ee exis he coec inepeaion is ha in he absence o "ienical" ims he sae aiable shoul no be he sock pice bu he oal equiy alue o he waan-issuing im. An alenaie appoximaion ha seems o wok quie well is wha has come o be known as "opion-like" waan aluaion: A numbe o empiical suies see o example Bensoussan ouhy an Galai 99 chulz an Taumann 994 ienius 996 hae suggese ha hee is no nee o ollow he exbook eamen o alue waans. Insea base on hei empiical esuls an simulaions hey sugges ha waans coul jus as well be alue as.. Fo example chulz an Taumann 989 conclue: "To obain waan alues wih accepable accuacy ajusmens o he Blackcholes omula ae no neee excep pehaps o eep-ou-o-he-money waans." In hei suy is assume o be he Black- choles opion picing omula. Along he same lines is he conclusion o ienius 996: " he waan pices obaine using he sock pice meho ae ienical o he esuls aising om he exbook meho." Alhough "opion-like" waan aluaion is only an appoximaion an has no heoeical ounaion i appeas o wok quie well. Accoing o el 994: "This esul can be explaine by looking a he moiicaions o he pecise waan aluaion moel in elaion o he oiginal Black-choles omula: The sock pice is eplace by he equiy alue pe shae o common sock. The sana eiaion o he euns on common sock is eplace by he sana eiaion on he im's equiy 3 The enie omula is muliplie by he iluion aco. The ac ha only maginal ieences exis can be aibue o he ac ha he eecs om eplacing by moiicaion an by moiicaion ae ouweighe by he muliplicaion by he iluion aco moiicaion 3." 8

10 Bu is hee a heoeical jusiicaion o suppo his appoximaion? The heoeical peequisie o "opion-like" waan aluaion o wok well is no ha o ine. imply he unelying isk-neual isibuions behin he opion picing uncions an D i.e. behin he pocesses o he sock pice o a non-waan-issuing an a waan-issuing im especiely shoul behae in such a way so ha aking expecaions o he iscoune payo o he waan oe he wo isk-neual isibuions shoul pouce nea ienical alues. Inee lae in he pape ecion 5 we will illusae ha espie o he ac ha he isk-neual isibuions o a waan-issuing im an a non-waan issuing im ae ieen aluaion by aking expecaions o he iscoune payo oe he wo ieen isibuions pouces waan pices ey close o each ohe o a lage numbe o cases. Noe howee ha we will also poie coune-examples as a wo o cauion whee opion-like appoximaion migh no always wok well an esul in poo waan pices an lage mispicing eos..3 Insighs aising om he Texbook Teamen aluaion o aans THEOREM : Assuming no abiage he ollowing elaionship beween he wo opion picing uncions an D mus hol: D.3. o eaanging.3. an wiing i in ems o : D D.3. This elaionship is inepenen o isibuional assumpions. PROOF: Equaing equaions..6 an..9 we ge: D.3.3 o D.3.4 Now equaions.. an..6 imply:.3.5 imilaly equaions.. an..9 imply: D.3.6 Then soling.3.5 an.3.6 in ems o 9

11 D.3.7 ubsiuing.3.7 in.3.4 we ge.3.. imilaly subsiuing.3.6. in.3.3 we ge.3.. EXAMPLE : onsie an opion picing omula linea in a shae on he equiy alue o he im ; i.e. a b. Then om equaion.3. oling o D we ge: D a b D a b D Å D b b b Hence D is also linea in. imilaly i we sa om an opion picing omula linea in ; i.e. D a b using equaion.3. we ge oling o we ge a b a b Å D b b b Hence is also linea in. This example migh seem ou o place a his poin bu i isn. Fo example as is well known he Black-choles omula o an a-he-money opion is a linea uncion o he unelying o moe eails see Remak below. Moeoe he alue o an opion on a sock ha ollows a binomial pocess is a linea uncion o he sock pice i.e. he binomial moel. Fo he geneal case i.e. a non-linea ieeniable opion picing omula gien o D one can use Taylo seies expansions o in an opion picing uncion o D o especiely as a powe seies see o example Bule an chache 986 an nigh an achell 997. Assume o example ha we know. Then poie ha he egulaiy coniions ae saisie we can wie as a conegen Taylo seies; i.e. i β i whee is he poin o expansion assume o simpliciy i is zeo. Now he unknown D can also be wien as a polynomial uncion i.e. j D j whee is he poin o expansion assume o simpliciy i is zeo In boh hese cases expansions ae wien in ems o he sana eiaion.

12 an he s ae he unknown coeiciens o D ha we nee o calculae. Then om j equaion.3. we hae: Hence D βi i i i D k β i i ki D j k ki j j βi j i i j i i Equaing coeiciens we obain D as a conegen Taylo seies in..4 peciying a Pocess o he ae aiable I is now ime o become moe speciic. To alue he opion whose payo is gien in equaion..5 i.e. whose payo is: [ ] T one mus assume a pocess o. The lieaue on waan picing has mainly concenae on hee pocesses o he alue o he im. Appa om Black an choles's Geomeic Bownian Moion which assumes consan olailiy ox an Ross's 976 onsan Elasiciy o aiance E moel see Noeen an olson 98 Lauebach an chulz 99 chulz an Taumann 989 an Meon's 976 Jump iusion moel see eme an Roenel 993 hae also been use o moel he alue pocess. In his pape we will no go ino a eaile eiew o he lieaue on waan picing. An exhausie suy ha eiews he empiical eseach une alenaie pocesses has aleay been conuce by el 994. Hee i suices o quoe one o he conclusions o el: "Thee is no conclusie eience o eplace iien coece moels in which a consan olailiy is assume i.e. Blackcholes 973 like moels by moe complicae moels such as he Jump Diusion o he E moel." In he onsan Elasiciy o aiance moel E he assumpion o a consan olailiy is eplace by he assumpion o a consan elasiciy o aiance. In his moel i is geneally assume ha he elasiciy aco is eine in a way ha he olailiy eceases as he sock pice inceases. pecial cases o he E moel ae he quae Roo moel which assumes ha he olailiy is inesely elae o he squae oo o he sock pice an he Absolue moel which assumes ha he olailiy is inesely popoional o he sock pice. The Jump Diusion moel eelope by Meon 976 also ops he assumpion o consan olailiy. The moel assumes a wo pa sochasic pocess geneaing sock euns: a small coninuous pice moemens geneae by he same pocess as assume by Black an choles an b lage inequen jumps geneae by a Poisson pocess. O couse when hese moels ae use o waan aluaion he sae aiable is he equiy alue o he im an no he sock pice.

13 AUMPTION : The unobseable shae on he oal equiy alue o he im ollows a Geomeic Bownian Moion: Z.4. whee : The expece ae o eun on he alue o he im s asses. : The sana eiaion o he ae o eun on he alue o he im s asses. Z: A sana Bownian moion. Assumpion an equaions hen imply ha: B.4. B whee... enoes he Black-choles opion pice an a shae on he oal equiy alue o he im. peciically: B e.4.3 whee ln e an.4.4 Also T : The ime o mauiy o he ousaning waans : The execise pice o he waans : The isk-ee ae o inees.... : The sana nomal cumulaie isibuion uncion. REMAR : I he waan is cuenly a-he-money i.e. e as is oen he case wih execuie sock opions see Noeen an olson 98 hen some consieable simpliicaions occu in he aluaion omula: Equaion.4.4 euces o an he Black-choles aluaion omula.4.3 simpliies o: B ATM.

14 NOTE : Remak can be inepee as a special case whee he execise pice is sochasic so ha he elaionship e hols cuenly. In ohe wos he esul is only locally insananeously applicable o alenaiely applicable a iscee anom imes..5 The Pocess Goening he ock Pice o he aan-issuing Fim In he peious secion we assume ha Geomeic Bownian Moion is he pocess goening he equiy alue o he im. ha is hen he pocess goening he sock pice? oling equaion.. o we ge: Also om equaion.4. we hae.5. B Then he ollowing epesenaion o he sock pice mus hol: Now since B.5. B > hee is a unique one-o-one elaionship beween an an heeoe equaion.5. can be inee o ia a saighowa numeical meho e.g. Newon-Raphson. e wie.5.3 whee is he inese uncion o he elaionship gien in.5.. In geneal is no known explicily. Howee when he waan is a-he-money he aboe equaion can be wien in analyic om: LEMMA : Gien ha he equiy alue o he im ollows a Geomeic Bownian Moion hen he pocess o he sock pice is gien by e Z.5.4 3

15 whee ln e an φ... enoes he sana nomal pobabiliy ensiy uncion. PROOF: ee ecion in he Appenix. OROLLARY : uppose we hae an a-he-money waan i.e. e. 3 Then gien ha he equiy alue o he im ollows a Geomeic Bownian Moion he pocess o he sock pice is also goene by a Geomeic Bownian Moion: PROOF: ee ecion in he Appenix. Z The pocess gien in Lemma epesens he pocess o he sock pice. I is a pocess wih sochasic i an aiance paamees. Noe howee ha alhough he paamees o he pocess ae sochasic hey shoul only exhibi he sochasic behaiou allowe by he uncional epenence on he sock pice an ime. In ohe wos he unelying pocess o he sock pice is o he om: Z. This equaion escibes he mos geneal se up ha goes beyon he case o a puely eeminisic ime-epenen olailiy an i bu sill allows isk-neual aluaion wihou inoucing ohe heging insumens apa om he unelying isel. Thus he pocess gien in Lemma belongs o he popula class o sochasic olailiy moels calle "esice sochasic olailiy moels" see o example Rebonao 999 o "leel-epenen olailiy moels". These moels hae o couse he aanage ha hey pesee make compleeness. Hence a isk-neual measue o he iscoune sock pice exiss an as sae in Lemma he alue o he waan will be gien by: Q T e E \ I e \ T RN T T.5.5 Fom his i is obious ha knowlege o he isk-neual ensiy RN T \ is suicien o alue he waan. To obain he isk-neual ensiy one nees is o apply he ameon- Main-Gisano change o measue o make he iscoune sock pice a maingale see Duie 996 Baxe an Rennie 997 Bjok 998. One can in heoy obain he ansiion pobabiliies o he sock pice by soling he sochasic ieenial equaion DE o Lemma he isk-neual ansiion pobabiliies can similaly be obaine by soling he sochasic ieenial equaion aising une he iskneual measue ha makes a maingale. This is howee oo ha a ask. Alenaiely one 3 This coollay is o heoeical inees only an shoul be inepee in conjuncion wih Noe in he peious page. 4

16 can make o wihou soling he DE. Insea shiing om i o a secon oe paial ieenial equaion PDE is also possible. Inee une some egulaiy coniions o he sock pice s i an iusion coeiciens he ansiion pobabiliies can be obaine as unamenal soluions o olmogoo s backwa equaion o inee olmogoo s owa Fokke-Planck equaion. 4 ill howee he backwa an owa equaions hae been explicily sole only in a ew simple cases e.g. Black-choles Onsein-Uhlenbeck o ox an Ross s onsan Elasiciy o aiance pocesses. These hae usually been oun by aking he Fouie an Laplace ansomaion o he ansiion pobabiliies. In geneal one mus ely on numeical mehos such as soling numeically he PDE i.e. he olmogoo s equaions o peoming Mone alo inegaion o he DE i.e. equaion.5.4. Finally a numbe o appoximaions hae also been suggese. Fo example Jaow an Ru 98 hae popose eplacing he unknown ensiy o he DE by an appoximae ensiy ypically aing ee skewness an kuosis o he lognomal ensiy so as o allow o epaues om he Black-choles omula. Howee he isaanage o his appoach is ha he appoximae ensiy ignoes he unelying pocess o he sae aiable. Moe ecenly Ai-ahalia 999 popose a meho o obaining a close om appoximaion o he ansiion ensiy. In paicula he poposes ansoming sanaising he DE o say equaion.5.4 o an DE wih uni iusion say P p P Z. The poin o making he ansomaion om o P is ha i is possible o consuc an expansion o he ansiion ensiy o P. Ai-ahalia hen suggess appoximaing he ansiion ensiy o P by using a Hemie polynomial expansion aoun a Nomal ensiy uncion. Once his is one one obains an appoximaion o he ansiion ensiy o as a non-linea ansomaion o ha o P. ha we inen o o in ecion 3 o eie he isibuion o he sock pice o a waan-issuing im can be iewe as simila in spii o Ai-ahalia 999. Howee in ou case we ake aanage o he elaionship beween he sock pice o a im an is equiy alue pe shae so ha ou "sanaise" DE is he equiy alue pocess. By assumpion his is Geomeic Bownian Moion hus hee is no nee o ansom o uni iusion an heeoe he ansiion ensiies o ou sanaise DE ae aailable in he well known lognomal om. Noe howee ha alhough ou analysis assumes lognomaliy o he alue pocess i is obious ha ohe cases can also be iscusse..6 Ineing he alue olailiy an Di o he Fim In ecion.4 equaion.4. we hae shown ha he alue o a waan can be calculae as: B 4 Fo uhe eails an o he egulaiy coniions see o example Anol

17 Le us biely illusae how an can be inee so ha hey can be use in he aluaion omula. e sa wih equaion.. i.e.. lealy i he waans ae ae i.e. is obsee an one beliees ha he ae pice coneys inomaion one can ine an iecly om he aboe equaion. Howee i he waans ae no ae e.g. he case o execuie shae opions o one wans o use he "heoeical" alue o he im o in he "heoeical alue" o he waan he ollowing poceue is employe: Fom equaion.5. aboe we hae ha he ollowing epesenaion o he sock pice mus hol: B.6. The pape by chulz an Taumann 994 pesens a mehoology o aiing a boh he unobsee equiy alue an olailiy o a waan-issuing im. The auhos use equaion.6. ogehe wih he ac ha he elasiciy o a sock gies he pecenage change in he sock s alue o a pecenage change in he im s equiy alue; i.e. ε.6. whee is he obseableesimable sock olailiy an ε epesens he sock elasiciy wih espec o he alue o he im. Relaionship.6. is a sana esul in opion picing heoy whee he sock s elasiciy gies he pecenage change in he sock s alue o a ε pecenage change in he im s alue; see Jaow an Ru 983 p.. They hen appoximae he elasiciy by ε. hee. Hence hey obain a sysem o wo equaions B.6.3 which can hen be sole simulaneously o he wo unknown agumens an. e ae able o inepenenly eiyeie equaion.6.3 om ou esuls o he peious secion. Remembe ha we hae shown ha he pocess o he sock pice is o he om: Z. Then he pocess o he eun o he sock pice will be gien by: Z 6

18 whee is he sock eun olailiy. In paicula om equaion.5.4 we hae ha: e Z Hence Bu his is ienical o equaion.6.3 aboe. I is woh epeaing hee ha such a speciicaion o he sock pice pocess an subsequenly olailiy implies ha he sock eun olailiy o a waan-issuing im will be sochasic an will epen among ohe hings on he leel o he sock pice an on ime. O pu a bi ieenly he sock eun olailiy will epen on he moneyness o he waans an on hei ime o mauiy. In ac ou eie pocess o he sock pice impliespouces a olailiy smile o smik simila o he one obsee in opions makes. Thus jus incopoaing waansexecuie opions on he sock pice pocess o he ims ha happen o issue hem coul help euce some o he obsee biases beween heoeical an obsee opions pices. Fo a eaile iscussion on he non-saionaiy an geneal behaiou o sock olailiy we ee he eae o chulz an Taumann 994. Finally i is woh noing ha ae haing obaine alues o an saighowa o also obain he gowh ae i o he im. e know ha i is e Hence soling o he i o he im we ge: e..7 The Non-Taabiliy o he alue o he Fim Fo he geneal case whee he waans o he im ae no ae i.e. he popula case o execuie shae opions he alue o he im is clealy a non-aable quaniy. Does his mean ha pas o ou analysis ha hae been base on no-abiage agumens ae inali? lealy no. e illusae below why his is he case see also he seminal papes o Haison an eps 979 an Haison an Pliska 98: 7

19 e hae ha he non-aable is moelle wih he sochasic ieenial: Z. Alhough is non-aable a eeminisic uncion o is aable 5. In paicula B e In Lemma aboe we hae shown using Io s omula ha has ieenial incemen: e Z As is aable we can wie own is make pice o isk. Assuming ha he iscoun ae is consan a we hae ha e e This is he make pice o isk o he sock pice o a waan-issuing im. Noe ha is a consan an heeoe he ameon-main-gisano bouneness coniion T E P [exp ] < is auomaically saisie. This is suicien o he exisence o he isk- neual measue ha makes he iscoune equiy alue a maingale he isk-neual measue is he measue une which he iscoune sock pice is a maingale i.e. a aable. In ohe wos wha we ae saying is ha using he maingale epesenaion heoem we can consuc ou o an a iskless cash bon. Inee he make pice o isk is simply anohe way o wiing he change om naue s measue P o he isk neual measue Q. The behaiou o une he isk neual measue Q is 5 e also acknowlege he epenence o on ime. Fo cleaness o he agumen howee we ignoe ha epenence o he ime being. 8

20 Z e ~ Z e Z ~ is he Bownian moion une Q ~ Z e B bu now ~ Z which o couse makes he behaiou o he iscoune sock pice a maingale as equie. ince is he change o measue om P o Q we can also wie he behaiou o une he isk-neual measue: Z ~ Z Thus i we hae claims on hey can be pice ia he nomal expecaion oue using his isk neual DE o. 3. THE DITRIBUTION OF THE TO PRIE FOR A ARRANT-IUING FIRM In his secion we eie he "heoeical" isibuion o he sock pice o a waanissuing im when he unelying isibuion o he oal equiy alue o he im is assume o be lognomal. e o his by exploiing he monooniciy o he sock pice wih espec o he equiy alue o he im an applying a non-linea ansomaion: Fom Assumpion we hae ha: Z. Fom his i ollows ha a shae on he alue o he im is lognomally isibue. Is pobabiliy ensiy uncion is hen gien by: ln \ exp 3. π 9

21 whee epesens he alue o he im pe shae a ime. 6 Remembe now om equaion.5. ha he ollowing epesenaion o he sock pice mus hol: B. ince hee exiss a one-o-one elaionship beween an see he agumens pesene o he eiaion o equaion.5.3 we can sole he aboe equaion o hus obaining as a uncion o ; i.e.. I we now sa om he isibuion o he equiy alue i.e. \ an consie he ansomaion we obain \ \ 3. π ln exp whee e ln This is he "heoeical" isibuion o he sock pice o a waan-issuing im. I elecs he poenial o iluion om he ousaning waans an alhough he paamees ae usually no iecly obseable hey can be inee using numeical ouines. To ine he i an olailiy o he im we will use he poceue o ecion.6. PROPOITION : The isk-neual isibuion o he sock pice exiss an can be shown o be equal o: RN π ] [ln exp \ In paicula o he case when he isk-neual isibuion has he ollowing simple om: ln exp \ π RN This isibuion can be use o alue waans ia equaion I T T RN T T Q e E e \ \ 6 Noe hee ha we will use o enoe pobabiliy ensiy uncions geneally an no one speciic pobabiliy ensiy. The agumen o as well as he conex in which i is use will ieniy he paicula pobabiliy ensiy being consiee.

22 PROOF: ee ecion in he Appenix. THEOREM : As he ime o mauiy o he waans ens o ininiy he sock pice o he waan-issuing im coneges in isibuion o he lognomal isibuion: lim Λln o lim Λln Λ enoes he lognomal isibuion wih paamees an. PROOF. By assumpion we hae ha: ~ Λ ln. e hae also shown ha B he ollowing epesenaion o he sock pice mus hol:. B Bu lim. Hence a he limi. The esul now ollows om he popey o he lognomal isibuion ha i X ~ Λ an b an c ae consans whee c > say a c e hen b cx is Λ a b b. In ou case we hae ha an ~ Λ ln. Hence c an b. Theeoe i mus be he case ha: lim Λln ln Howee a he limi we also hae ha. Hence lim Λln Finally i is no ha o show ha lim use equaion.6.3 combine wih he ac ha an lim ollows iecly om. Noe o couse ha he esul also ollows om ou eie isibuion o equaion 3.. Jus ealuae lim \ o conim he esul. The impoance o heoem is ha ceeis paibus ims ha oe long-hoizon execuie shae opions ae less likely o sue iniially om non-nomal isibuions han hose wih sho hoizon opions. Now accoing o coollay an example we mus hae ha he isibuion o he sock pice o an a-he-money waan-issuing im mus also be lognomal. Le us conim ha his is inee he case: I he waan is a he money i.e. e we hae:

23 B ATM Also Then ATM \ \ ln exp π ln exp π Hence i is inee he case ha ln ~ Λ. 3. The isibuion o he waan alue Le us now obain an expession o he isibuion o he waan alue. e sa om \ an emembeing ha B B we consie he ollowing ansomaion: B B 3.. whee. Noe ha B enoes he inese o he Black-choles opion picing omula wih espec o he alue o he im. In geneal B is no known explicily. An excepion occus when he waan is a-he-money. Then B \ \ B B π ln exp

24 whee ln B e Again i he waan is a-he-money i is no ha o show ha he isibuion o he waan alue will also be lognomal wih ~ Λln Noe ha so a we hae coniione on an. Howee hese paamees ae ypically unobseable an hee ae agumens ha he numeical ouines pesene in secion.6 ae no aequae o coecly ine he ue alues o hese paamees. In he ollowing secion we pesen a mehoology whee we can elax he assumpion ha he iusion paamees o he equiy alue pocess shoul be obseableesimable. 4. A BAYEIAN FRAMEOR e now eelop a Bayesian amewok whee we elax he assumpion ha an ae obseable. Employing a Bayesian appoach allows us o accoun o he unobseabiliy o he i an iusion paamees since hese paamees ae eae as anom aiables which can eenually be inegae ou o eliminae ia a non-linea ansomaion. Le us is ieniy pio isibuions o he i an he olailiy paamees o he unobseable alue pocess o he im. In ieniying hese isibuions we ollow sana Bayesian mehoology as pesene in Raia an chlaie 96 Zellne 97 an moe ecenly in Bauwens Lubano an Richa 999 an Dasinos an achell : AUMPTION 3. hen he aiance o an inepenen nomal pocess is assume known bu he mean is a anom aiable he mos conenien isibuion o he naual conjugae o he likelihoo o he sample is he nomal isibuion. The coniional isibuion o he expece ae o eun is heeoe gien by: whee m is a hypepaamee. \ m π m exp 3

25 4 AUMPTION 4. e can assign an Inee-Gamma- isibuion wih hypepaamees as he pio isibuion o. Is pio pobabiliy ensiy uncion is hen gien by: exp \ η η η Γ o exp \ η η η Γ Alhough η m ae hypepaamees which ae also unobseable i is much easie o obain esimaes o hese om analyss oecass om long-un alues o company aa gowh ages se by each company ec han obaining iecly esimaes o he i an iusion paamees o he alue pocess. Noe hee ha when he isibuions ae coniional on any pio paamees i.e. η an m ano on we will no explicily coniion on hem an we shall ee o hem as unconiional. e.g. we will wie \ η. 4.. Deiaion o he Unconiional Disibuion o he ock Pice o a aan- Issuing Fim Le us is obain he join ensiy o i an olailiy. Diecly om Assumpions 3 an 4 we ge: \ Now muliplying he aboe equaion wih equaion 3. we ge: \ Finally Dasinos an achell hae shown ha: Γ ] [ln 4 exp exp m η η π η Now sa om an consie he ansomaion: B B

26 5 whee B enoes he inese uncion o he Black-choles opion pice wih espec o equiy olailiy. This is he so-calle implie olailiy o he opion pice. In geneal B is no known explicily. Howee when he waan is a-he-money he aboe equaion can be wien as B. The Jakobian o he ansomaion is gien by: ega J whee φ e ega ln e heeoe ge Γ ] [ln 4 exp exp m ega J B B B B B η η π η whee φ e ega B B ln. Now om equaion.. we hae ha. I we nex consie he ansomaion: whee J we ge J Γ ] [ln 4 exp exp m ega B B B B η η π η Finally

27 6 m ega B B B B Γ ] [ln 4 exp exp η η π η 4. Deiaion o he Disibuion o he alue o a aan To eie he isibuion o he alue o he waan hee ae a numbe o alenaies. In secion 4. aboe we eie boh an. I is heeoe obious ha o. Howee o spee an simpliciy in he numeical calculaions ha nee o be peome we ecommen he ollowing: Remembeing ha B B sa om an consie he ansomaion: B whee ega J. Hence Γ ] [ln 4 exp exp m J B B B η η π η whee B e ln Finally. 4.3 Deiaion o he coniional isibuion o he sock pice To eie he coniional on he waan alue isibuion o he sock pice we nee one inal calculaion:

28 \ whee he numeao is eie in secion 4. an he enominao in secion 4.. O couse he cos o emoing he epenence on he laen paamees by employing he Bayesian appoach is ha mos calculaions mus be peome numeically base on he quasi-analyic expessions poie aboe. 7

29 5. EMPIRIAL BEHAIOUR OF THE TO PRIE DITRIBUTION 5. ompaing he ock Pice Disibuion o a aan-issuing an a Non-aan- Issuing Fim: An Illusaion o Risk Managemen In ecion 3 equaion 3. we hae eie he heoeical isibuion o a waanissuing im. e epouce i hee o eeence: exp ln \ 5.. π whee ln e e also hae by assumpion ha he sock pice o a non-waan-issuing im is lognomally isibue: ln \ exp 5.. π In Table below we compae he wo isibuions by means o epoing summay saisics i.e. mean sana eiaion skewness an kuosis o he aily weekly an monhly sock pice isibuions o a non-waan-issuing im i.e. a lognomal isibuion an o waan-issuing ims wih ieen iluion acos 5% 5% an % an ieen egees o moneyness o hei waans. The calculaions ae base on paamees alues: cuen sock pice: sock eun olailiy: 5 % ime o mauiy o waans: yeas isk-ee ae: 5% i ae: 5% iien yiel:. The equiy olailiy an equiy i o he waan-issuing ims ae calculae using he poceue ouline in ecion.6. e ae paiculaly ineese o obsee how he skewness an kuosis alues o he sock pice isibuions o he waan-issuing ims eiae om he especie benchmak alues o skewness an kuosis o he lognomal isibuion. e eine he so-calle pecenage eiaion om he benchmak skewness an kuosis alues o he lognomal isibuion as: aan-issuing im s kewnes ouosiss alue - Lognomal kewness o uosis alue Pecenage Deiaion Lognomal kewness o uosis alue Ou esuls ae exhibie in Table. 8

30 TABLE ummay saisics o he aily weekly an monhly sock pice isibuions o:. A im wih no waans i.e. he unelying sock pice isibuion is lognomal. A waanexecuie opion issuing im wih iluion aco 5% an ieen egees o moneyness. 3. A waanexecuie opion issuing im wih iluion aco 5% an ieen egees o moneyness. 4. A waanexecuie opion issuing im wih iluion aco % an ieen egees o moneyness. The numbes in paenheses enoe he pecenage eiaion o he skewness an kuosis alues o he waanissuing-im s sock pice isibuion om he especie benchmak alues o he lognomal isibuion. ock Pice Disibuion: Mean. De. kewness % De. E. uosis % De. Daily Lognomal In-he-money waans 5% %.4.5% 5% %.48.% % % % Nea-he-money waans 5% %.38-5.% 5% %.3 -.% % %.3 -.5% Ou-o-he-money waans 5% % % 5% %.9-5.5% % %. -7.5% eekly Lognomal In-he-money waans 5% %.3.% 5% %.4.6% % % % Nea-he-money waans 5% %.9-3.5% 5% %.6-9.% % %.55 -.% Ou-o-he-money waans 5% % % 5% % % % % % Monhly Lognomal In-he-money waans 5% %.8.6% 5% %.973.8% % % % Nea-he-money waans 5% % % 5% % % % % % Ou-o-he-money waans 5% % % 5% % % % % % 9

31 I is clea om able ha he exisence o waans in he capial sucue o a im oes aec is sock pice isibuion paiculaly so o ims wih moeae o high iluion acos. In paicula egaing he mean an sana eiaion o he sock pice hee is no signiican ieence beween waan-issuing ims o any iluion an moneyness an non-waan-issuing ims. Howee when i comes o he skewness an kuosis alues we obsee a clea paen. Fims wih in-he-money waans en o hae simila o slighly highe alues o skewness an signiicanly highe alues o excess kuosis when compae wih he lognomal isibuion. Fims wih nea-he-money waans en o hae lowe alues o skewness an excess kuosis han hei especie non-waan-issuing counepas. Finally ims wih ou-o-he-money waans en o hae signiicanly lowe alues o skewness an excess kuosis when compae wih non-waan-issuing lognomal ims. Neeless o say hese eecs ae magniie o all cases he highe he iluion aco. An inuiie inepeaion o hese esuls is ha ims wih ou-o-he-money waans en o be iskie han ims wih in he money waans. Inee ou esuls sugges ha he log-pice an eun o a im wih ou-o-he-money waans will hae a longe an hinne le ail han he nomal isibuion while a im wih in-he-money waans a shoe an ae ail. To obain he log pice o log eun isibuion o a waan-issuing im one nee only ake he log o log ansom o he isibuion o equaion 5... Fo alue-a-risk calculaions his is cucial. Fo example i a nomal appoximaion is use o he log-pice an eun isibuions o all ims we will en o calculae a ar ha is oo low o ims wih ou-o-he money an nea-he-money waans an oo high o ims wih in he money waans. Inee paciiones oen inepe a im wih in-he-money execuie sock opions as a goo inicaion ha he im has been meeing is gowh ages an ceeis paibus is expece o o so in he uue. On he ohe han he pesence o ou-o-he money execuie opions can be inepee as a signal o unepeomance o inancial isess. Typically waans ae issue ou-o-he-money an execuie sock opions ae issue nea-he-money. Boh insumens ae issue wih ey long mauiies o example execuie sock opions las o as long as 5 yeas. e o no pesen isibuional esuls o ims wih ey long mauiy waansexecuie opions since base on ecion 3 Theoem we hae ha he isibuion o he sock pice o hese ims coneges o he lognomal as he ime o mauiy inceases ineiniely. Insea we moel he sock pice isibuion o hese ims when hei waans ae wihin a ew yeas o mauiy. 3

32 5. ompaing he Risk-Neual ock Pice Disibuion o a aan-issuing an a Non-aan-Issuing Fim: An Applicaion o Execuie Opionaan aluaion In ecion 3 Poposiion we hae eie he isk-neual isibuion o a waanissuing im. e epouce i hee o eeence: 5.. RN \ exp ln π This isibuion can be use o alue waans ia equaion Q D e E \ I e RN \ 5.. To ae his appoach has nee been use o alue waans ue o he lack o knowlege o RN \. aluaion o waans has been o couse caie ou using he sana alue o he im appoach: B 5..3 Neeless o say ha equaions 5.. an 5..3 pouce ienical waan alues. They ae ae all equialen appoaches o waan aluaion. Now opion-like waan aluaion suggess ha insea o using he isibuion o equaion 5.. one can obain a ey goo appoximaion o he alue o he waan by using a Black-choles amewok an aluing he waan as i i was ienical o a call opion on he sock o he im. This basically means using he lognomal isk-neual isibuion in B equaion 5.. aboe o equialenly using. The lognomal isk neual isibuion i.e. he isk-neual isibuion o a non-waan-issuing im is gien by: ln RN \ exp 5..4 π In Table below we compae he wo isk-neual isibuions by means o epoing summay saisics i.e. mean sana eiaion skewness an kuosis o he sock pice isibuions o a non-waan-issuing im i.e. a lognomal isibuion an o waanissuing ims wih ieen iluion acos 5% 5% an % ieen egees o moneyness o hei waans an ieen mauiies yeas an 3 monhs. 3

33 TABLE ummay saisics o he isk-neual isibuions o he sock pice o:. A im wih no waans i.e. he unelying isk-neual isibuion is lognomal. A waan-issuing im wih iluion aco 5% ieen egees o moneyness an ieen mauiies 3. A waan-issuing im wih iluion aco 5% ieen egees o moneyness an ieen mauiies 4. A waan-issuing im wih iluion aco % ieen egees o moneyness an ieen mauiies ock Pice Disibuion: Mean. Deiaion kewness Excess uosis Lognomal yeas In-he-money waans: 5% % % Nea-he-money waans: 5% % % Ou-o-he-money waans: 5% % % Deep-ou-o-he-money waans: 5% % % Lognomal 3 monhs lose-o-mauiy-ou-o-hemoney-waans: 5% % % The calculaions ae base on paamees alues: uen sock pice: ock eun olailiy: 5% Time o mauiy o waans: yeas o he las case o lose-o-mauiy-ou-o-he-money waans a ime o mauiy o 3 monhs was use insea Risk-ee ae: 5% Diien yiel:. The isk-neual lognomal isibuion is gien in equaion 5..4 The paamee speciic i.e. speciic on iluion moneyness ime o mauiy ec isk-neual isibuion o he sock pice o a waan-issuing im is gien in equaion 5... I is obious om Table ha he isk-neual isibuions o waan an non-waanissuing ims ae makely ieen. Geneally he log-sock pice isibuion o high iluion waan-issuing ims exhibis highe mean lowe sana eiaion an make skewness an excess kuosis when compae o he log-sock pice o a non-waan-issuing im i.e. a nomal isibuion. Thus he nex quesion ha naually aises is how much his will aec he picing o waans. In Table 3 below we epo he waan alues obaine using he heoeical appoach an he opion-like appoximaion appoach. e compae hese alues by wha we call he pecenage mispicing eo. This is calculae as: 3