Abstact he textbook teatmet fo the aluato of waats takes as a state aable the alue of the fm ad shows that the alue of a waat s equal to the alue of a call opto o the equty of the fm multpled by a dluto facto. hs appoach howee apples oly fo the ot so ealstc case whee the fm ssues oly a sgle waat. By a sgle waat we mea waats wth a sgle execse pce ad tme to matuty. hat happes howee the eal wold whee copoatos ssue multple waats ad execute stock optos? hat happes fo example whe fms ssue waats of dffeet execse pces dffeet matutes ad dffeet dluto factos? alug each type of waats depedetly of the othes s clealy appopate ad wll esult mspcg. I ths pape we dee dstbuto-fee ad dstbuto-specfc fomulae fo fms that ssue waats wth dffeet matutes dffeet stke pces ad dffeet dluto factos ad fo fms that ssue waats of the same matuty but dffeet stke pces ad dffeet dluto factos. he dstcto we make betwee waats ad execute stock optos s smply a matte of whethe the cotact s taded o ot. e use the tem waat to coe both cases. eywods: aluato waats execute stock optos captal stuctue dluto. JEL lassfcato: G G.
ON HE ALUAION OF ARRAN AND EXEUIE O OPION: PRIING FORMULAE FOR FIRM IH MULIPLE ARRAN/EXEUIE OPION HEOFANI DARINO AND EPHEN E. AHELL * Faculty of Ecoomcs ad Poltcs Uesty of ambdge Apl. INRODUION aats execute/employee stock optos ad exchage taded call optos ae all examples of cotget clams. oceptually they ae all athe smla stumets. hey do afte all pode the holde wth a call opto o the udelyg. Howee despte the supefcal smlaty waats ad execute optos dffe a umbe of mpotat ways fom alla exchage taded optos. hese dffeeces tally betwee waats ad stadad call optos ad the ey ecet yeas betwee execute stock optos EOs ad exchage taded optos hae sped a geat deal of academc ad pactcal teest that has spaed the subject ot oly fom a puely asset pcg/aluato pespecte o fom a puely copoate face/agecy theoy pespecte but has also esulted a amalgamato of the two the sese that aluato of waats ad EOs has occasoally ot bee fee of cosdeatos such as copoate polces maageal cetes stockholdes atttudes etc. Futhemoe as teest ad moe mpotatly ssuace of these copoate stumets patculaly EOs has suged oe the yeas ad all ths amdst wdespead demad fo moe facal taspaecy copoate accouts assessg coectly the tue cost/alue to stockholdes/stakeholdes has ee bee moe mpeate. e fst befly outle a few fom a log lst of featues that dstgush waats ad EOs fom stadad exchage taded optos ad ca potetally complcate *heo Dasos gatefully ackowledges facal suppot fom the A.G. Leets Foudato the ebuy cholashp Fud at the Uesty of ambdge ad the Ecoomc ad ocal Reseach oucl ER. e thak asos Myloas fo useful dscussos.. Dasos e-mal: td@eco.cam.ac.uk..e. atchell e- mal: tee.atchell@eco.cam.ac.uk Accodg to Hull : aats ae call optos that ofte come to exstece as a esult of a bod ssuace. hey ae added to the bod ssue to make t moe attacte to estos. ypcally a waat lasts fo a umbe of yeas. Oce they hae bee ceated they sometmes tade sepaately fom the bods to whch they wee ogally attached. Execute stock optos ae call optos ssued to executes to motate them to act the best teests of the compay s shaeholdes. hey ae usually at-the-moey whe they ae fst ssued. Afte a peod of tme they become ested ad ca be execsed. Ulke waats ad exchagetaded stock optos they caot be sold. hey ofte last as log as o 5 yeas.
the aluato. e the cocetate oe patcula featue amely the captal stuctue o dluto effect of waats ad EOs whch podes the bechmak fo ths pape. o stat wth waats ofte cota specal posos that allow the ssug compay the ght to call the waats; see chultz 99 as well as the possblty to alte the matuty ad/o execse pce; see Logstaff 99 Howe ad e 99. he same also apples to EOs whee esettg the tems of aleady ssued optos that hae goe udewate.e. deep-out-of-the-moey s a commo pheomeo amogst youg deelopg copoatos. hs usually oles specfyg a ew stke pce whch at the tme of esettg s ofte set equal to the cuet stock pce. Extedg the matuty of the optos o cacelg a execute's exstg optos ad gatg hm o he ew optos wth a lowe stke pce although ot as fequet ae also the ageda. Amog othes Bee udaam ad Yemack ate ad Lych exame the epcg of EOs ad allocate t to poo fm-specfc pefomace. he empcal edece hace uma ad odd suggest that EOs ae usually epced whe the stock decles by about 5%. Achaya Joh ad udaam estgate the optmalty of esettg EOs ad fd that a copoate polcy that allows/accommodates fo some esettg s almost always optmal. he default ths study wll be that all the afoemetoed specal featues whch we heeby efe to as specal execse posos ae exogeous. Howee f equed t s also possble to edogeze them the aluato. Bee udaam ad Yemack ote fo example that a EO that wll be epced.e. whose stke pce wll chage to a lowe stke pce the fst tme the stock pce falls below a pe-specfed bae ca be alued as a potfolo of a dow-adout call wth the tal stke pce ad a dow-ad- call wth the ew lowe stke pce. Mog o ow fom the specal execse posos of the stake-ssues to alue ehacg execse stateges of the stake-holdes Emauel 98 ostatdes 984 ad patt ad tebez 988 hae examed execse stateges by waat-holdes to cease the alue of the waats e.g. sequetal execse ealy execse ad the actos of the waats-ssues to eutalze ths e.g. eestmet polces. Alog the same les maageal cetes asg fom the ssuace of EOs to cease the stock pce cease sk educe dded yeld see Johso ad a apete ad cetes tmg the execses of EOs; see apete ad Remmes ae also futful aeas fo eseach. Aga fo the puposes of ths pape we teat alue-ehacg o sometmes alue-educg maageal stateges as exogeous factos. ug to matuty ssues both waats ad EOs hae much loge matutes tha exchage taded optos compae fo example -5 yeas ad -5 yeas espectely wth up to yea fo stadad optos. Lautebach ad chultz 99 ad Rubste 995 ae good efeeces detfyg some of the dffcultes assocated wth applyg stadad opto models to log-tem optos. he esults peseted ths study ae dstbuto fee ad thus applcable to a wde age of opto pcg models cludg
models such as the ostat Elastcty of aace E of ox ad Ross 976 o Meto s 976 Jump-dffuso whch mght be moe sutable fo alug log-tem optos. Fom a facal ecoomcs pot of ew the theoetcally most teestg featue of waats ad execute stock optos as opposed to stadad calls s the captal stuctue effect o the ssug fm. he waats ad execute optos ae execsed ew shaes ae ssued ad ay execse poceeds ae pad to the fm. As a esult the teests of the fm's exstg shaeholdes ae dluted cludg possble dluto wealth otg ghts ad the udelyg stock pce. he souce of ths dluto s the game that s played betwee the stakeholdes ad exstg shaeholdes. A way to deal wth the so-called dluto effect ogally suggested by Black ad choles 97 populazed by Gala ad chelle 978 ad also dscussed Lautebach ad chultz 99 chulz ad autma 994 s to alue waats as call optos o the alue of the fm ot the stock pce multpled by a dluto facto. hs famewok has subsequetly become the stadad teatmet fo waat aluato ad appeas almost eey textbook o deate secutes. Oe the past decade empcal lteatue o waat pcg see fo example Besoussa ouhy ad Gala 99 eld 994 deus 996 has suggested that thee s o eed to follow the textbook teatmet to alue waats.e. the alue of the fm appoach. Istead based o the empcal esults ad smulatos these studes ecommed opto-lke waat aluato. hs smply meas alug the waat as f t was detcal to a call opto wthout olg the aluato pocess the typcally uobseable alue of the fm ad gog ay dluto. upsgly such a appoxmato woks ey well fo a lage umbe of cases. ox ad Rubste 985 ad Igesoll 987 ecogze that whe dluto s suffcetly small the t may be goed. chulz ad autma 994 ad Dasos ad atchell hae llustated that opto-lke aluato wll deed pce coectly ad effcetly -the-moey ea-the-moey ad log matuty out-of-the-moey waats ad EOs. Howee they also fd that fo deep-out-of-the-moey ad ea matuty out-of-the-moey waats geeal ths appoxmato wll sgfcatly oepce waats wth the oepcg a ceasg fucto of the dluto facto. he pupose of ths pape s to exted the theoetcal famewok fo waat ad EO aluato so that t coectly accouts fo the dluto effect the pesece of multple waat o EO ssuaces. Both the exstg theoetcal famewok ad the opto-lke aluato appoxmato outled aboe ae oly appopate fo the case whe fms ssue just a sgle waat. By a sgle waat we mea detcal Regadg the dluto facto thee appea to be o specfc lmts placed upo the total umbe of optos gated. Each buch of shae optos to be awaded s put befoe the shaeholdes the geeal meetg fo the appoal. Oe tme the total gated ca se to a cosdeable pecetage ofte well excess of % of the ssued captal patculaly so fo hgh-tech fms.
outstadg waats wth a sgle execse pce ad tme to matuty. Fo the moe ealstc case whee fms ssue multple waats wth dffeet execse pces ad dffeet matutes applcato of the cuet textbook appoach wll esult oepcg. Ideed the exstg famewok does ot take to accout the tedepedece betwee waats ad the potetal dluto effect of each ssuace o the emag waats. he fomulae deed ths pape copoate the potetal dluto effect of all the outstadg waats o each patcula ssuace thus adjustg dowwads the tue alue/cost of these stumets. Futhemoe just because opto-lke aluato s ofte a good appoxmato techque fo the sgle waat/eo case does ot a po mply ad ghtly so that t wll poe useful fo appoxmatg the alue of multple waat ssuaces. By deg the theoetcal fomulae fo the aluato of multple ssuaces we pode the coect bechmak agast whch to check the qualty of ths appoxmato. hs le of equy s patculaly useful sce copoatos the aual epots accodg to the FAB madates ofte calculate the alue/cost of the EOs usg opto-lke aluato. e belee that a exteso of the exstg famewok s madated sce aely pactce do compaes ssue oly a sgle Execute tock Opto o waat. Fo example Maquadt 999 exames 58 Fotue fms oe a yea peod ad fds a aeage of 7 cotacts pe fm. Futhemoe a casual glace at the accouts of ay compay that gats optos to ts executes/employees eeals a aety of matutes ad execse pces. he dstcto we make ths pape betwee waats ad execute shae optos s smply a matte of whethe the cotact s taded o ot. Moeoe we take the tem execute stock optos to cota the popula subset of employee shae opto schemes whch hae bee successfully toduced amogst copoatos dug the past few yeas. Geeally we shall use the tem waat to coe all cases.e. waats execute optos ad employee optos. Issues of o-tasfeablty delayed estg ad suboptmal execse polces of Execute tock Optos EOs ae ot dscussed ths pape. Fo these ad a few othe dffeeces betwee exchage-taded optos ad EOs we efe the eade to Rubste 995. Hee t suffces to say that the educto alue of a execute opto due to the o-tasfeablty costat s usually hadled my multplyg the alue that would othewse be obtaed.e. the alue of a taded waat by oe mus the pobablty that the execute/employee wll leae the fm befoe execse s possble. he default ths study wll be that the pobablty of pematue leae s zeo t s staghtfowad howee to modfy ths assumpto. Regadg ow the delayed estg costat o execse polcy of EOs geeal t s deed the case that most opto plas do ot pemt employees to execse It s woth metog hee that waats ae usually ssued deep-out-of-the-moey whle EO s ae typcally ssued at-the-moey see fo example Noee ad olfso 98. Hall ad Muphy estgate the ecoomc atoale behd ths appoach ad fd that by settg the execse pce at- o eathe gat-date maket pce copoatos maxmze the cetes of the employees ad executes. 4
the gated optos utl afte a pedefed peod of tme has elapsed fo example a execute opto has typcally a matuty of yeas; howee though delayed estg execse s usually ot pemtted fo a peod afte gat typcally yeas. I othe wods EOs ae ethe Euopea o Ameca. I ts Exposue Daft Accoutg fo tockbased ompesato FAB allows alug EO s as Euopea but eques usg the so called expected lfe of the opto stead of the actual tme to expato. apete 998 exames the execse polces of maages whch ca be execute-specfc whe the ddual pefeeces edowmets hedgg ad tasfeablty costats come to play. he shows that ealy execse of a EO s ot cosstet wth execse pattes obseed the data. Executes hold optos log eough ad deep eough to the moey befoe execsg to captue a sgfcat amout of the potetal alue. Detemple ad udaesa 999 fd that EOs may be execsed pematuely ee whe the udelyg asset pays o ddeds. I ay case as dcated by Rubste 995 a EO has two alues. he oe s hghly pesoal depedg o the pefeeces ad facal ccumstaces of each employee. he othe whch s the oe that should be used by the copoato ts exteal facal statemets s the alue of the opto accodg to the effect ts exstece has o the alue of the udelyg stock. hs s the oto of alue that we use ths pape. As a fal ote EOs ae patculaly popula amogst hgh-tech fms. uch copoatos pomse apd gowth but also pay lttle o o ddeds. see amogst may othes Mcosoft opoato Oacle sco ystems AoL me-ae etc. hs has the addtoal adatage that Ameca-type EOs ca fo the puposes of aluato be cosdeed as Euopea. he pla fo the pape s as follows. I ecto we outle the textbook teatmet fo waat aluato. e show that the pce of a waat ca be ealuated as a call opto o the equty alue of the fm multpled by a dluto facto equal to /. epesets the ato of the total umbe of ew shaes ssued upo execse of the waats to the total umbe of exstg shaes. Howee we clam that ths famewok apples oly fo fms wth a sgle waat/eo ssuace whch pactce s aely the case. I ecto we exted the theoetcal famewok to accommodate fo multple waats. I patcula secto. we dee pcg fomulae fo fms that ssue waats of dffeet matutes dffeet stke pces ad dffeet dluto factos whle secto. we pode fomulae fo fms wth waats of commo matuty but dffeet execse pces ad dffeet dluto factos. he deed fomulae ae dstbuto fee ad ae thus applcable to a wde age of pocess assumptos fo the state aable. I ecto 4 we make the deed fomulae opeatoal ude the ey wdely-used assumpto of Geometc Bowa Moto fo the alue pocess. ecto 5 cotas a empcal llustato whee we alue the outstadg EOs of sco ystems. e fd that applcato of the sgle waat aluato techque the multple settg of sco ystems esults sgfcat oepcg. O the othe had Black-choles opto-lke aluato esults udepcg. hs e-efoces exstg coces o the optmalty of 5
usg stadad umodfed opto pcg models to accout fo the cost of stock-based compesato. ecto 6 cocludes.. A FRAMEOR FOR ARRAN PRIING. he extbook eatmet fo the aluato of aats: Fms wth a gle aat Issuace aluato of waats dates as back as the aluato of taded optos. I fact accodg to Black 979 the celebated opto aluato model of Black ad choles 97 was ogally deeloped fo the pcg of waats. Howee waats ad execute stock optos ae wtte by compaes o the ow stock. he they ae execsed the compay ssues moe shaes ad sells them to the stakeholdes fo the stke pce. hs subsequetly dlutes the equty of exstg shaeholdes. he textbook teatmet fo waat aluato thus madates that the waats should be egaded as call optos o a shae of the total equty alue of the fm whee equty s defed as the sum of the alue of ts shaes ad the alue of ts waats. 4 e theefoe stat wth the followg smplfyg assumpto: AUMPION : he waat-ssug fm s a all equty fm wth o outstadg debt. hs the mples that stocks ad waats ae the oly souces of facg that the compay s usg. Hece at cuet tme t the compay has a total equty alue of: N whee N: he umbe of outstadg shaes. : he cuet pce pe shae. : he umbe of outstadg waats : he cuet pce of a waat o a shae. mlaly the alue of the fm pe shae total equty pe shae / N s: 4 Alteately aluato of waats ca also be pefomed by pcg them as call optos o the stock pce of the waat-ssug fm. hs of couse eques kowledge of the dstbuto of the stock pce of the waat-ssug fm whch wll be dffeet both fom the dstbuto of the alue of the fm ad fom the dstbuto of the stock pce of a o-waat ssug fm see Dasos ad atchell fo moe detals. 6
whee / N s the dluto facto. Moe geeally t t ad t wll be the alues of the aboe ettes at tme t. uppose ow that each waat ettles the holde to puchase oe shae of the fm at tme fo a stke pce of pe shae. 5 ΠAE : he waats ae execsed at matuty. If the outstadg waats ae execsed at tme the compay ecees a cash flow of ad the total equty alue ceases to. hs alue s the dstbuted amog N shaes so that the pce pe shae mmedately afte execse becomes / N Hece f the waats ae execsed the payoff to each waat holde s N N N N N Of couse the waats wll be execsed oly f >. ΠAE al: he waats ae ot execsed at matuty. If the waats ae ot execsed the o dluto of equty takes place. At matuty the pce pe shae wll be ad the payoff to the waat holdes wll be zeo he waats wll ot be execsed whe. e hae theefoe llustated that the payoff to a waat holde at matuty wll be 5 e assume that each waat ettles the holde to puchase oe shae of the fm. It s staghtfowad to modfy ths assumpto ad assume that each waat ettles the holde to puchase p umbes of shaes. Fo smplcty the default ths study wll be p. 7
But ths s the payoff of / call optos o the total equty alue of the fm pe shae. Hece the alue of a waat at cuet tme t ca be obtaed as: exp τ f RN / f RN \ 4 whee f RN / s the codtoal sk-eutal dstbuto of the alue of the fm pe shae. tg the aboe symbolc otato suppessg ay subscpts we hae 5 whee deotes a call opto pcg fucto s the tme to matuty of the waat ad epeset the equty olatlty ad sk-fee ate espectely. 6 hs the mples see also equato that the followg epesetato fo the stock pce of a waat-ssug fm must hold: 6 I othe wods the stock pce of a waat/execute opto ssug fm s ge as a lea combato of the equty alue pocess ad a opto o the equty alue pocess. heoetcally as fst dcated by Gala ad chelle 978 waat aluato ca also be pefomed usg the stock pce dstbuto of the waat-ssug fm. ymbolcally ths meas alug waats as: D 7 whee D epesets a call opto aluato fucto. deotes the stock etu olatlty. 6 Fo claty we ote hee that whe we efe to equty olatlty o stock olatlty we do t eally mea the absolute olatlty of the equty o the stock but athe the olatlty of the ate of etu o the equty o stock espectely. 8
alug waats usg equato 5 o equato 7 wll poduce detcal esults. he two equatos epeset afte all equalet appoaches to waat aluato. Note howee that D s dffeet fom. he two call opto pcg fuctos ae dffeet the sese that s deed based o the pocess followed by the alue of the fm whle D depeds o the pocess of the stock pce of the waat-ssug fm. Dasos ad atchell dee the dstbuto of the stock pce of a waat-ssug fm ad show that t s makedly dffeet fom the dstbuto of the stock pce of a owaat ssug fm patculaly so fo hgh to modeate dluto factos. Howee they also fd that despte of the fact that the sk-eutal dstbutos of a waat-ssug fm ad a o-waat-ssug fm ae dffeet aluato by takg expectatos of the dscouted payoff of the waats oe the two dffeet sk-eutal dstbutos poduces waat pces ey close to each othe fo a lage umbe of cases ee whe the log-stock pce dstbuto of the waat-ssug fm exhbts maked skewess ad kutoss. he sgfcace of ths esult s that fo a lage umbe of cases waats ca be alued usg the followg opto-lke aluato appoxmato 7 : whee 8 D Exceptos occu fo deep-out-of-the-moey ad close to matuty out-of-the-moey waats geeal. I such cases the opto-lke aluato appoxmato of equato 8 wll oepce waats wth the pcg eo ceasg the hghe the dluto facto. I ay case the famewok peseted thus fa apples oly fo the ot so ealstc case whee the fm ude cosdeato ssues oly a sgle waat. hat happes howee the eal wold whee copoatos ssue multple waats ad EOs? Do the same fomulae apply fo fms that ssue waats ad EOs wth multple execse pces ad matutes? e pode the aswe to ths questo the followg secto. 7 o aod ay msudestadg we ote hee that the atue of the so-called opto-lke aluato appoxmato s that t takes the dstbuto of the stock pce of a o-waat-ssug fm as a appoxmato to the dstbuto of the stock pce of a waat-ssug fm. Futhemoe to make thgs moe specfc the pcg fucto s usually take to be the Black-choles model. Howee f s Black-choles the the pesece of waats D wll ot be Black-choles. he atue of the appoxmato s thus that D s appoxmated by the Black-choles model. 9
. EXENDING HE FRAMEOR FOR ARRAN PRIING: FIRM IH MULIPLE ARRAN Most f ot all compaes ssue moe tha oe type of waats/execute stock optos. By ths we mea that compaes ssue q types q of waats each wth ts ow execse pce ad tme to matuty. Moeoe fo each type of waats thee s a umbe of outstadg waats. Ude such a specfcato the compay s cuet alue of the fm pe shae s ge by: 9 whee N s the dluto facto of each type of waats. / : he pce of a type- waat o a shae at cuet tme t. hs the mples the followg epesetato fo the stock pce of the waatssug fm: q q Howee ths tme we caot wte the multple-waat case as a aalogue of the sgle waat case. I othe wods the followg elatoshp does ot hold: q hat s we caot pce each type of waats depedetly of the othes as fo q. alug each type of waats as a sgle ssuace usg equato wll esult oepcg. e eed a aluato fomula that takes to accout the tedepedece betwee waats ad the potetal dluto effect of each ssuace o the alue of the othe waats. o smplfy exposto we wll cosde two subcases. I secto. we fst cosde the case whee the fm ssues waats of dffeet matutes ad dffeet stke pces whle secto. we cosde the case whee the copoato ssues waats of commo matuty but wth multple execse pces.
. Fms wth waats of dffeet matuty ad dffeet stke pce o stat wth assume fo smplcty a fm wth two waats of dffeet matutes ad ad dffeet stke pces ad. 8 thout loss of geealty take <. he the cuet alue of the fm pe shae s ge by: whee / N ad / N ae the dluto factos. ΠAE : he fst-type waats ae execsed at matuty. If the fst-type waats ae execsed at tme the compay ecees a cash flow of ad the total equty alue ceases to. hs alue s the dstbuted amog N shaes so that the pce pe shae mmedately afte execse becomes / N Hece f the fst-type waats ae execsed the payoff to the holdes at matuty s: N N N N N By a smla agumet poded that the fst-type waats hae bee execsed at tme we hae that f at tme the secod-type waats ae execsed the compay wll ecee aothe cash flow of. he pce pe shae mmedately afte execse ow becomes: / N Hece f the secod-type waats ae execsed the payoff to the waat holdes s N 4 N N N N 8 Note hee that the stke pces ca also be the same.
he payoffs equatos ad 4 wll be ald f ad oly f the fst-type waats hae bee execsed at tme. e expess the pobablty that the aboe payoffs ae ald as P ob >. ΠAE : he fst-type waats ae ot execsed at tme. he of couse at matuty: 5 aluato of the secod-type waats ow educes to alug a sgle waat ssuace. e ca theefoe show that the payoff of the -matuty waats wll be: 6 hese payoff stuctues wll hold f ad oly f the fst-type waats hae ot bee execsed at tme. he pobablty of ths happeg s atually - P ob >. e ca ow dee pcg fomulae fo the two waats. Fo the -matuty waats ths s staghtfowad. e just hae usg equatos ad 5 that the cuet alue s: O symbolc otato exp f RN \ d 7 he alug the -matuty waats we must take to accout the potetal dluto effect of the -matuty waats. e theefoe hae usg equatos 4 ad 6: P ob > exp f RN \ d P ob > exp [ ] f RN \ d
O symbolc otato: P ob > 8 P ob > e also get the followg epesetato fo the stock pce of ou waat ssug-fm: P P ob ob > > 9 Expessos fo the case of q dffeet waats ca also be deed a smla mae. I fact the aboe fomulae ae ald fo a fm wth q-types of waats wth matutes q < < < < < <....... o dee the alue of the waats wth matuty we eed to calculate pobabltes o moe pecsely poducts of pobabltes. he aluato fomula fo the -matuty waats wll thus cosst of 4 tems ad s ge by: P P ob ob > > P P ob ob > > P P ob ob > > P P ob ob > > I geeal to fd the aluato fomula fo the -matuty waats fo ay we defe > f f δ δ...
4 he thee ae possble alues of... δ δ. Let be the set of all such alues. Also let P ob p >. Hece the fomula wll cosst of tems ad s ge by: p p δ δ δ δ hs s the aluato fomula fo a fm wth waats of dffeet matuty ad dffeet o commo stke. e make pactcal use of ths fomula late ou empcal applcato of secto 5 whee we alue the multple EOs of sco ystems. e ow tu to deg the fomula fo a fm wth waats of commo matuty but dffeet stke pces... Fms wth waats of commo matuty but multple execse pces Aga fo ease of exposto cosde fst a fm wth waats wth execse pces ad ad commo tme to matuty. thout loss of geealty we assume that <. Œ AE : Both waats ae execsed at matuty. If both waats ae execsed at matuty the compay ecees a cash flow of ad the total equty alue ceases to. hs alue s the dstbuted amog N shaes so that the pce pe shae mmedately afte execse becomes N Hece f the waats ae execsed the payoff to the holdes s: N N N N
mlaly N N Of couse both waats wll be execsed f ad oly f > ad >. ce we hae < ths the mples that both waats wll be execsed f ad oly f > ΠAE : Oly oe waat s execsed at matuty If oly the waat wth the lowest stke pce s execsed the usg a smla pocedue as aboe we ca show that the payoff to the waat holdes s: ad N N 4 he waat wth the lowest execse pce wll be execsed f ad oly f >. ΠAE al: Noe of the waats ae execsed he fal case whch s edudat s whe oe of the waats ae execsed. he the payoff to the holdes s zeo. hs wll be the case f. e ca ow dee pcg fomulae fo the two waats. Let us fst obta a aluato fomula fo. he fst waat has two dffeet ozeo payoffs depedg o whethe oe o both waats ae execsed: exp [ f / d RN 5 5
6 ] \ d f RN hs the mples that 6 he calculatos equed to ae to equato 6 fom equato 5 ae exhbted the Appedx. o obta a aluato fomula fo we ote that the secod-type waats hae a o-egate payoff oly whe both waats ae execsed. Hece t follows mmedately that: d f RN \ ] [ exp o symbolc otato 7 It s ow staghtfowad to ty ad geealze the fomulae. Fo example fo a fm wth waats wth execse pces < < ad commo tme to matuty usg the same methodology as aboe we get: 8 9
Geeally fo q-types of waats wth commo matuty ad multple stke pces < < <... < <... < q we hae that fo ay : q j j j j j [ ] q whee we use the coeto that whe > q. j j 4. PEIFYING A PROE FOR HE AE ARIABLE o fa ou aalyss has bee dstbuto fee ad the esults peseted aboe ae ald fo a age of pocess assumptos fo the alue of the fm. Howee to make the fomulae opeatoal we must ow specfy the law that goes the eoluto of. I othe wods we must assume a pocess fo the state aable so that the call opto pcg fucto obtas a specfc fuctoal fom ad ca theefoe be ealuated. he lteatue o waat pcg has maly cocetated o thee pocesses fo the alue of the fm. Apat fom Black ad choles s Geometc Bowa Moto whch assumes costat olatlty ox ad Ross s 976 ostat Elastcty of aace E model see Noee ad olfso 98 Lautebach ad chultz 99 chulz ad autma 989 ad Meto s 976 Jump Dffuso model see eme ad Roefeldt 99 hae also bee used to model the alue pocess. he esults peseted aboe wll be ald ude ay specfcato. Of couse the fuctoal fom of wll deped o the patcula pocess chose. It s ot the am of ths pape to go to a detaled eew of the lteatue o the most appopate alue pocess fo waat pcg. A exhauste study that eews the empcal eseach ude alteate stochastc pocesses has aleady bee coducted by eld 994. Hee t suffces to quote oe of the coclusos of eld: hee s o cocluse edece to eplace dded coected models whch a costat olatlty s assumed.e. Black-choles lke models by moe complcated models such as the Jump Dffuso o the E model. Fo opeatoal coeece ad sce t s the absolute stadad the dusty we theefoe assume heeafte that the alue pocess s a logomal dffuso: 7
AUMPION : he total equty alue of the fm pe shae s goeed by Geometc Bowa moto: d µ dt db whee µ : he expected ate of etu o the alue of the fm s assets : he stadad deato of the ate o the alue of the fm s assets. B : A stadad Bowa moto. Assumpto mples that a sk-eutal wold: Z exp whee Z B B ;.e. Z s a stadadzed omal aable. As a bypoduct of assumpto we ca ow calculate explctly usg equato the pobablty that P ob >. Remembe that ths pobablty appeas the pcg equato fo multple waats wth dffeet matutes;.e. equato. e theefoe hae that p P ob > P ob exp Z > l P ob Z > l P ob Z > l Φ whee Φ deotes the stadad omal cumulate dstbuto fucto emembe that Z s a stadad omal adom aable. Futhemoe the pcg fucto δ that also appeas equato takes the famla fom 8
B δ B whee deotes the Black-choles opto pce fo a Euopea call. e hae ow a fully opeatoal waat pcg fomula ad apat fom ad the emag agumets of the fomula ca be obseed the maket o extacted fom the aual accouts of each fm. Howee t s also possble to estmate ad usg obseed stock maket data fo ad. o do that we just eed to sole a system of two olea smultaeous equatos. Remembe that fom equato we hae the followg epesetato fo the stock pce of a waat/eo ssug fm: q whee we ow kow that each... q s ealuated usg the fomula of equato. It s also well kow see fo example chulz ad autma 994 Dasos ad atchell that whch ou case becomes q 4 Equatos ad 4 ca be soled smultaeously fo the two ukow agumets ad. ug ow to the fomula fo q-types of waats wth commo matuty ad multple stke pces < < <... < <... < q we hae that ude Black-choles assumptos fo ay : B 5 q B j j j j j [ ] j j 9
hs follows staghtfowadly fom equato. I ths case as well the latet paametes ad ca be estmated usg the umecal pocedue outled aboe whee of couse ths tme we substtute equato 5 fo the alue of each... q 5. AN EMPIRIAL APPLIAION As a empcal applcato of the theoetcal famewok peseted thus fa we ow tu to alug the Execute tock Optos of sco ystems. sco ystems s oe of the leadg techology compaes ad has stock optos ssued to about half ts employees aouced Jauay 9. At July 8 the compay s EOs wee classfed to 5 majo categoes accodg to stke ad matuty ad the aggegate dluto potetal was appoxmately 5%. able exhbts all the eleat fomato as extacted fom the ompay s aual epot. ABLE sco ystems: Ifomato o Outstadg Execute tock Optos at July 8 Numbe of EO Issuaces: Matuty Yeas Execse Pce pe hae $ Numbe Outstadg mllos --------------------------------------------- ---------------------------------------------------- --------------------------------------------------------------- --------------------------------------------------- 4.45 4.97 6.% 6.64.98 55.5% 7.5 7.45 8 4.6% 4 4 7.54 4 55.85 4 4.9% 5 5 7.77 5 69.5 7 5.4% otal 6.66 9.4 6 4.5% a he last ow the table as epeseted by otal s a attempt to teat the multple EO ssuaces as a sgle epesetate ssuace wth matuty ad execse pce ge by a weghted aeage of the ddual stkes ad matutes espectely. pecfcally the sgle epesetate matuty s ge by 6/6*4.4555/6*6.648/6*7.54/6*7.547/6*7.776.66 ad the sgle epesetate stke by 6/6*4.9755/6*.988/6*7.454/6*55.857/6*69.59.4. b At July 8 thee wee 74 mllo oday shaes ssue equed to calculate the dluto factos. Addtoal fomato extacted fom stock maket data ad the fm s accouts clude that at July 8 the shae pce of sco ystems was $9.6 the expected futue dded was d % the sk-fee teest ate was 5.4% ad the expected futue stock etu olatlty was 4.8%. sco uses pojected olatlty ates whch ae based upo hstocal olatlty ates teded to futue yeas. e wsh to alue the compay s 9 ee Busess. he Hdde ost of tock Optos Dad Futelle Febuay.
EOs usg: the Multple aat aluato Famewok Multple-F toduced ths pape the stadad gle aat aluato Famewok gle-f ad the Black-choles aat aluato Famewok B-F. he Multple-F: Repoducg a pogammable aalogue of equato combed wth assumpto of secto 4 we hae that the aluato fomula fo a fm wth q Execute tock Optos of matutes <... < q ad of stkes... q s ge fo ay... q by j p δ j p δ j B δ j δ j l j whee p Φ ad δ mod[ ] j Rouddowwad. o make the fomula opeatoal we eed to obta estmates fo the alue of the fm ad olatlty. o do that we follow the pocedue outled aboe secto 4 utlzg the fact that we hae fomato o ad.e. we sole umecally the smultaeous equatos ad 4 usg a Newto-Raphso teate pocedue. e obta the followg alues: $. ad 6.%. It s woth emphaszg that to obta these estmates we substtuted the Multple-F fomula fo each... q equatos ad 4. he gle-f: he gle-f costtutes the stadad textbook teatmet fo the aluato of waats. I a multple waat settg t takes the followg fom: fo a fm wth q Execute tock Optos of matutes <... < q ad of stkes... q we hae that fo ay... q B mod[x z] epesets modulo dso. hs etus the alue that s the emade of the tege dso of x by z.
whee the latet paametes ad ae feed oce aga by solg smultaeously equatos ad 4. hs tme howee we substtute the gle-f fomula fo each... q. I othe wods we sole the followg system of equatos q B q B e thus obta: $. 9 ad 6.%. he B-F: he Black-choles aat aluato Famewok o optolke waat aluato famewok s popula amogst copoatos sce t s cosstet wth the FAB madates fo accoutg fo the alue/cost of stock-based compesato. It oles alug EOs ad waats as stadad call optos. Hece the fomula fo a fm wth q Execute tock Optos of matutes <... < q ad of stkes... q s smply ge fo ay... q by B I ths case all the paametes oled the aluato ae extactable fom the aual accouts ad the facal makets. sco states ts aual epot that the fa alue of each opto gat s estmated o the date of gat usg the Black-choles opto pcg model. Howee t ackowledges that because employee stock optos hae chaactestcs sgfcatly dffeet fom those of taded optos ad because chages the subjecte put assumptos ca mateally affect the fa alue estmate the exstg models do ot ecessaly pode a elable sgle measue of the fa alue of the ompay s optos. he Multple-F podes the bechmak agast whch to check the accuacy of the gle-f ad the B-F. he latte two ae of couse appoxmatos to the tue alue/cost of EOs the pesece of multple-ssuaces. I able we epot the alues of the compay s EOs usg all thee aluato famewoks. e the ealuate the qualty of the gle-f ad the B-F appoxmatos by epotg the pecetage mspcg eo asg fom the applcato. hs s calculated as [gle-f Multple-F / Multple-F] ad [B-F - Multple-F / Multple-F] espectely.
ABLE aluato of sco s EOs ude the Multple- gle- ad B- aat aluato Famewoks ad Pecetage Mspcg Eo asg fom the use of the gle-f ad the B-F appoxmatos. $ alue / ost % Mspcg Eo Multple-F gle-f B-F of gle-f of B-F ------------------ ------------------------------------ ------------------------------------------ ------------------------------------ ----------------------------------------------- ------------------------------------------------- 5.8.4% -4.8%.8 4.9% -.% 5.4.6% -.6% 4 4 4 4.4 4.6% -6.% 5 5 5 5.68 6.5% -9.% otal 6. 6. 6.5 N/A.5% a he last ow the table as epeseted by otal ges the $ alue/ost pe EO whe oe attempts to teat the multple EOs as a sgle ssuace. It coespods to the paamete alues ge the last ow of table. It s clea fom the table that use of the textbook gle-aat aluato Famewok wll esult sgfcat oepcg wth the pecetage mspcg eo becomg pogessely hghe as the umbe of teeg EO ssuaces ceases. Ideed the moe EO ssuaces thee ae befoe each patcula ssuace ca be execsed the lage the falue to accout fo the lkelhood of dluto of the fm s equty po to execse of the ssuace ude cosdeato. hs uaodably leads to oepcg. I aggegate ow f the fm eoeously used the gle-f stead of the Multple- F to accout fo the total cost/alue of ts EOs t would oeestmate the tue cost/alue by appoxmately $48.6 mllo. ug ow to the Black-choles aat aluato Famewok we obsee that ts applcato wll also mspce the fm s EOs but ths tme the opposte decto.e. ths tme we obsee udepcg of sco s EOs. Despte of the fact that the B-F has poed to be a accuate appoxmato sgle-waat settgs ts pefomace a multple settg s ufotuately ot ealy as good the case of sco pcg eos aged fom.% to 9.%. If t wee to be used the B-F appoxmato would udeestmate the total cost/alue of the fm s optos by appoxmately $85.5 mllo. I the last ows of tables ad we hae attempted to teat the multple EO ssuaces of sco ystems as a sgle epesetate EO ssuace wth stke ad matuty ge by a weghted aeage of the ddual stkes ad matutes espectely ad alue t accodgly. I ths case the Multple-F ad the gle-f cocde ad the alues obtaed fo ad ae $9.94 ad 6.% espectely. he theoetcal alue o ae at ths esult smply just subtact the total cost of sco s optos asg fom the use of the Multple-F.e. 5.95*6.7*55 5.6*8.64*4.95*7 $9.5 mllo fom the total cost asg fom the use of the gle-f.e. 6.*6.7*55 6.*8 4.7*4.4*7 $9694. mllo.
of ths epesetate EO s the foud to be $6.. osstet wth what has bee sad so fa the B-F appoxmato ths sgle-waat settg pefoms ey well gg a alue of $6.5 pe outstadg EO a mspcg eo of oly.5%. Attemptg howee to assess the total cost/alue of the fm s EO usg ths sgle epesetate EO settg s deftely appopate. he total cost/alue of the compay s EOs ude ths appoach amouts to appoxmately $6.487 bllo a udestatemet of a staggeg $.7 bllo whe compaed to the theoetcal cost of $9. bllo asg fom the use of the Multple aat aluato Famewok. see also footote. Ideed the message gog though fom ths last execse s that t s ot geeally a good dea to appoxmate multple EO ssuaces by a epesetate sgle ssuace alue t accodgly ad subsequetly assess the total cost/alue of the compay s optos ude such a specfcato. 6. ONLUION Oe the yeas teest ad moe mpotatly ssuace of execute stock optos as a fom of stock based compesato has suged. As a esult t s ow a commo pheomeo fo copoatos to ssue a lage umbe of Execute tock Optos schemes each wth ts ow execse pce ad matuty. he t comes to alug copoate stumets such as execute optos ad waats oe of the dffcultes that mmedately ases s the captal stuctue effect o the ssug fm. I sgle-ssuace settgs ths has ee eally bee a poblem sce ot oly thee s a well deeloped theoy to accout fo the so-called dluto effect but also t has bee demostated by aous studes that opto-lke aluato esults ealy detcal pces fo these stumets as the textbook alue of the fm appoach. Howee a estgato o the applcablty ad sutablty of the cuet aluato methods multple-ssuaces settgs despte the ceasg populaty has ot so fa bee pefomed. I ths pape we hae exteded the theoetcal famewok of Gala ad chelle 978 to accout fo multple waat ad EO ssuaces ad deed dstbuto fee fomulae fo fms wth waats ad EOs of seeal matutes ad stkes. e hae llustated that whe appled multple settgs the textbook sgle ssuace appoach ca esult sgfcat oepcg. Lkewse we hae also show that opto-lke aluato s ot that good a appoxmato multple settgs ad wll typcally esult udepcg. hs e-efoces exstg coces o the optmalty of usg stadad umodfed opto pcg models to accout fo the cost of stock-based compesato. Fally fo the puposes of assessg the total cost of a fm s optos the pactce of teatg multple ssuaces as a sgle epesetate ssuace wth stke ad matuty ge by a weghted aeage of the ddual stkes ad matutes espectely s potetally ey dageous ad wll poduce hghly msleadg alues. I the example cosdeed the text ths leads to a udepcg of appoxmately %; 4
5 thus udeestmatg the mpact of EOs o the fm s alue. Lkewse wth aeage pecetage gas o losses of the ode of % of captal applcato of the sgle waat aluato famewok o of the Black-choles opto-lke famewok ca hae sgfcat effects o dffeet goups of shaeholdes be they employees o shaeholdes. he multple waat aluato famewok toduced ths pape should be used stead. 7. APPENDIX tatg fom equato 5 we eed to ae to equato 6. e fst epoduce equato 5: d f RN / [ exp 5 ] \ d f RN d f d f RN RN / / [ exp d f RN \ ] \ d f RN d f RN / [ exp / d f RN d f RN \ ] \ d f RN ] [ P exp ob P <
6 P ] [ P exp ob > whee ad P deote call ad put opto pcg fuctos espectely. Hece we hae: exp P P Now usg Put-all paty the aboe expesso smplfes to: exp exp exp ome staghtfowad algeba ad we get equato 6:
8. REFERENE Achaya.. Joh ad R. udaam : O the optmalty of esettg execute stock optos Joual of Facal Ecoomcs 57 65-. Besoussa A. M. ouhy ad D. Gala 99: Black-choles appoxmato of waat pces Pape Peseted at the 9th aual meetg of the Euopea Face Assocato Lsbo Potugal August 99. Black F. 989: How we came up wth the opto fomula Joual of Potfolo Maagemet 4-8. Black F. ad M. choles 97: he pcg of optos ad copoate labltes Joual of Poltcal Ecoomy 8 67-654. Bee M. R. udaam ad D. Yemack : Alteg the tems of execute stock optos Joual of Facal Ecoomcs 57-8. apete J. 998: he execse ad aluato of execute stock optos Joual of Facal Ecoomcs 48 7-58. apete J. : Does opto compesato cease maageal sk appette? Joual of Face 55 -. apete J. ad B. Remmes : Execute stock opto execses ad sde fomato Joual of Busess 74 5-4. ate M. ad L. Lych : A examato of execute stock opto epcg Joual of Facal Ecoomcs 6 7-5. hace D. R. uma ad R. odd : he epcg of execute stock optos Joual of Facal Ecoomcs 57 9-54. ostatdes G. 984: aat execse ad bod coeso compette makets Joual of Facal Ecoomcs 7-97. ox J. ad. Ross 976: he aluato of optos fo alteate stochastc pocesses Joual of Facal Ecoomcs 45-66. ox J. ad M. Rubste 985: Optos Makets Petce Hall Eglewood lffs. 7
Dasos. ad. atchell : he mpled dstbuto fo stocks of compaes wth waats ad/o execute stock optos DAE Depatmet of Appled Ecoomcs okg Pape No. 7 Uesty of ambdge. Detemple J. ad. udaesa 999: Notaded asset aluato wth potfolo costats: a bomal appoach Reew of Facal tudes 85-87. Emauel D. 98: aat aluato ad execse stategy Joual of Facal Ecoomcs -5. Gala D. ad M. chelle 978: Pcg of waats ad the alue of the fm Joual of Face -4. Howe J. ad P. e 99: he aluato effects of waat extesos Joual of Face 5-4. Hall B. ad. Muphy : Optmal execse pces fo execute stock optos Ameca Ecoomc Reew 9 9-4. Hull J. : Optos Futues & Othe Deates Fouth Edto Petce-Hall Iteatoal. Igesoll J. 987: heoy of Facal Decso Makg Rowma ad Lttlefeld aage. Johso. ad Y. a : Idexed execute stock optos Joual of Facal Ecoomcs 57 5-64. eme J. ad R. Roefeldt 99: "aat Pcg: Jump-Dffuso s. Black- choles" Joual of Facal ad Quattate Aalyss 8 55-7. Lautebach B. ad P. chultz 99: Pcg waats: a empcal study of the Black- choles model ad ts alteates Joual of Face 45 8-9. Logstaff F. 99: Pcg optos wth extedble matutes: aalyss ad applcatos Joual of Face 95-957. Maquadt. 999: he cost of employee stock opto gats: a empcal aalyss wokg pape New Yok Uesty. 8
Meto R. 976: Opto pcg whe udelyg stock etus ae dscotuous. Joual of Facal Ecoomcs 5-44. Noee E. ad M. olfso 98: Equlbum waat pcg models ad accoutg fo execute stock optos Joual of Accoutg Reseach 84-98. Rubste M. 995: O the accoutg aluato of employee stock optos Joual of Deates 5-5. chultz P. 99: alls of waats: tmg ad maket eacto Joual of Face 68-696. chulz G. ad. autma 994: Robustess of opto-lke waat aluato Joual of Bakg ad Face 8 84-859. deus J. 996: "aat Pcg - Is Dluto a Deluso?" Facal Aalysts Joual eptembe/octobe 77-8. patt. ad F. tebez 988: aat execse ddeds ad eestmet polcy Joual of Face 49-56. eld. 994: "aat Pcg: A Reew of Empcal Reseach" eter Dscusso Pape ees No 944 lbug Uesty he Nethelads. 9