Stock option grants have become an. Final Approval Copy. Valuation of Stock Option Grants Under Multiple Severance Risks GURUPDESH S.

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1 Valuaion of Sock Opion Gran Under Muliple Severance Rik GURUPDESH S. PANDHER i an aian profeor in he deparmen of finance a DePaul Univeriy in Chicago, IL. gpandher@depaul.edu GURUPDESH S. PANDHER Execuive ock opion (ESO) gran have a number of imporan feaure ha do no conform o aumpion of he Black-Schole model. Thi aricle develop a rik-neural model for valuing uch opion where he holder i expoed o muliple everance rik like erminaion wih caue or wihou caue, or moraliy, wih varying caue-coningen ex-everance payoff and ock holding rericion. The Black-Schole model ignificanly overeimae he co of ESO o he firm by a much a 8% o 39% for a everance rae of 5%, and he bia i inverely relaed o volailiy. The everance even i modeled uing a flexible doubly ochaic Poion proce ha permi he rich informaion rucure of he ae variable and he everance even o be endogenouly capured in he valuaion. Valuaion i accomplihed uing a muli-everance binomial ESO model and a mulieverance parial differenial equaion. Sock opion gran have become an increaingly dominan porion of he oal compenaion of CEO and enior execuive. Such a grea ranfer of value from hareholder o corporae execuive ha raied call for corporae expening of ock opion gran. Advocae argue ha opion gran repreen a co of acquiring managerial ervice whoe marke value hould be an adumen o earning in he ame way ha inere and depreciaion reflec a firm co of deb and expendiure of phyical capial. Financial Accouning Sandard Board (FASB) guideline ugge he ue of Black- Schole (BS) [1973] pricing o meaure he co of ock opion graned by a corporaion. In fac, he Black-Schole mehod ignificanly overeimae he value of ock opion by a much a 8% o 39%, a a cumulaive everance rae of 5%, becaue i ignore he everance rik employee face and he cauedependen ex-everance value of opion plan. Execuive ock opion (ESO) have a number of imporan feaure ha and ouide he aumpion of he Black-Schole model. Fir, he holder of he ock opion face ubanial everance rik; dimial prior o opion expiraion (ypically, en year a iue) may lead o lo of he opion award, or immediae exercie, or reducion in mauriy, depending on he ype of employmen exi. Second, ock opion plan may impoe a ock holding rericion, uch a 1 monh, on opion exercied early (hi may be circumvened by regiering he exercied ock o a financial inermediary a a dicoun o he marke price, of abou 10%-30%). The holding rericion may be plan-pecific, or may arie from regulaion uch a Rule 144 of he U.S. Securiie and Exchange Commiion (SEC), which reric he abiliy of corporae inider and affiliae o ell hare in heir firm. Third, here are rericion on veing before which ock opion may no be exercied. We hu need a rik-neural mehod o value ock opion ha price he muliple everance rik, he caue-coningen everance payoff, and he ock holding rericion WINTER 003 THE OURNAL OF DERIVATIVES 5

2 embedded in opion gran. For hi purpoe, we develop a muli-everance binomial ESO model, and alo idenify an ESO muli-everance parial differenial equaion (PDE) ha offer an alernaive mehod of valuaion. The implemenaion of he muli-everance ESO model i illuraed by valuing a ock opion gran expoed o hree ource of everance: wih caue, wihou caue, and moraliy. The example alo inveigae he effec of volailiy and exercie price on opion valuaion and he early exercie rae. A benchmark, reul are compared wih he Black-Schole (BS) model and an alernaive way of accouning for opion lo rik by exogenouly aduing he BS value by he probabiliy of non-everance. I. VALUATION APPROACHES AND MODELS Two general approache have been adoped o model execuive exercie deciion and valuaion of he co of ESO o a firm. In he fir approach, execuive elec a policy of opion exercie ha maximize heir expeced uiliy, ubec o rericion on hedging and opion ale (Lamber, Larcker, and Verrecchia [1991], Huddar [1994], Marcu and Kulailaka [1994], and Huddar and Lang [1996]). The uiliy framework, however, require informaion and modeling of unobervable variable uch a he execuive rik averion, wealh holding, and he impac of employmen change. In an alernaive rik-neural approach, early opion exercie i modeled a a random opping ime wih a Poion exercie rigger a in ennergren and Nalund [1993]. Carpener [1998] how ha a reduced-form American call opion model perform a well a or beer han he more complicaed rucural model in predicing exercie paern for a ample of 40 firm. Cuny and orion [1995] and Rubinein [1995] allow exercie probabiliie o depend funcionally on he value of he ock, and Brenner, Sundaram, and Yermack [000] conider he valuaion of ock opion whoe exercie price may be ree. Carr and Lineky [001] ue a ock price barrier rigger in he ineniy proce o model he execuive diverificaion and liquidiy conideraion. Thi aricle make hree conribuion. Fir, he propoed ESO valuaion model incorporae muliple everance rik wih ex-everance value ha depend on he execuive mode of exi from he firm. Oher model aume a ingle everance payoff for he ESO, and hi rericion fail o price he caue-dependen everance rucure of acual ock opion gran. Second, he everance even i modeled a a doubly ochaic Poion probabiliy proce in he muli-everance binomial ree and he PDE gain proce. Thi pecificaion allow he ock price and oher ae variable o affec he everance probabiliie in a very flexible and ochaic fahion, allowing complex and rich informaion rucure o be endogenouly capured in he valuaion. Finally, he rik-neural valuaion approach find near-arbirage-free price ha are independen of he opion holder peronal rik and wealh aribue. Thi i an aracive feaure of he model from he perpecive of a corporaion aemping o find he co of i employee ock opion gran. II. CREDIT RISK MODELING There are ome parallel in our mahemaical formulaion of he everance even and he defaul even in reduced-form credi rik model for fixed-income ecuriie (ee Arzner and Delbaen [1995], arrow and Turnbull [1995], Duffie and Singleon [1995, 1997], Duffie, Schroder, and Skiada [1996], arrow, Lando, and Turnbull [1997], Lando [1998], Madan and Unal [1998], and oher). Depie he imilariie, here are imporan difference beween he wo eing (credi rik veru everance rik) ariing from he ype of underlying ae, he ource of he rik, he opion plan everance-coningen payoff, and holding rericion. For ock opion, he underlying ae i company ock a oppoed o he po rae of inere or forward rae in credi rik model. Second, he credi rik in fixed-income claim (e.g., corporae bond, wap) perain o he diconinuaion rik of he corporaion or he counerpary, while everance rik affec an individual employee and i eenially inernal o he firm. Third, he ock opion gran are expoed o muliple ource of everance and exhibi a wide array of everance-dependen value (e.g., forfeiure of opion, immediae exercie, reducion in mauriy). III. ESO PRICING UNDER MULTIPLE SEVERANCE RISKS We fir decribe he doubly ochaic Poion probabiliy repreenaion for he everance even under muliple caue, and hen exend he andard binomial opion pricing model in Cox, Ro, and Rubinein [1979] (CRR) o incorporae he ESO muli-everance and caue-coningen ex-everance feaure. The correponding PDE i idenified, and he effec of ock holding conrain i alo dicued. 6 VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

3 Severance Rik and he Doubly Sochaic Poion Repreenaion Sock opion award ypically have very long mauriie; en year i a common mauriy a iue. During hi ime, a ock opion holder may exi he firm for a number of reaon: dimial wih caue, dimial wihou caue, moraliy, and o on. If erminaion occur due o caue, he opion may be forfeied compleely, while in he cae of dimial wihou caue or moraliy he opion mu be exercied immediaely or wihin a hor period of ime (e.g., hree monh). Therefore, he ESO holder i expoed o ubanial everance rik. Le be he random fuure ime a which everance occur. Then, he everance even i convenienly repreened by he urvival indicaor I [ >u], which ake he value of 1 prior o a everance even a ime u and 0 hereafer. There are Œ{1,,, } poible caue of everance, and repreen he fuure random ime a which everance reul from caue. The fuure ime of everance (from any caue) i given by min( 1,,, ). The everance even i modeled a he fir umpime of a doubly ochaic Poion proce where everance hazard rae funcion h h (,): [0, ] Æ[0, ) depend on he random ock price. The informaion e under which probabiliie of fuure even are compued i repreened by G H D which comprie informaion on boh he ock price evoluion (H ) and he everance even (D ). * Under muliple everance rik, he probabiliy of non-everance by ime compued a ime < i given by: Pr( > G) Pr( > G) E( I[ > ] G ) 1 Ê EÁ E( I[ > ] H D ) G Ë Ê E Ú Áe Ë 1 - ( hu hu ) du G where he law of ieraed expecaion i applied a he hird equaliy uing he fac ha (H D ) à (H D ) G. The condiioning argumen allow he everance probabiliy a fuure ime o evolve wih he ae variable revealed a ha ime (e.g., ock price). The ouer expecaion hen average over he uncerainy in he fuure value of hee ae variable. 1 (1) Muli-Severance Binomial ESO Model Rik-neural argumen are applied o conruc he muliple-everance binomial ESO model (MSB-ESO). Le 0 be he iniial ock price, and le, Œ[( 0 u d 0 ), ( 0 u -1, d 1 ),, ( 0 u 1 d -1 ), ( 0 u 0 d )], 1,, n, repreen he ock price in he binomial ree where n i he number of ime ep up o mauriy T (in year) of lengh D T/n. A each node, he ock price eiher move up by he facor u exp( D ) or down by he facor d 1/u, where i he ock reurn volailiy. Furher, given he coninuouly compounded rik-free rae r and dividend yield d, he rik-neural probabiliie for he up and down movemen of he ock price a each node are given by p exp[(r d)d] d/(u - d) and 1 p, repecively. In he conex of he binomial model, he probabiliy of urvival over he nex ime ep under Equaion (1) become: Pr ( > D( + 1) > D, ) - (, ( ))... (, ( )) Ê ( h1 u D h u D + 1 ) D pe Á -( h1 ( d, D( + 1)) h ( d, D( + 1) ) D Ë + ( 1 - pe ) Le W (, I [ > D] ) W (, I [ > D] ;, r, K, T ) repreen he ESO call opion value when expoed o muliple everance rik wih exercie price K. W (, I [ > D] 1) repreen he urvived value of he ESO a ime D, and W (, I [ > D] 0, ) i he ex-everance value of he ESO when everance occur due o caue Œ {1,,, }(we aume ha he everance even occur a he end of he period [D( 1), D]). The correponding hazard rae of everance a ime D i h h (, D): [0, )Æ[0, ). ESO plan ipulae differing erm for opion under variou mode of everance. We noe wo example. Severance wih caue: Lo of opion. If he ESO holder i erminaed due o caue, he ock opion award uually become null and void. Then, he ex-everance payoff a he end of period [D, D( + 1)] may be wrien a: W+ 1( + 1, I[ > D( + 1) ] 0, ) 0 Severance wihou caue or moraliy: Reducion in ESO mauriy. If he employee leave volunarily or i erminaed wihou caue, he holder reain he veed opion bu may be required o exercie hem wihin a pecified period T (e.g., hree monh), hereby reducing () (3) WINTER 003 THE OURNAL OF DERIVATIVES 7

4 he mauriy of he ESO. Le V( +1, T ) repreen he value of an American call opion wih ime o mauriy T. Then, he value of he ock opion upon everance a he end of period [D, D( + 1)] i: W (, I 0, ) V(, T ) [ > D( + 1) ] + 1 S (4) The expecaion in Equaion (7) i aken under he doubly ochaic Poion probabiliy proce, and he condiioning argumen of Equaion (1) and () are ued in he econd and hird equaliie. The new quaniy W +1 ( +1, I [>D(+1)] 0) repreen he expeced value of he ESO over he everance ae, and i given by: Alernaively, if he opion holder mu exercie immediaely upon everance, he coninuaion value of he ock opion a each node of he binomial ree change o Rik-neural valuaion of coningen claim i baed on he principle ha if he price rik of he derivaive ecuriy can be dynamically eliminaed unil expiraion by holding poiion in he underlying radable ae, hen o rule ou arbirage he hedged poiion mu earn a reurn equal o he rik-free rae r. Equivalenly, he derivaive arbirage-free value may be deermined by aking an expecaion of he coningen claim under a pecial rik-neural probabiliy meaure. We can apply rik-neural argumen o our eing becaue alhough he opion holder canno freely ell he ock or he ESO, he corporaion i free o hedge he opion uing i ock direcly or hrough a financial inermediary. Working backward hrough he binomial ree, he ESO value a ime in ae i given by: where he coninuaion value W c (, I [>D] 1) of he ock opion i given by: c -rd W (, I ) E e W (, I ) G W+ 1( + 1, I[ > D( + 1) ] 0, ) max( K, 0) W(, I 1) [ > D] [ ] [ > D] c max - K, W (, I 1) [ > D] [ > D( + 1)] D -rd Ee ( EW ( + 1( + 1, I[ > D( + 1)] ) HD( + 1) DD ) GD ) ( ) e ( ) -rd (5) (6) Ï p e -hu (, D( + 1) D È ) W + ( u, I[ > D( + )] ) Í - hu (, D( + 1) D Í ) Î+ ( 1 - e ) W + ( u, I[ > D( + )] ) Ì hd p e - (, D( + 1) ) D È W + > + 1( d, I[ D( 1)] 1) + ( 1 - ) Í - hd ( D + Í, ( 1) ) D Î+ - + > + Ó ( 1 e ) W 1( d, I[ D( 1)] 0) (7) hd W+ 1( + 1, I[ > D( + 1)] 0)  W+ 1( + 1, I[ > D( + 1)] 0, ) h for +1 Œ ( u, d). Thi follow from calculaing he opion condiional expeced value under everance a he end of period [D, D(+1)]. Thi i given by where  W + ( +, I[ > D( + )] 0, )Pr( D( + 1)) Pr( D( + 1)) Pr( > D( + 1)) hd ( + 1), Pr( > D( + 1)) h D( + 1) D( + 1), D( + 1) Severance Probabiliie and Eimaion Can we ue everance probabiliie (or hazard rae) baed on he company hiorical employmen daa in ESO valuaion? The company human reource deparmen ha hiorical employmen urnover daa o allow eimaion of annual everance probabiliie of i managerial aff, and hee can be diaggregaed by caue. An imporan iue here i wheher uch empirical everance probabiliie may be ued in he muli-everance binomial ESO model when he rik-neural opion pricing framework require rik-neural probabiliie of everance. We conider hi from he viewpoin of boh he opion wrier (he corporaion) and he ESO holder (he execuive). From he perpecive of he corporaion, i may be argued ha employee everance rik i relaively well diverified (ennergren and Nalund [1993], Cuny and orion [1995], and Carr and Lineky [001]). A corporaion wih many enior execuive, conan managerial urnover, and a ready pool of poenial replacemen ha relaively low expoure o he everance ump rik ariing from any ingle employee. Therefore, on an aggregae level, he company employee everance rik i no y- h h ( + 1), D( + 1) (8) 8 VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

5 emaic and i relaively diverified (hi i imilar o he CAPM argumen ha he marke hould no compenae diverifiable ecuriy rik in equilibrium). Thi may alo be a reaonable pracical aumpion on empirical ground. For example, Marquard [1999] examine 58 Forune 500 firm for over 1 year, and find an average of 17 ESO gran per firm. Thi line of reaoning implie ha, from he company perpecive, one may rea he firm empirical everance probabiliie a rik-neural everance probabiliie for he purpoe of valuing i ock opion gran. Thee argumen for he firm do no apply o he opion holder, and a uiliy framework i neceary for complee analyi. Thi approach, however, require informaion and aumpion on unobervable uch a he execuive rik averion and wealh endowmen ha poe ubanial problem in implemenaion. Here, we can aer ha he rik-neural MSB-ESO model offer an eay-ocompue upper bound (much lower han Black-Schole) on he value of a ock opion gran o he execuive. A company can readily eimae from i employmen urnover daa he annual probabiliie q ha a manager will involunarily leave he firm from each caue. Thee probabiliie can be convered o annualized hazard rae of everance (e.g., for q 0.05, h 1n(1 q ) ), or, alernaively, one can ue more refined aiical echnique uch a Cox proporional hazard modeling o eimae hazard funcion in erm of covariae, including he ock price. For everance relaed o moraliy, ageand ex-pecific acuarial life able publihed by he Sociey of Acuarie may be ued o quanify hi probabiliy diribuion. Sock Holding Rericion and Dicouned Exercie If he employee exercie an award prior o expiraion, ome opion plan preven he opion holder from elling he ock for a cerain period (e.g., 1 monh). Thi rericion impoe an addiional rik, a he ock price can change over he holding period. In he rik-neural world, however, he held ock i expeced o grow a he rik-free rae over he holding period, and i preen value i u he curren value of he ock. Therefore, he holding rericion doe no affec he ESO. Raher han wai one year o ell he ock, he opion holder may exercie he opion and regier he ock in he name of a financial inermediary a a (1 l)% dicoun (ypical dicoun fall in he 10%-30% range). The effec of he dicoun ale alernaive on opion valuaion are incorporaed by ubiuing for Equaion (6): c W(, I 1) max l - K, W (, I 1) [ > ] [ > ] ESO Muli-Severance Rik PDE An alernaive mehod of valuaion i o olve he muli-everance rik-adued parial differenial equaion (PDE) repreening he ESO. Meron [1976] conider opion pricing wih diconinuou ump in he ock reurn proce. Thi framework i no exacly applicable here becaue everance rik i independen of ump in he ock price, and everance affec he ESO value direcly (no he ock price). Furher, he ESO ex-everance value depend on he mode of everance. Varian of a ingle-diconinuiy rik PDE are conidered by ennergren and Nalund [1993] and Rubinein [1995] under a conan hazard pecificaion and by Carr and Lineky [001] who allow he conan hazard rae o ump when he ock price hi a predefined barrier. Thi work, however, conider only a ingle aggregae caue of everance and reric he ESO o one ex-everance value. We now obain a muli-everance ESO PDE. We ake o repreen coninuou ime, and X repreen he correponding ock price (a oppoed o in he binomial model). We make he andard BS aumpion ha he ock price obey a diffuion proce a follow under he rik-neural probabiliy meaure Q: dx ( r - q) X d + X db Q ([ ] ) (9) (10) where r i he rik-free rae (drif under Q), q i he dividend payou rae, i he reurn volailiy, and B Q i andard Brownian moion. To avoid a dramaic change from he earlier noaion, we underand W (X, I [>] 1) W(, X, I [>] 1):[0, µ) {0, 1} Æ [0, µ)o be wice-differeniable wih repec o he fir wo argumen. Mahemaically, he rik-neural pricing condiion i ha he dicouned value of he derivaive claim mu be a maringale under he rik-neural probabiliy meaure Q: { { } } < < < Q -r E d W( X, I ) e G 0, [ > ] 1 0 T (11) The erm of d[w (X, I [>] 1)e r ] are obained by applying he generalized form of Iô formula (ee Proer [1990, p. 71]), and he expecaion in Equaion (11) i made uing he law of ieraed expecaion: WINTER 003 THE OURNAL OF DERIVATIVES 9

6 E( G ) E( E( H D) ) G) , , , ). In he implemenaion, he annual probabiliie are convered o he relevan everance hazard rae by caue. The reuling muli-everance rik ESO PDE i given by: The valuaion of he ESO i conidered according o everal model: W + 1 W X È ÍÂhW ( X, I[ > ] 0, ) Í Î Í ( h h ) W ( X, I > + ( r - q) X W - rw ( X, I [ > ] 1 [ ] 0 1) (1) for 0 < T where W T (X T, I [>T] 1) (X T K) + i he ock opion erminal value; W (X, I [>] 0, ) i he ESO ex-everance value when riggered by caue 1,, ; and h h (X, ): [0, ]Æ[0, ] i he correponding hazard rae funcion of everance. (Deail are in he appendix.) The ESO may be valued by olving he muli-everance rik ESO PDE (1) uing numerical mehod (e.g., explici finie difference, implici finie difference, Crank- Nicholon). The early exercie and dicouned exercie feaure can be eaily applied a each ieraion in he finiedifference procedure. IV. APPLICATION AND NUMERICAL STUDY We ue he muli-everance binomial ESO (MSB- ESO) model o price a ock opion gran under hree poenial caue of everance. The numerical udy alo inveigae he valuaion bia from uing alernaive model (Black-Schole, exogenou adumen o BS) and he effec of volailiy on he bia and early exercie. ESO Valuaion Under Three Severance Rik The ESO gran coni of 500,000 hare wih a mauriy of en year and a veing period of hree year. The iniial marke price of he ock i X 0 $50; he annualized reurn volailiy i 50%; and he ock i nondividend-paying. We conider hree caue of ob everance ( 3): wih caue ( 1), wihou caue ( ), and moraliy ( 3). The execuive annual probabiliie of everance are aken o be % for everance wih caue (q 1 0.0) and 3% for everance wihou caue (q 0.03). The probabiliie of moraliy (q 3, 1,, 10) a repored in he acuarial life able publihed by he Sociey of Acuarie for an inured man aged 50 are ued: q 3 ( , , , , , , 1. Black-Schole call opion value (no everance rik adumen).. MSB-ESO model wih he caue-dependen everance payoff: A. Lo of ock opion plan upon everance for caue 1. B. Immediae exercie upon everance for caue. C. Three-monh opion expiraion upon everance for caue Severance probabiliy-adued Black-Schole. Valuaion i conidered under boh he 1-monh ock holding rericion a well a he 90% dicoun ale alernaive. Scenario 3 refer o a imple approach of aduing he Black-Schole opion value by he probabiliy of non-everance: n  1 Pr( > T > ) exp( - h D) A repored in Exhibi 1, valuaion under everance rik i ubanially lower han under he Black-Schole (BS) model. A comparion of column (1) and () how ha BS overeimae he opion value by beween 8% and 39%, leading o ignifican overvaluaion of he ock opion gran. Black-Schole give a valuaion of $16,801,06 while he muli-everance binomial ESO model (MSB- ESO) yield a value of $1,57,99 wih a veing period of hree year (econd panel). Meanwhile, he probabiliyadued BS mehod undervalue he opion gran by 19% o 5%. Exhibi and 3 graph he reul. I i clear ha ignoring everance rik, a occur in Black-Schole pricing, lead o dramaic overvaluaion of he ESO value (op curve in Exhibi and 3), while he imple adumen o he Black-Schole value by he probabiliy of urvival (along he line of ennergren and Nalund [1993]) lead o ignifican undervaluaion. Thi adumen ignore he addiional value from early exercie under everance rik expoure a well a he everance-coningen payoff of he ock opion gran. 30 VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

7 E XHIBIT 1 Comparion of Execuive Sock Opion Valuaion under Muliple Severance Rik and Exercie/Veing Rericion Type of Sock Holding Rericion No Rericion / Sock Holding 90% Dicoun Sale Time o Veing/ Share/ BS Bia 0 Yr Veing 500,000 hr BS Bia 3 Yr Veing 500,000 hr BS Bia 0 Yr Veing 500,000 hr BS Bia 3 Yr Veing 500,000 hr BS Bia Effec of Volailiy and Exercie Price To conider he effec of ock volailiy and rike price on ESO valuaion, we mainain he ame parameer eing a in Exhibi 1, bu ue wo level of volailiy ( 0% and 50%) and hree level of exercie price (0.8X 0, X 0, and 1.X 0 ). Numerical reul for veed opion wih no ock holding rericion (ame a 1- monh holding) are repored in Exhibi 4. Again, Black-Schole (BS) valuaion and he probabiliy adumen for everance lead o ignifican overand undervaluaion, repecively, bu he exen of he bia depend on he volailiy and moneyne of he opion. An increae in volailiy from 0% o 50% reduce he BS pricing bia from he range of 30%-41% o 5%- 30%. Converely, he undervaluaion bia in he probabiliy-adued Black-Schole increae from 18%-4% o 4%-7% a volailiy rie. Therefore, he ue of Black- Schole for opion expening will lead o a greaer overvaluaion bia for lower-volailiy blue-chip ock han he more volaile Nadaq echnology ock. We alo oberve ha he BS bia increae uniformly wih he opion exercie price. Column () alo repor he early exercie rae for he MSB-ESO model. Thi i he fracion of he oal ae in he binomial ree where early exercie occurred. An increae in volailiy increae he early exercie rae. A ock volailiy increae from 0% o 50%, he (1) Black-Schole Valuaion ($16,801,06) 8% ($16,801,06) 34% ($16,801,06) 33% ($16,801,06) 39% () MSB-ESO (Severance Rik Valuaion) ($13,14,349) ($1,57,99) ($1,66,487) ($1,053,34) (3) Black-Schole Probabiliy- Adued ($9,806,745) -5% ($9,806,745) -% ($9,806,745) -3% ($9,806,745) -19% BS Bia i he Black-Schole overvaluaion from ignoring everance rik and i compued a he percenage difference beween he BS and MSB-ESO value. early exercie rae increae from 8%-33% o 38%-40%. While volailiy increae he coninuaion value of he opion under everance, i alo ha he offeing effec of raiing he immediae payoff from early exercie. The empirical reul ugge ha he laer effec dominae, yielding a poiive relaionhip beween volailiy and early exercie. Furher, early exercie rae fall a he exercie price rie. Effec of Ex-Severance Payoff To examine he impac on ESO valuaion of pecific ex-everance opion value independenly, we mainain he ame parameric eing a before. Now here i only one ource of everance ( 1), and he annual probabiliy of everance i e a q 5% (everance hazard rae of h ). The ock opion are valued uing hree model: 1. Black-Schole call opion value (no everance rik adumen).. MSB-ESO valuaion wih ex-everance payoff: A. Lo of ock opion plan. B. Immediae exercie. C. Three-monh opion expiraion. 3. Severance probabiliy-adued Black-Schole. WINTER 003 THE OURNAL OF DERIVATIVES 31

8 E XHIBIT Sock Opion Valuaion o Mauriy by Valuaion Model No Rericion on Sock Holding E XHIBIT 3 Sock Opion Valuaion o Mauriy by Valuaion Model 90% Dicoun Sale Exhibi 5 how ha he greae impac on opion valuaion occur when opion are lo upon everance (.A). In hi cae, he Black-Schole overvaluaion bia increae from 36%-41% o 51%-59% when volailiy fall from 50% o 0%. When opion mu be exercied wihin hree monh of everance (.C), he bia in pricing drop o 13%-19% for 50% and 14%-6% for 0%. The early exercie rae how ha early exercie i opimal only when opion are lo upon everance (.A), while here i no early exercie for immediae exercie (.B) and hree-monh expiraion (.C). For cenario.a, a deailed breakdown of early exercie rae a differen ime in he life of he ock opion (rike a he money) i given in Exhibi 6 and 7. 3 VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

9 E XHIBIT 4 Effec of Volailiy and Exercie Price on ESO Valuaion and Early Exercie Exhibi 6 how ha under he no-ock holding rericion (and 1-monh ock holding), a he volailiy drop from 50% o 0%, he inerval over which early exercie occur horen from year o year. The ame horening for he 90% dicoun ale feaure i from year o year (Exhibi 7). I i clear from he wo graph ha volailiy increae boh he rae of early exercie and he pan of he early exercie ime zone. V. CONCLUSION Sock Srike Moneyne (1) () MSB-ESO (Severance (3) Black-Schole Volailiy (m) Black-Schole Rik Probabiliy K mx0 Valuaion Valuaion) Adued 0.8 Opion Value BS Bia 4.6% -7.3% Early Ex. Rae 39.7% 50% 1.0 Opion Value BS Bia 7.8% -5.4% Early Ex. Rae 38.6% 1. Opion Value BS Bia 30.6% -3.7% Early Ex. Rae 37.6% 0.8 Opion Value BS Bia 9.9% -4.% Early Ex. Rae 3.6% 0% 1.0 Opion Value BS Bia 36.0% -0.6% Early Ex. Rae 30.0% 1. Opion Value BS Bia 41.0% -17.7% Early Ex. Rae 7.8% Early Exercie Rae i he percenage of node in he binomial ree where early exercie occurred. BS Bia give he bia in Black-Schole valuaion due o ignoring everance rik. Thi aricle develop a rik-neural model for valuing execuive ock opion expoed o everance rik from muliple ource wih varying caue-coningen everance payoff and ock holding rericion. Thee feaure of execuive ock opion gran are no preen in he Black-Schole model. In he cae of opion expening, he model offer he furher advanage ha valuaion doe no depend on he rik averion and endowmen of he opion holder. Thi i an aracive feaure of he model for a corporaion aemping o find he oal everanceadued value of he opion i ha graned. Valuaion may be performed by implemening eiher he muli-everance binomial ESO model (MSB-ESO) or by numerically olving a muli-everance PDE. Boh repreenaion accommodae he opion varying caueconingen payoff. Modeling he everance even uing a flexible doubly ochaic Poion proce wih ochaic hazard parameer enable incorporaion of complex ineracion beween ae variable and he everance even during valuaion. The model alo endogenouly value he execuive early exercie deciion, a everance rik diminihe he ESO coninuaion value. The numerical udy how ha valuaion by Black- Schole a uggeed in FASB guideline reul in ignifican overvaluaion, a hi mehod ignore he everance rik faced by he opion holder. The Black-Schole approach inflae he opion expene by he range of 8% o 39% for a-he-money opion wih a cumulaive everance rae of 5%, while an exogenou probabiliy adumen for opion lo o he Black-Schole formula lead o undervaluaion of 19% o 5%. Furher, volailiy reduce he BS overvaluaion bia and increae boh he rae of WINTER 003 THE OURNAL OF DERIVATIVES 33

10 E XHIBIT 5 Relaive Effec of Severance Payoff on ESO Valuaion and Early Exercie Sock Volailiy 50% 0% Srike Moneyne (m) K mx (1) Black- Schole (No Severance Adumen) () MSB-ESO (Severance Rik Valuaion) (3) Prob.- Adued Black- Schole (1) BS Call Opion (.A) Opion Lo (.B) Immediae Exercie (.C) Three-monh Expiry (3) Prob. Ad. To A Opion Value Early Ex. 40.3% 0 0 BS Bia 35.9% 1.7% 1.7% 67.0% Opion Value Early Ex. 41.4% 0 0 BS Bia 38.9% 16.0% 15.9% 67.0% Opion Value Early Ex. 40.3% 0 0 BS Bia 41.4% 18.7% 18.6% 67.0% Opion Value Early Ex. 40.0% 0 0 BS Bia 50.8% 13.7% 13.7% 67.0% Opion Value Early Ex. 37.0% 0 0 BS Bia 55.8% 0.3% 0.% 67.0% Opion Value Early Ex. 34.7% 0 0 BS Bia 58.9% 6.0% 6.0% 67.0% E XHIBIT 6 Early Exercie Rae over Time under Opion Forfeiure upon Severance No Rericion on Sock Holding 34 VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

11 E XHIBIT 7 Early Exercie Rae over Time under Opion Forfeiure upon Severance 90% Dicoun Sale early exercie and he pan of he exercie ime horizon. Therefore, Black-Schole valuaion will overvalue employee ock opion gran on blue-chip ock o a greaer exen han more volaile Nadaq echnology companie. APPENDIX Muli-Severance ESO PDE -r The componen of dw are obained by ( XI, [ > ] 1) e applying he generalized form of Iô formula (Proer [1990, p. 71]). Thi give: -r { [ > ] } dw( X, I 1) e { } -r dw ( X, I 1) e - rw ( X, I 1) e [ > ] [ > ] Ï W 1 W Q W + X + (( r - q) X + XdB ) Ì e ÈW( X, I[ > ] 0, ) - + Í - rw( X, I[ > ] 1) Ó ÎW( X, I[ > ] 1) (A-1) where DW W( X, I - W X I [ > ] 0, ) (, [ > ] 1) i he ump in ESO value due o everance a he momen [, + d], and he remaining erm repreen he diffuive change. From he doubly ochaic Poion repreenaion of Pr( > G ), <, given by Equaion (1), he inananeou everance meaure i: -r -r (A-) Similarly, he inananeou everance meaure due o caue 1,, i given by (A-3) Taking he condiional expecaion of Equaion (11) under he law of ieraed expecaion E( G ) E( E( H D) ) G) wih < < T where ( H D) Ã ( H D) G ) yield: Q -r E d W( X, I 1) e G E { { [ > ] } } Q Pr ( < G - > ) [ 1 Pr( G )] Ê Ú ( hu hu ) du EÁ( h1... h ) e G Ë Pr ( <, G ) Ê Ê Ï W 1 W Á Á + X + (( r-q) X Á Á Á Á Q W + XdB ) + Á Á Ì Á Á Á Á W( X, I[ ] 0, ) - W( X, I[ Á Á Ë Ë Ó - rw( X, I[ > ] 1) Ê EÁhe Ë - Ú ( hu hu ) du [ > > ] 1) ] G -r e H D G WINTER 003 THE OURNAL OF DERIVATIVES 35

12 Q E Ê Ï Á W W + 1 X r q X W Á Á + ( - ) Á rw X I 1 X W Á X db Q -r - - (, [ > ] ) + Ú ( hu hu ) du Ì e e G Á Á Á È ÍÂ hw ( X, I[ > ] 0, ) - Á + 1 Í Á Î Í( h h ) W ( X, I[ > Ë Ó ] 1) where he condiional expecaion of he ump componen DW i E Nex, applying he no-arbirage maringale condiion in Equaion (11) give he required rericion on he evoluion of he ESO value proce aed in Equaion (1). ENDNOTES The auhor hank he referee, Rangaraan Sundaram, for very helpful commen and uggeion on he final verion of hi aricle. He alo hank Fred Ardii, Avi Bick, Rober Grauer, Rober arrow, Peer Klein, and Carl Luf for beneficial commen and dicuion. Paen are pending on hi model. * More formally, in coninuou ime, {D, 0 T} i he filraion, D (I [>u],0 u ) for he defaul proce, and he urvival indicaor I [>] i D -meaurable. Similarly, {H, 0 T} i he filraion, H (X u, 0 u ) for he ock proce, and X i H -meaurable. REFERENCES Ê È Á ÍÂhW( X, I[ > ] 0, ) - e Ú 1 Í Á Ë Î Í( h h ) W ( X, I[ > ] 1) 1 Q - ( hu hu ) du G Arzner, P., and F. Delbaen. Defaul Rik Inurance and Incomplee Marke. Mahemaical Finance, 5 (1995), pp Black, F., and M. Schole. The Pricing of Opion and Corporae Liabiliie. ournal of Poliical Economy, 81 (1973), pp Brenner, M., R. Sundaram, and D. Yermack. Alering he Term of Execuive Sock Opion. ournal of Financial Economic, 57 (1) (000), pp Carpener,.N. The Exercie and Valuaion of Execuive Sock Opion. ournal of Financial Economic, 48 (1998), pp Carr, P., and V. Lineky. The Valuaion of Execuive Sock Opion in he Ineniy-Baed Framework. European Finance Review, 4 (001), pp Cox,., S. Ro, and M. Rubinein. Opion Pricing: A Simplified Approach. ournal of Financial Economic, 7 (1979), pp Cuny, C., and P. orion. Valuing Execuive Sock Opion wih an Endogenou Deparure Deciion. ournal of Accouning and Economic, 0 (1995), pp Duffie, D., M. Schroder, and C. Skiada. Recurive Valuaion of Defaulable Securiie and he Timing of Reoluion of Uncerainy. The Annal of Applied Probabiliy, 6 (1996), pp Duffie, D., and K. Singleon. An Economeric Model of he Term Srucure of Inere Rae Swap Yield. ournal of Finance, 5 (1997).. Modeling Term Srucure of Defaulable Bond. Review of Financial Sudie, 1 (1995), pp Huddar, S. Employee Sock Opion. ournal of Accouning and Economic, 18 (1994), pp Huddar, S., and M. Lang. Employee Sock Opion Exercie: An Empirical Analyi. ournal of Accouning and Economic, 0 (1996), pp arrow, R., D. Lando, and S. Turnbull. A Markov Model for he Term Srucure of Credi Rik Spread. Review of Financial Sudie, 10 (1997), pp arrow, R., and S. Turnbull. Pricing Derivaive on Financial Securiie Subec o Credi Rik. ournal of Finance, 50 (1995), pp ennergren, L., and B. Nalund. Commen on Valuaion of Execuive Sock Opion and he FASB Propoal. The Accouning Review, 68 (1993), pp VALUATION OF STOCK OPTION GRANTS UNDER MULTIPLE SEVERANCE RISKS WINTER 003

13 Lamber, R., D. Larcker, and R. Verrecchia. Porfolio Conideraion in Valuing Execuive Compenaion. ournal of Accouning Reearch, 9 (1991), pp Lando, D. On Cox Procee and Credi Rik Securiie. Review of Derivaive Reearch, (1998), pp Madan, D., and H. Unal. Pricing he Rik of Defaul. Review of Derivaive Reearch, (1998), pp Marcu, A., and N. Kulailaka. Valuing Employee Sock Opion. Financial Analy ournal, 50 (1994), pp Marquard, C. The Co of Employee Sock Opion Gran: An Empirical Analyi. Working paper, New York Univeriy, Meron, R.C. Opion Pricing when Underlying Sock Reurn are Diconinuou. ournal of Financial Economic, 3 (1976), pp Proer, P. Inegraion and Differenial Equaion. New York: Springer-Verlag, Rubinein, M. On he Accouning Valuaion of Employee Sock Opion. The ournal of Derivaive, 3 (1995), pp To order reprin of hi aricle, pleae conac Aani Malik a amalik@iiournal.com or WINTER 003 THE OURNAL OF DERIVATIVES 37