(Real-)Options, Uncertainty and Comparative Statics:

Transcription

1 obas Brg * / Sascha H. Mölls / mo Wllrshausn Ral-Opons, Uncrany an Comparav Sacs: Ar Black an Schols msakn? Summary: h purpos of hs papr s o analyz h nflunc of uncrany on h valu of ral opons whl allowng for a possbl chang n h valu of h unrlyng ass. W show ha h proposon of a srcly posv nflunc of uncrany os no hol, f h valu of h unrlyng ass changs u o a varaon of h sanar vaon. Only f h unrlyng rsk s unsysmac or h bnng rlaon bwn rsk an rurn s nglc, h srcly posv ffc of uncrany can b ran. In all ohr cass, h nflunc bcoms ambguous. In aon, w scuss h consquncs of our rsuls on a mor conomc lvl o convy an unrsanng of whn h procur al wh woul b nca. Fbruary 9 Kywors: Ral Opons; Uncrany; Invsmn/Uncrany-Rlaonshp; Rsk; Unrlyng JEL: G3, G3, O3 * obas Brg, chncal Unvrsy of Munch, Arcssrass, D-89 Munch, Grmany. Sascha H. Mölls Corrsponng Auhor, Chrsan-Albrchs-Unvrsy of Kl, Insu for Busnss Amnsraon, Char of Fnancal Accounng an Aung, Olshausnsrass 4, D-498 Kl, Grmany. E-Mal: molls@bwl.un-kl., phon , fax mo Wllrshausn, PrcwarhousCooprs, Mar-Cur-Srass 4-8, D-6439 Frankfur, Grmany.

2 Ral-Opons, Uncrany an Comparav Sacs: Ar Black an Schols msakn? Summary: h purpos of hs papr s o analyz h nflunc of uncrany on h valu of ral opons whl allowng for a possbl chang n h valu of h unrlyng ass. W show ha h proposon of a srcly posv nflunc of uncrany os no hol, f h valu of h unrlyng ass changs u o a varaon of h sanar vaon. Only f h unrlyng rsk s unsysmac or h bnng rlaon bwn rsk an rurn s nglc, h srcly posv ffc of uncrany can b ran. In all ohr cass, h nflunc bcoms ambguous. In aon, w scuss h consquncs of our rsuls on a mor conomc lvl o convy an unrsanng of whn h procur al wh woul b nca. Kywors: Ral Opons; Uncrany; Invsmn/Uncrany-Rlaonshp; Rsk; Unrlyng JEL: G3, G3, O3

3 . Inroucon Bas on h work of Myrs 977, h ral opon-approach has brokn nw groun n h mos vrs aras of applcaon snc h n of h 98s. Whn hs approach, a narly unanmous consnsus has crysallz n h lraur on h posv ffcs of ncrasng uncrany on h valu of ral opons, rcly opposng h funamnal prncpls of no-classcal nvsmn an fnancng hory. 3 h work of Black an Schols 973 provs h horcal founaon of hs assron, showng ha h valu of an opon ncrass as h volaly of h unrlyng sock grows, prov all ohr paramrs rman unchang. Howvr, s parcularly ffcul o concv usng ral opons of how h valu of h unrlyng rgularly calcula as h prsn valu of fuur cash flows DCF-valu shoul no rac o an ncras n volaly. In hs cas, on rahr has o assum a rucon of valu. hus, wo counrvalng ffcs nflunc h opon valu, whos n rsulng nncy can vary. h arcl a han sks o xamn hs ffrn n ffcs mor closly. hs problm rlas rcly o h crqu al wh unr h rm nvsmn/uncrany-rlaonshp 4 concrnng h mporanc of uncrany whn h ral opon-approach. h scusson whn hs crqu concnras on h probably of xrcsng h opon, hrby rfung h prvalng prcpon ha growng uncrany las o a clraon n nvsmn. In, h quson as o how a chang n h unrlyng nfluncs h corrsponng opon valuaon rmans nglc, an omsson whch lavs a furhr an prhaps all h mor sgnfcan aspc largly unsu up unl now. h sngl xcpon, Davs, os ak hs facor no consraon an lkws coms o a conracory rsul: h valu of h growh opon ns o fall wh an ncras n uncrany. 3 4 Cf. for prvous praccal-orn lraur Smh/rans 995, p. 48; sbrg 995, p. 43; Amram/Kulalaka 999; Coplan/Ankarov ; Damoaran 3, p. 4; Damoaran o.j., p. 7 ff. Cf. for purly scnfc ramns McDonal/Sgl 986; Pnyck 988; Paock/Sgl/Smh 988, p. 54; Wllams 993; sbrg 994; rgorgs 996; O/hompson 996 an Schwarz/Moon. For an mprcal suy s Haushalr al.. ry fw xcpons an lkws rfrncs of ohr nrpraons can b foun n Emry al. 978; Jagannahan 984; McDonal/Sgl 985 an 986, p. 7; Kulalaka/Pro 998; rgorgs 999 an Dx/Pnyck 994. hs rcommnaon [o nvs n projcs wh hghr varanc] s n rc conras o h prscrpons rv from h raonal Markowz 95-obn 958 fnancal porfolo an Hllr 963 ral nvsmn mols. Cf. Emry al. 978, p Cf. Sarkar ; Cappucco/Moro ; Davs an Lun 5.

4 hs arcl sks o concpually gnralz Davs's fnngs n consraon of h prvously scuss lraur an o xpan from hr no ohr opon yps. hs combnaon wll succ n rvng clos-form soluons n h cas of Europan nry an x opons, soluons on h bass of whch clar sparaons concrnng h valu ffcs ar possbl. h prncpl conclusons wll b as follow: In accor wh Davs, h analyss of h nry opon rvals a valu rucon n hs opon yp vs-à-vs ncrasng uncrany. For h x opon, howvr, on os no obsrv h rvrs mpac. Rahr, h cas of h x opon coms o an nnsfcaon of h raonal valu ffc. From hs, h snsvy of h ral opon valu clns consrably rlav o h somhow opaqu uncrany paramr. Also, h analyss of uncrany n Mron's 974 mol always las o h ngav nflunc of hs paramr on quy valu as long as h rang of h frm s br or qual o h BB - rang class. Fnally, on can subsqunly rsolv h larnng paraox of h ral opon-approach,.. h quson as o why a frm wh flxbls woul b nrs n larnng abou h fuur gvn ha s opon valu rss wh h gr of uncrany. hs conclusons shoul nrch subsqun scusson wh furhr aspcs an nhanc h accpanc of h ral opon-approach va a br horcal founaon. hs arcl s organz as follows: In Chapr, h sup of h mol wll b luca, whrby h ffc of volaly on h prsn valu of cash flows as wll as on h opon componn wll b ngra. Bulng upon ha, Chapr 3 wll al wh h comparav sacs for pus an calls of boh Europan an Amrcan yps on h bass of hs mof approach. Chapr 4 wll conan hr xampl applcaons whch shoul clarfy h alr conomc mplcaons whn h nw framwork. h arcl wll n wh a concluson an an oulook for fuur aras of applcaon.

5 . Drvaon of h Mof w 3. Mahmacal Drvaon In h followng, wll b shown by xampl how a chang n h rsk srucur of h unrlyng affcs h valu of h classcal nry an x opon. 5 o hs n, h cash flow procss assoca wh h ponal nry an x wll b frs rv no a prsn valu rprsnng h unrlyng of h rspcv opon yp. h rvaon hus pns on prvalng procurs whn h framwork of ral opon-hory. 6 h cash flow assoca wh h xrcs of h opon uncran an follows a gomrc Brownan moon of h form: 7 CF s assum o b CF = α CF + CF B, CF = c = known, wh rf α an volaly as wll as h common assumpons n rlaon o probably spac an flraon. In accoranc o h objcv, h procss s now compl by a furhr rsk componn, whch can b unrsoo as an ncrmnal accrual of uncrany: 8 CF = α CF + CF B + ε CF B, CF c. = Gvn h assumpon of a compl capal mark, h xsnc of raabl scurs X can b ascran, whch ar capabl of hgng h rsk B of h procss, Cf. for a smlar rvaon Wllrshausn al. 7, p. 36 ff. h rasonng can also funamnally b carr ovr o h cas of fnancal opons. In h framwork of h ral opon-approach h c.p.- assumpon s, howvr, lss problmac snc h capal mark paramrs as wll as h qulbrum upon h capal mark s no affc by paramr varaons n gnral. An analogcal ranson coul also b on rlav o h rskfr nrs ra. Cf. Dx/Pnyck 994. Lkws, Sarkar approachs h problm rfrrng o cash flows; howvr, hs mofcaon of h hory appars n Dx/Pnyck 994 as wll. As an alrnav o h valuaon bas on h hory of Marngals h Dynamc Programmng-approach s ofn chosn. h ncluson of so-call jump procsss woul also b possbl as a mar of prncpl. hs aspc was howvr alray scuss brfly n Dx/Pnyck 994 as wll as n Mölls/Wllrshausn/Krag 5 an can hus b carr ou n accoranc wh hs houghs. A hs pon, w wll forgo furhr complcang h analyss wh such jump procsss. h nsnua gomrc Brownan moon has no qualav ffc on h subsqun rsuls. I was chosn srcly for provng an xpn comparson bwn h rsuls sclos hr an h classcal rsuls. o ha h cash flow scrbs a cash flow ra ha s accorng o h sanar wrn wh capal lrs. Whou an ssnal lmaon of gnraly, s assum ha B an B ar complly uncorrla. h argumn changs margnally, whn hs assumpon s lf ou. In hs cas, a rvson of h bassransformaon woul n o b carr ou. Cf. lsn 999, p. 37 ff.

6 =,. 9 I s assum ha h 4 X follow a sochasc ffrnal quaon of h form : X = µ X + X B. Prov hs, h proposon from Harrson an Plska yls h xsnc of a fn marngal Q, o whch h normalz prc procsss of ach raabl sock ar marngals. Wh h valu of Q known, h prsn valu of h cash flows can b rmn whn hy ar gnra sarng n m.. whn nrng h projc: r Q CF s CF = E s CF rs =, whrby 3 = µ µ 6 Rqur 4748 Rurn 4 r r + + ε α = r + θ + εθ α, 3 r whch corrspons o h sarng pon of Sarkar an Davs. h rqur ra of rurn can b nrpr n h sns of an nsananous an saonary CAPM. 4 In so ong, r sans for h rsk fr nrs ra an θ nos h mark prc of rsk of h rspcv rsk class B, whch s prsum o b npnn of h amoun of rsk of h cash flows. Whn nrprng h prcng quaons, hr ffrn cass can b I s possbl o forgo hs assumpon an choos an alrnav approach. Cf. Davs, p. 4 ff. as wll as Lun 5, p. 6 f. I s no ncssarly h cas ha = or ε =. If s aonally assum ha h volals of h procsss corrspon o on anohr, hs woul affc h ra of rurn shorfall cf. h xplanaons furhr blow. Howvr, hs assumpon, whch s foun n much lraur on h subjc, s only slom nabl. Coplan/Ankarov, p. 94, gv an spcally monsrav xampl: h valu of a gol mn wll faur a hgh corrlaon wh h prc of gol, bu h volals connc wh h objcs ar n gnral complly snc. In prncpl, s possbl o prouc h prfacor by rsrbung h wghng n h uplcang porfolo. Snc h rsk class canno b abanon n ong so, h xpc rurn ~µ of h rarrang porfolo shoul b rmn accorng o h CAPM va ~ µ r µ r + = cf. also Øksnal, p. 54 ff. A frs glanc, h ra of rurn shorfall woul hn b ruc o ~ µ α, bu h subsuon of ~µ n corrsponnc o h prov quaon wll µ r subsqunly rsul n h valu = r + α, as hs arcl wll show blow. Cf. Harrson/Plska 98 an 983 an Dlban/Schachrmayr 994. Cf. Scon 6.. h rasonng shows ha + α pcs xacly h rqur coss of capal. Snc CF follows a gomrc Brownan movmn, s wll fn. h rmnology chosn n hs passag follows Dx/Pnyck 994, p. 48 f.

7 sngush concrnng h aonal rsk componn Purly unsysmac rsk θ = : 5 An aon of purly unsysmac rsk os no chang h rqur ra of rurn as wll as h prsn valu. In hs cas, all h conclusons of classcal ral opon-hory rman n ffc. B : Sysmac rsk an an ncras of cash flow θ >, CF : Wh an aon of sysmac rsk an a corrsponng smulanous ncras of h cash flow CF, h prsn valu rmans unchang. hs suaon can, as a gnral rul, b no as a nw nvsmn. If a frm has h choc bwn svral projcs ha all rqur h sam prsn valu.. amoun of nvsmn, hn s o b assum ha wh h choc of h rskr projc h gnra cash flows woul b hghr. In hs cas, h classcal vw of ral opon-hory also rmans n ffc. 3 Sysmac rsk whou an ncras of cash flow θ >, CF : Wh an aon of sysmac rsk whou a smulanous rs n cash flow CF, h prsn valu clns. hs suaon crcumscrbs a projc alray n forc. If an nvsmn s alray acv an h volaly of h cash flows changs lar on, a prcaon of valu follows. In such a sng, Davs hghlghs ha n ral suaons coul nhr b assum ha h prc of rsk woul b ngav or zro, nor coul h ral ra of growh corrsponngly b corrc by rasng h varanc. 5 Exprss n anohr way, h valu of rss wh an upak of furhr rsk whn h vlopmn of h man cash flow rmans consan an assumng a posv mark prc of rsk n ha rsk class 6 θ >. 7,8 h rlaon a pon = furhr mpls Davs, p. 6 ff. gvs numrous ral xampls of such a crcumsanc. hs s h ralsc cas. In mos ral projcs, a posv ba valu s assum, hus mplyng ha h assoca mark prc of rsk n h corrsponng rsk class s posv. I s mpl ha hrough a chang n h valu of h valu α os no chang. h cash flow of h projc alon os no hrfor mrg from a compl mark. I shoul no b assum ha hghr uncrany auomacally rsuls n a hghr ra of rurn n h vlopmn of h cash flow. hus, a frm can ancpa h man growh of cash flows an h apprasal of such growh os no chang mrly accorng o an ncras n volaly cf. Davs. Hghr ras of rurn, whch a projc wh hghr volaly mus xhb, com abou hrough a rucon of h prsn valu. Alrnavly,.. corrcng for rf, wo projcs wh complly snc cash flow profls woul b compar. I shoul com as no surprs ha h subsqun conclusons of such a cas woul san n conracon o h funamnal ns of nvsmn hory.

8 6 ha an ncras n c.p.,.. wh a hol consan bcaus s obsrvabl CF, rsuls n a lowr valu of h unrlyng. For hs rason, hghr uncrany works o cras h valu of h opon's unrlyng. hs accors wh h funamnal ns of nvsmn hory. Usng h Io Formula, h valu procss rsuls n h followng: = r + B = r, Q CF + B + ε B, Q, Q + ε B, Q, = CF /. As h xprsson shows, h volaly srucur of h prsn valu procss rsuls from h volaly of h vlopmn of h cash flow cf.. A complly uncoupl obsrvaon of boh uncrany srucurs shoul hrfor also no b ffc for mor gnral cash flow procsss. h alraon of h prsum volaly n h unrlyng n favor of a furhr rsk componn ε has va h cash flow procss an ffc upon h prsn valu of h cash flows. Consqunly, h prsn valu of h unrlyng slf funamnally sans n funconal cohrnc wh h paramr as wll as ε. In summary, can b assr ha u o holng h prsn valu consan a = h classcal vw mplcly mpls furhr assumpons whn carryng ou comparav sacs. 9 A fxaon of h prsn valu nals nohng ohr han an ncorporaon of purly unsysmac valuaon-rrlvan rsk θ =, prov ha h unrlyng cash flow profl α s no suppos o bcom complly supplan. hs corrspons graphcally spakng o a horzonal ranson of h mos vrs rsk classs cf. Fgur. Accorngly, h classcal vw can b ran, f h rs n rsk mrgs n h form of unsysmac rsk. If, howvr, sysmac componns gan a foohol, n h n only h prsn valu rucon or cash flow procss xchangs woul rman. hus, a conflc wh classcal nvsmn hory os no xs. 8 9 Sarkar an 3, Cappucco/Moro, Davs an Lun 5 all com o h sam assrons. Dx/Pnyck 994, p. 334 warn of h uncrcal us of comparav sacs rfrncng hos assumpons ofn hl mplc. I s mporan o no ha h sncon bwn rsk avrson an rsk nuraly s no ssnal for h purpos scuss hr. Rahr, h form of h projc rsk s mor saln.

9 7 Fgur : Classcal vs. nw vw whn h CAPM sysmac rsk unsysmac rsk. Illusraon of h Rsuls In hs scon, h abov rsuls wll b llusra for h classcal nry an x opon. For h valu of an Amrcan call opon upon h unrlyng wh srk prc I an nfn maury, h followng s val: wh β A, C I, für < = für β = r / + r / + r / > β β β A = * I / * =. I β h lf llusraon n Fgur graphcally plos hs opon for wo snc valus of. I wll bcom clar ha an ncras of uncrany < wll hav h opon valu rs. Wh a fx unrlyng, hs las rcly o h assron ha uncranly has a β Cf. Dx/Pnyck 994, p. 36 ff. In h abov mol hs corrspons o an aon of,.. ε B =.

10 8 srcly posv nflunc on h valu cf. h lf llusraon. Amly, such a concluson pns csvly on h assumpon ha a rs n h volaly of cash flows os no nflunc h DCF-valu. Howvr, as was shown n Scon., hs s only a rasonabl assumpon unr cran prmss, u o h rlaon CF CF = = r + θ + εθ α. Whou hs assumpons h valu racs o a chang n h volaly srucur n pnnc wh h rsk class. A projc rsk θ = complly uncorrla wh h mark rsuls n h aformnon consllaon =. In h cas θ fol- lows ha >, an fnally, whn θ, < s val. I s vn < ha, pnn on, h cumulav valu of h opon can jus as wll rs as fall cf. h llusraon o h rgh n Fgur, whr only h ralsc cas θ s agramm. 3 Fgur : Ol vs. nw prspcv rlav o h Amrcan call > > 3 Accorng o h nw approach an ncras n h paramr no longr ncssarly las o a lar nry, snc an ncras n c.p. causs a cln n h nry hrshol. Cf. Sarkar for a mor labora scusson on a comparabl quson. Sarkar also chooss h abov pcon of, bu a scusson of how a chang of h paramr whn h comparav sacs affcs hs assumpon rmans absn.

11 9 h unvrsal valy of h samn suggs by h llusraon a h lf n Fgur s n no cas nabl, howvr. Rahr, a funamnal conflc bwn h wo scrb ffcs prsss: Growng uncrany nals boh a fall n h DCF-valu as wll as a rlav rs n h valu of flxbly. An ssnally homognous pcur rsuls n h cas of an Amrcan pu opon wh nfn maury cf. Fgur 3. For a projc rsk ha s posvly corrla o h mark θ >, boh ffcs ac n h sam, valuapprcang rcon. hs s no rmarkabl nsofar as pu opons rcly nsur agans srssng conons. Fgur 3: Illusraon of h nw prspcv rlav o h Amrcan pu 3. Comparav Sacs In hs chapr, h crqu ra n h prvous scons wll b analyz quanavly. Whn h praccal framwork of comparav sacs-analyss s always assum ha an ncras of h facor scrbs h growh n uncrany. In lgh of h abov lbraon, hs corrspons o h aon of a scon an ncal Brownan moon,.. B = B. Bcaus of hs, h problm of h form of h rsk sysmac or unsysmac from whch h ncras n uncrany orgnas, rucs o h quson of h yp of rsk hs parcular Brownan moon B xhbs.

12 Four classcal opon yps wll b xamn 4 : Frs, h smpl Europan call an pu opon wll b al wh, follow by h Amrcan nry an x opon wh nfn maury. h frs wo cass rsul n clos-form soluons, whl n h lar wo suaons w rvr o numrcal rsuls u o prvalng complxy an ambvalnc concrnng h rsulng ffcs. h procur slf ffrnas from promnang approachs 5 n on csv aspc: I os no s h valu as fx, bu rahr rmns hs paramr hrough h mplc quaon CF CF = = r α + θ for ach choc of an hn calculas h opon valu usng h mof unrlyng. 6 Only h valu CF s assum o rman consan bcaus s obsrvabl an hrfor corrspons o an nvaran vn ra = CF. Morovr, n h followng h xrcs of h call opon wll b nrpr as an nvsmn an h xrcs of h pu opon wll b vw as a snvsmn. 3. Analyss of Europan Call Opons hs xamnaon aks as s pon of parur h rlaonshp of h Europan call opon valu C wh srk prc I an maury accorng o h Black/Schols-Formula on a vn-payng sock: 7 C = I, r whrby nos h cumulav sanarz normal srbuon ln / I + r + / = an =. Assumng ha s complly npnn from h valu of, h paral rvaon of h call opon on a vnpayng sock wh maury rsuls n h followng: Cf. for an analogcal analyss Wllrshausn al. 7, p. 38 ff. For xcpons s Sarkar un Davs n parcular. hus, h so-call opon la rmns h chang n h opon valu. Cf. Hull 5. Cf. for h rvaon Hull 5. o ha as h currn sock prc s always grar han zro. nos h rvaon of h cumulav sanarz normal srbuon.

13 C = >. 4 hs wll-sablsh rlaon founs h assron ha growng uncrany n h form of posvly nfluncs opon valus. h assumpon, howvr, ha slf s a funcon of slghly changs h pcur: 9 C = θ h lang componn of h rvav s novl. I s nuc by h rucon of h prsn valu an splays h ambvaln valu ffcs. h rm θ + s lss han zro for posv valus of θ an, an can xc h valu wh a suabl choc of paramrs. In hs cas, ncrasng uncrany woul hav a ngav nflunc. h abov formula maks furhrmor clar ha h mpac of h prsn valu complly sparas slf from h ncras n flxbly as xprss by h scon summan n an av mannr. As a comparson wh h quaon n Formula 4 splays, h fnal rm rmans complly unalr n s form. 3 hs rsuls rval h n for cauon whn nrprng paral rvavs. Drvaon 4 mrly sas ha n wo projcs wh h sam DCF-valu on mus choos h on whch xhbs a grar rsk, snc h ral opons connc wh hs projc ar mor valuabl. 3 If on obsrvs changs n rsk whn a projc, hn no gnral concluson on h alraon of valu n h ral opon can b oban cf. 5. Insa, such a cas rqurs a mor prcs xamnaon of h corrsponng crcumsancs. 3 A hs pon s appropra o rmn as Davs aply pus whhr h ral opon s n h mony or ou of h mony. 33 In h frs cas, an ncras n Cf. Scon 6.. As shown n Scon 6., hs s npnn of h spcal form of h funconal pnncy. o agan ha h n prsn valu-rul n hs cas woul rsul n an nffrnc rgarng h cson-makng. h uncononal assron of a posv nflunc, as has bn ras n ohr works cf. Foono n clar sncon from nav DCF-mhos, shoul b vw xrmly crcal n lgh of h abov xplanaon. Huchzrmr/Loch prsn a sngl xcpon, brakng own an nrprng h nflunc of uncrany an rsk vry prcsly on h bass of a bnomal srucur. rgorgs 999, p. 37 f., rfrncs hs aspc n a fgur by sngushng h ffc of ncrasng uncrany no h nflunc on h sac P an h opon prmum. Mahmacally spakng, h cas ffrnaon s rflc n h varous valus of h cumulav normalz srbuon: Bng n h mony rsuls n a hghr valu for h cumulav normalz srbuon, whl bng ou of h mony lns slf o a lowr valu.

14 uncrany works promnaly owar a cras n h prsn valu, snc h opon characr s no parcularly pronounc n hs rang. From hr, h opon valu ns o cras n sum. Howvr, snc h opon characr gans n mporanc rlavly wh ach rucon n h valu of h unrlyng, h cln n valu has a snc absolu mnmum. h oppos rsul appars n h scon cas. Hr, h opon s ou of h mony an h opon valu s ssnally born upon h opon characr. An ncras n uncrany aonally rass h valu as wll as mosly compnsas for h rucon n h prsn valu. Fgur 4 splays hs suaon graphcally. h graph o h lf shows h bhavor for h cas of bng ou of h mony. In h classcal suaon rprsn a θ =, snc rmans consan a hs pon 34, h valu of h ral opon ncrass as xpc n a srcly monoonc fashon. Howvr, wh an ncras n θ, h ffc on h valu bcoms ambvaln. A compnsaon of h prsn valu ffc no longr occurs n cran suaons. h nflunc rvrss slf. h graph o h rgh scrbs h bhavor n h alrnav rang. h classcal vw θ = ancpas a rs n valu. h mof vw, n conras, ns o sa a cras n valu, whch urns ou all h mor clarly wh an ncrasng mark prc of rsk n rsk class θ. hs confrms h rsuls of Davs n h Europan cas as wll. Fgur 4: Europan call pnn on ou of h mony vs. n h mony 34 Aon of purly unsysmac rsk. hr s no ncssy for a rucon n h prsn valu.

15 3 3.. Analyss of Amrcan Call Opons h cas of an Amrcan call opon wh nfn maury only margnally ffrs from s Europan counrpar. 35 Fgur 5 shows h valu of h call opon pnn on wh CF hl consan n boh cass of bng n h mony an ou of h mony. I can b onc agan obsrv hr ha wh an ncras of h mark prc of rsk n h gvn rsk class rprsn by ρ Mark x, θ, whrby ρ nos h coffcn of Mark corrlaon h ffc of a posv nflunc bcoms mga o h pon ha hs ffc bcoms rvrs cf. h graph o h rgh n parcular. hs rvrsal alray occurs n complly ralsc chocs of paramrs. hus, h lows curv n h graph o h rgh rfrs o a mark prc of rsk of h rsk class a a lvl of. 8, whch wh a rsk-fr nrs ra of r = 5%, a mark rsk of = 8%, an a % mark ra of rurn yls a corrlaon of h projc rsk wh h mark porfolo of approxmaly.3 % 5% Accorngly, wha has hus far bn scuss s no a fancful 8% consruc. As sn alray n h Europan cas, h rspcv mal run of h funcon shows h ambvaln bhavor of valu spcally clarly. Smallr valus of rsul n an ancpa ngav nflunc on h valu. If h gr of uncrany ncrass furhr, howvr, hs posvly affcs h valu of h opon a frs. In hs cas, h ncras n flxbly compnsas for h rucon n h prsn valu whch accompans h growh n rsk. Hr, hs gnral concluson concurs wh h classcal vw. A furhr xpanson of uncrany fnally las o y anohr rvrsal of h nncy of nflunc. As xpc, h conclusons of Davs ar confrm for h Amrcan nry an growh opon. Aonally, h mporanc of h mark prc of rsk shoul b hghlgh, nsofar as h abov scusson shows how snsvly h nflunc of uncrany racs o h mark prc of rsk of h unrlyng In hs cas, h problm of h opmal nvsmn hrshol.. h valu of h unrlyng a whch an nvsmn s opmal as wll as s assoca nvsmn probably com furhrmor no play. Among h gvn conons, hs appls for h hrshol = [ β / β ] I cf. Dx/Pnyck 994 for a pcon of β n parcular. If on accouns for h fac ha β = β, s val, wh h rmnaon of h nvsmn probably on mus allow for h chang n no only h srbuon bu also va h opmal hrshol h varabl. hs aspc wll no b furhr scuss hr snc hs suaon has alray bn al wh n h no works an h xamnaon hr sks o focus on h valu ffc. I coul b clam, for xampl, ha olkswagn xhbs a on-yar corrlaon of approxmaly.7 wh h Grman DAX. In such a cas, h rsk class woul hn rach approxmaly %, clarly hghr han h 8% rfrnc valu.

16 4 Fgur 5: Amrcan call pnn on ou of h mony vs. n h mony 3.3 Analyss of Europan Pu Opons As was h cas wh h Europan call opon, h Europan pu opon P for a vn-payng sock wh maury an srk prc I rsuls n a clos-form soluon n rgar o h nflunc of ncrasng uncrany on h valu. h samn P = 443 > θ + + > s val as long as θ,, >. 37 hs rsul ffrs from h ypcal pcon as alray n h cas of h nry opon by h xsnc of a furhr summan. Du o h pu call pary hs summan rsmbls h abov formula, bu also now xhbs a posv sgn ha rflcs an vn srongr mpac of an ncras uncrany on h valu han assum by h prvalng vw. Rsrcvly shoul b no ha h nflunc on h valu coul bcom ambvaln hr as wll f h growh n uncrany arss from a rsk ngavly corrla wh h mark,.. for θ <. Howvr, hs suaon woul hav mor of a horcal rahr han praccal rlvanc. 38 On can onc agan obsrv ha h valu ffcs avly spara from ach ohr, lavng h flxbly componn complly nac sp h changng funconal rlaon. For a graphcal rprsnaon of hs suaon s rfrnc o h rsuls of h followng scon Cf. Scon 6.. h cas θ = rsuls, as xpc, n h classcal prsnaon of a srcly posv nflunc on valu.

17 5 3.4 Analyss of Amrcan Pu Opons Concrnng h valu ffc h cas of h Amrcan pu opon rsuls n a rahr unsurprsngly homognous pcur. umrcal analyss no only confrms h prvalng vw, bu also rvals an nflunc o b all h mor ncsv unr h nw prspcv. h hg n rgar o poor mark vlopmns ha splay h conomc krnl of h pu opon bcom ncrasngly manngful unr h mof vw: Rsk-avrs nvsors punsh h hghr lvl of uncrany by lowrng h mark valu. Accorngly, h unrlyng of h opon bcoms wors n rgar o s n prsn valu, wha s o h fnal bnf of pu opon's holr. h graph o h lf n Fgur 6 shows h ffrnc bwn h wo vws θ = vs. θ. In hs parcular confguraon, h sncon bwn bng n h mony vs. ou of h mony bars only margnal ffrncs. Wh ha n mn, an xplc comparson bwn h wo wll no b carr ou. Howvr, on shoul no ha h holr of a ral pu opon x opon also ns o own h unrlyng slf, rsrcng h sgnfcanc of an sola analyss of h opon's valu. For hs rason, h graph o h rgh chars h mor rlvan valu ffc for h sum of h pu opon an h unrlyng ruc for h srk prc. 39 hs sum rsuls howvr n narly ncal parns compar o h cas of h Amrcan call opon no abov u o h pu call pary. 4 Fgur 6: Amrcan pu as wll as sum of h pu an ruc unrlyng pnn on 39 4 h valu was ruc by E n orr o us h pu call pary. h sum of h pu plus h unrlyng s grar by amoun E. I shoul b no ha h pu call pary s only an nal approxmaon n h Amrcan cas. For hs rason, h bhavor of h valu s no complly ncal.

18 6 4. Applcaon of h Rsuls h consquncs of h rsuls n h prvous scons wll b scuss n hs chapr on a mor conomc lvl n orr o convy an unrsanng of whn h procur al wh hr woul b nca. Hnc, h followng scons wll focus on hr aras. Frsly, h quson wll b clarf as o how a fals smaon of h paramr woul affc h valu of a ral opon. In lgh of h prvous lbraon, wll hrby b shown ha h hgh snsvy of h calculus bcoms mga. Gong furhr, hs chapr wll onc agan challng h valuaon of quy for a lvrag frm whn Mron's 974 mol. h lar xampl wll srv o clarfy a funamnal nconssncy whn h ral opon-approach,.. f [ral] opons ar n gnral ncrasng funcons of uncrany whras larnng rucs uncrany, why woul w wan o larn? Marzoukos/rgorgs, p.. h classcal vw can only xplan hs paraox wh gra ffculy. 4. Uncrany abou h Dgr of Uncrany In orr o calcula a ral opon valu on has o quanfy h hgh of h paramrs nvolv a h bs. o hs n, h smaon of h rsk class θ = θ ρ as wll as of h ral xpc growh ra of cash flows can b on Mark Mark, mor asly n comparson o an appropra ascranmn of h parcularly rlvan paramr. 4 Snc h classcal vw xhbs a hgh snsvy n hs rspc, h applcaon of h ral opon-rasonng ofn ncounrs ffculs n pracc. 4 Howvr, h agrams n Chapr 3 show ha h corror n whch h valu of h ral opon flucuas accorng o varyng ralsc valus of obsrvably conracs unr h nw approach. h graph on h lf of Fgur 5 plos jus such a chang n h margn from approxmaly [,] o [,6] wh a θ of. an o [,] wh a θ of.8. hs corrspons o a rucon of 6% an 9% rspcvly. 43 In hs way, h hgh Cf. Backr/Homml 4, p. 7 ff. for a gnral scusson of hs problm. Cf. for a smlar pcon Wllrshausn al. 7, p. 34 f. Cf. Backr/Homml 4, p. 7 ff. h prcnag valus ar bas on h followng calculaon: On mnus h nw nrval wh v by h classcal vw's nrval wh.

19 7 snsvy of h calculus bcoms mga as long as h gvn conons ar m. 44 Howvr, h abov consraons only apply o call-s yps of ral opons. As shown n Scon 3.3, h nw approach accnuas h snsvy of pu-lk opon yps. If s brough no consraon ha h ownr of a ral pu opon gnrally ns o own h unrlyng slf, h snsvy of h call carrs ovr o h sum of h pu an h unrlyng, as alray sn. 45 Consqunly, a rucon n snsvy can b uc hr as wll. A praccal applcaon of h ral opon-approach looks o b mor snsbl n lgh of hs. 4. alu of Equy n h Mron Mol h rsuls from Chapr 3 can b rawn upon hr o nvsga h nflunc of uncrany on h valu of quy n lm lably corporaons, hrby clarfyng h ssu as o whhr quy shoul rally always sk h maxmal rsk. In orr o o hs, h smpl sanar srucural mol of Mron 974 wll b call upon. 46 In hs mol, quy s unrsoo as a call opon on h frm's asss, snc h asss only hn ransfr ovr no quy f h b capal s amorz. 47 Unr h classcal vw, hs mol subsqunly assrs ha quy holrs prfr nvsmn projcs wh h gras possbl rsk bcaus of s call-lk valu srucur. Y, n ong so, h pnncy of ass valus on h unrlyng rsk an hrfor h lmnary rsk/rurn-rlaonshp bcom mplcly nga. Gvn h abov oucoms, h quy nvsor s fac wh a complx nvsmn cson, on whch mus accoun for h unrlyng as wll as h opon valu n qual masur. h nvsmn wh h hghs rsk hus os no always prov slf o b h mos avanagous. As h prvous lbraons show, h cson ns o sfavor h rskr projc so long as no aonal cash flow ffc rgsrs.. a largr α n h Corrsponng o h nw approach, h npu paramr shoul for xampl b ra as an nrval.g. bwn % an 3% gvn h paramrs α h cash flows' ral ra of growh an θ h rsk class of h projc, hrby lang o a rucon of h margn. Cf. h graph o h rgh n Fgur 3. h shfng aroun h consan E n hs mag os no affc h snsvy. Black/Schols 973 alray appl h opon prcng-hory o h valuaon of b, an a lar vlop furhr n Mron 974. Cf. Pro/Rosso 7 for an analyss of quy carv ous as sragc ral opons. o ha n h Europan cas h vws on quy as ownng a call or a pu ar quvaln. In h Amrcan cas quy has o b sn as a pu opon.

20 8 projc wh hghr uncrany. 48 In orr o unrpn h prvous assrons quanavly, h rsuls of a smulaon wll b prsn n h followng. For hs purpos, a rang class { Aa, A, Baa, Ba B} oghr wh h cumulav probabls RA RA, CumPD hav frsly bn slc bas on Mooy's 6. Subsqunly, a paramr vcor r,, θ ϕ =, f, an hn L / wr mplcly rmn n a way ha h cumulav probably of falur assoca wh h rang class s prcsly oban,.. P! ~ ln L / r + θ + /. 6 f [ Dfaul] = P[ L = = ] hs was rpa for all rang classs from Aa o B an for all paramr combnaons Ф={φ:, % r f %, % %, 5% θ 4%, 5% %}. Fgur 7 pcs h maxmal valus of h ga ha s, h rvav of h quy wh rspc o h volaly pr rang class an rsk class θ as wll as maury. h lf graph shows ha h ga for all rang classs Ba an br s always ngav,.. an ncras n volaly las o a lowr quy valu. Only wh vry poor cr sanngs rang class B a rs n volaly can la o a hghr quy valu. h graph o h rgh shows ha hs s h cas only wh longr maurs, howvr. hus, for xampl, vn for rang class B h ga always bcoms ngav gvn shor maurs of only a yar. All of hs obsrvaons confrm xsng bankng praccs concrnng cr rsk: On h on han, covnans as a rul spula lmaons on nw nvsmns CAPEX. As monsra n Scon., nw nvsmns conform o h classcal vw of opon prcng-hory, snc h nvsmn amoun s of fx sz an h cash flows ar rgularly hghr wh hgh-rsk nvsmns. Aonally, frms wh poorr cr sanngs ypcally only garnr shor-rm cr. h angr of a conflc bwn quy holrs an ous crors bcoms grar wh longr-rm cr, as h graph o h lf shows. Such a angr os no prss a br cr rangs snc hs opons pc vry p n h mony -opons ha ar omna by h prsn valu ffc. 48 Cf. Fgur 4. h homogny of nrss bwn quy an b n junk bons as posula by Llan/of 996 mus b crcally xamn n h sam way. hs arcl canno, howvr, carry ou an xhausv scusson of hs ssu.

21 9 Fgur 7: Maxmal ga accorng o h mof vw, xnng ovr all ralsc paramr combnaons as a funcon of h rang, rsk class θ, an maury of b n Mron's 974 mol Aa A Baa Ba B. Aa A Baa Ba B θ = 5% θ = % θ = 3% = =5 = Influnc of Larnng In lgh of h ral opon-approach, on mus ask wha sns maks o collc nformaon nn on a rucon of uncrany n a valu-bas sns. I s qusonabl, for xampl, whhr a frm ha carrs ou a mark suy on h commrcal succss of a ponal prouc woul b ong slf a valuang ssrvc by spllng uncrany n such a way. hs concluson woul hol whn h classcal vw n s pur form. A mor sophsca unrsanng of h problm ha aks no accoun h abov lbraons provs a ffrn an much mor snsbl pcur. A rucon of uncrany n h cash flows from a ponal markng of a prouc woul n fac prss h valu of flxbly. 49 A h sam m n conracon wh h classcal vw h ssoluon of uncrany woul morovr chang h unrlyng prsn valu, hrby alrng s lvl. h prsumpon of a sac DCF-valu npnn of s no avsabl n such a cas. Rsk-avrs mark acors woul rwar h aonal knowlg,.. h rucon of h margn, wh a rlavly hghr prsn valu. h acquson of nformaon woul vn nnsfy h hghr valuaon of h unrlyng, nsofar as nhanc knowlg an h ruc uncrany ar ofn accompan by a posv cash flow ffc. Hnc, h frm shoul hn always ak up h coss of rucng uncrany f h gan n h unrlyng woul compnsa for h loss n h opon valu an f h coss 49 h ssoluon of uncrany fnally maks h mporal progrsson of h DCF-valu ruly vsbl. In h xrm cas,.. wh compl crany of h cours of cash flow an hrby of h nr-mporal vlopmn of h DCF-valu, h opmal alrnav woul b apparn an rmn from h ous. Aonal flxbly n comparson o h smpl n prsn valu-rul woul b conomcally spakng worhlss.

22 gnra can b covr by h valu ffrnc. 5 hus, s vn hr as wll ha larnng abou fuur vlopmns an nvsmns o allva uncrany always maks sns f h corrsponng rsk s sysmac an f h gnra nformaon can ruc h margn of possbl sas. 5 Howvr, n h cas of purly unsysmac rsk, lk ha conjcur by h classcal vw, such an nvsmn bcoms conomcally snslss, snc a prsn valu ffc woul rman absn Concluson h prvous scusson has shown ha on canno sck o h swpng clam of a posv nflunc of ncrasng uncrany on h valu of ral opons, as s commonly argu n h prnn lraur. Rahr, concrnng h rsuls of comparav sacsanalyss on has o aop a mor nuanc prspcv ha accouns for h ranscon of valu-rlvan nrrlaons. hs arcl vlop hs nw vw an prsns a mof sanpon whch can brng h ral opon-approach n accor wh h funamnal ns of nvsmn hory. hus, a conflc bwn hs wo os no mrg. Qu o h conrary: Conssn wh classcal horcal approachs, can b subsana ha h valu of a ral opon pns vally on h form of unrlyng rsk. h corrlaon wh h mark porfolo mus hrfor also b sn as an ssnal masur for h rmnaon of valu whn h framwork of h ral opon-approach. h varanc alon can only cononally nform. Along wh hs funamnal nsgh, h rsuls furhr show ha h hgh snsvy of h ral opon-calculus vs-à-vs h somhow opaqu paramr mgas unr h mof prspcv. Praccal applcaon s suppos o fn hs crcumsanc spcally bnfcal snc hs paramr canno gnrally b rmn xacly u o ncompl nformaon. In aon, hs arcl's rsuls wakn h concluson of h classcal Mron 974 mol pranng o pronounc rsk propnsy among quy holrs. h nw prspcv only rcognzs hs phnomnon for unsysmac rsk Onc agan on coms own o h quson as o whhr an nvsmn n h ssoluon of uncrany xhbs a posv capal valu. Rspcvly h cash flow procss coul b alr n a way ha h nvsmn woul bcom profabl. Prsumably h nvsmn only nals a rucon of.

23 Rahr, h ang of valu-rlvan rsk ns o xpos quy sp s opon characr o a p n valu f a posv cash flow ffc os no accompany h rs n rsk, hrby cancllng ou h valu-crasng rsk/rurn-rlaonshp. Fuur conrbuons o h ral opon-approach shoul ncorpora h abov rsuls as wll as h crcal conclusons of h mnon works concrnng h nvsmn probably n hr ramns. In parcular, h analyss alng wh uncrany shoul show n al n whch form h lmnary rsk/rurn-rlaonshp fns rlvanc, allowng h rvaon of mor conclusv an o som xn br foun conclusons. 6. Mahmacal Appnx 6. Drvaon of h Prsn alu Formula Unr h assumpon of a compl capal mark, h xsnc of ra socks X capabl of hgng h rsk of h procss ha h X follow a sochasc ffrnal quaon of h form X B =, can b ascran. I s assum = µ X + X B. Gvn h assumpon of a compl capal mark an h proposon of Harrson an Plska 53, on can uc h xsnc of a unqu marngal masur Q, n rms of whch h sanarz prc procsss of ach raabl sock ar marngals. h procsss X can hus b wrn as: X = rx rx + µ X + X B = rx + X = rx + X B µ r 3, Q. θ + B A smpl zro xplmn was carr ou bwn h frs an scon ln, whl bwn h scon an hr ln, h proposon from Grsanov 54 was us, whrby: Cf. Harrson/Plska 98 an 983 an Dlban/Schachrmayr 994. Cf. Øksnal.

24 B = B, Q µ r. h procss of h mporal flucuaon of h cash flow CF unr Q rsuls n: CF = αcf + CF B + ε CF B, Q, Q = αcf + CF B θ + ε CF B θ, = α θ εθ CF + CF B + ε CF B = r CF + CF B, Q, Q + ε CF B, Q, Q whrby 55 6 Rqur 4748 Rurn 4 : = r + θ + εθ α. h valu θ µ r : = pcs as alray known from abov h mark prc of h corrsponng rsk class B. 56 If on obsrvs h mporal progrss of cash flows as a conngn clam, h prsn valu of h scoun cash flows sarng n m.. whn nrng h projc corrspons o h valuaon formula for any conngn clam: 57 r Q = E r = = CF s CFs r Q s CF = E rs Q CFs E CF = r s s s CF s =. r s s CF s r s r s CF CF s 55 rprsns h ra of rurn shorfall cf. McDonal/Sgl 984. θ ρ µ r 56 Mark θ can also b nrpr as = Mark, X, ha s, h rsk class corrla wh h mark prc Mark of rsk n a cran mannr. 57 I s assum ha cash flows ar accumula nfnly an ha > n orr o assur ru ngrably. h ranson from h frs o h scon ln aks plac hrough h gnralzaon of h proposon of Fubn cf. Duff 3, p In h scon ln was ulz ha CF suably sanarz wh r s s a marngal unr Q. Cf. Øksnal, p. 55.

25 3 6. Comparav sacs hs scon wll vrfy h prov formula for h paral rvaon. I s o rv C, whras s assum ha. I C r = Whn h prsnaon ffrnly han n h x h pnnc of h paramrs on ar mphasz. o as ' h rvav wh rspc o ach valu of rsuls n. ' ' ' ' ' I C r + = Prov ha / CF = an α θ + = r, follows ha n h frs par of h rvav, ' < + = + = CF θ θ θ gvn θ,, >. h scon par of h abov rvav ffrs from h classcal rvaon cf. Hull 5 among ohrs by h assum funconal pnnc for. Dsp hs chang, h classcal oucom s npnn of h concr form of h funconal pnnc prsrv, as h followng calculaons show:. ' ' ' ' ' ' = I I r r I s furhr val ha:, = = = whrby. ln r I + + = Subsung wc smplfs h rvav o:

26 4 ' '. Fnally, gvn ' = =, follows for h rvav aloghr: ' C = θ + +. As prvously no, h lar summan s xacly concoran wh h classcal formula. A novl, howvr, s h rs of h rvav ha s caus by h rucon of h prsn valu an ha splays h ambvaln valu nflunc. h Europan pu opon P of a vn-payng sock rsuls va h pu-call pary P = C + I r n h paral rvav: P C = ' + an lkws wh rcours o h abov rsuls: P = as long as θ,, >. 443 > θ + + >, 7. Ls of Lraur Amram, Marha/Kulalaka, aln 999: Ral Opons Managng Sragc Invsmn n an Uncran Worl, Boson 999. Backr, Phlpp./Homml, Ulrch 4: 5 Yars Ral Opons Approach o Invsmn aluaon: Rvw an Assssmn, n: Zschrf für Brbswrschaf, Supplmn 3, ol. 74, p Black, Fschr/Schols, Myron 973: h Prcng of Opons an Corpora Labls, n: Journal of Polcal Economy, ol. 8, May/Jun, p Cappucco, unzo/moro, Mchl : Commns on h Invsmn-Uncrany Rlaonshp n a Ral Opon Mol, Workng Papr o. 8/, Unvrsy of Paua. Coplan, homas E./Ankarov, lamr : Ral opons: A praconr s gu, w York.

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