OPTIONS EVALUATION - BLACK-SCHOLES MODEL VS. BINOMIAL OPTIONS PRICING MODEL

Transcription

1 Ya IX, o./00 37 OPIO EVALUAIO - BLACK-CHOLE MODEL V. BIOMIAL OPIO PRICIG MODEL Po. Ioan RECA, PhD Assis. Po. Maia-Miuna POCHEA, PhD un L. Angla-Maia FILIP, Ph Babş-Bolyai Univsiy, Cluj-aoa. Inouion A aiulaly imoan issu ha aiss whn i oms o oions is ixing hi valu. h mgn o uaniaiv hnius ha allow oaos o ollow h voluion o inanial asss i has mulili h ansaions on uus maks. Evaluaion oions hoy has is oos in Bahlis sah 900 who us Bownian moion o valua Fnh oions on govnmn bons. Only in h aly 70s oions valuaion mhos hav bgun o gain onsisny by mining a omula o alulaing h i o Euoan oions by Fish Blak an Myon hols. Blak an hols 973 a h ions in iing oion hoy. hy sa om h mis ha i oions a oly valua, h an b ainly no gain om h sal an uhas o oions an unlying asss. Using his inil, hy inou a omula o mining h hoial valu o an oion. his mol is h saing oin o mos uh sah. o Boai, Dml, Ghysls an os 000 min h i o Amian oions whn h asss suo ovis a sohasi ivin yil. hy show ha an Amian oion is woh h sum o h Euoan oion mium an h mium o xising h oion bo mauiy. Chan, Kuma an Rih 00 ovi oniions un whih h sana omula Blak-hols-Mon is vali, vn i h ivin is sohasi. o o his, hy assum ha h sn valu o uu ivins is obsvabl an a owa ona an b obain by aing hs aumula ivins. o obain an xlii omula o mining h oion i whn h mak is inoml, Gsk 978 us h CAPM Caial Ass Piing Mol o balan h isk mium in h onomy. Fom Blak-hols-Mon mol, Lioui A. 006 obain nw omulas o valua oions by onsiing h sohasi ivin yil. om o h hyohss Blak-hols- Mon mol a mov un h nw aoah. Unlik Blak an hols who us h inil o oninuous valuaion, Cox, Ross an Rubnsin sign h binomial mol o alulaing h i o an Amian oion, bas on h aoximaion o a oninuous oss wih a is on. his mol was sn in 979 in Oion Piing: A imlii Aoah. Mol summay onsiss in simulaion o unlying ass i voluion by iviing h im o mauiy in a ain numb o sho ios. mho is usul an vy oula o Amian all an u on a sok oviing ivins. h basi inil o his mol is ha h unlying ass i an ih inas o as in h nx io.

2 38 Finan Challngs o h Fuu. Blak-hols omula o mining h i o Euoan oions Blak-hols mol o mining h i o a Euoan oion is wily us in ai baus i uis knowlg o obsvabl aams: h unlying ass i, h sik i, h im o mauiy o h oion, h oninously omoun isk a an a aam o b sima innnly, h unlying asss volailiy. h mol is bas on a s o assumions o whih h mos siiv a: h unlying ass yil a nomally isibu, volailiy mains onsan houghou h li o h oion, h a no ansaion oss an i an boow mony a h isk a. aing i valuaion mol o Euoan oions by Blak an hols vlo in h 70s, i was wo omulas o mining h i a Euoan all oion an a Euoan u oion : 0 X X, 0 wh ln ln o o X / X / h union x is h umulaiv obabiliy isibuion union o a sanaiz nomal isibuion. In oh wos, i is h obabiliy ha a vaiabl wih a sana nomal isibuion wih avag 0 an sana viaion. 0 is h sn valu o h ass i h oion is xis, an X is h sn valu o h sik i i h oion is xis. 0 h sok i a im 0; X h sik i; σ h unlying asss volailiy; h oninously omoun isk- a; - h im o mauiy o h oion. is h mauiy, an is h momn o mauiy a oion issu =0. Pvious omulas a us whn h unlying ass os no gna anings. h amoun o ivins gna by h unlying ass o h oion as is i an, aoingly, h oion i. hus, h osing o ivins suls in low mium o all oions, an o inas o h mium o u oions. I h unlying ass gnas anings, omulas o mining h i o an oion a: 0 X wh ln ln o 0 o X X / X /, an is h annual ivin yil i h unlying ass is a sha o an inx an h isk- ass a o oign uny i h unlying ass is h xhang a. Exisn o an analyial soluion o h i o a Euoan oion allows analyzing how hi is son o hangs o vaiabls an aams is min. Is h Blak- hols mol assumions, h vaiabls: h unlying i an im o mauiy - an h aams: h unlying asss volailiy σ, h oninously omoun isk- a an h sik i X.

3 Ya IX, o./00 39 Oions i sons o h hangs o hs vaiabls a viually h snsiiviy oiins o h mium an main lmns o masuing h isk ha hs inanial asss involv an a us o in ais ov suh isks. In aiion, h iniaos ailia h vlomn o ash lows gna by ivaiv in h unlying ass aing, hniu whih an b usul i ain inanial oolio managmn sagis involv ivaivs. Givn h imoan o knowing h snsiiviy iniaos, w will oninu his way o mining hi valus in laion o in vaiabls an hi us. Dla Δ is h mos amous on Gk ls an i masus h oion i snsiiviy o vaiaions in h i o h unlying ass. Dla is alula as h is ivaiv o on oion i o a hang in h i o h unlying ass whn h oh aams main onsan. Paially, la is h numb o unis o h unlying ass w shoul hol o ah oion sho in o o a a isklss hg. Fo a Euoan oion on a nonivin-aying sok, i an b shown ha: h la o a all oion is always osiiv, ha is o say, a vaiaion in h i o h unlying ass imlis a vaiaion, in h sam iion, in h i o h all oion. On h oh han, h valu o a u oion ass i h i o h unlying ass inass, imlying, ho, a ngaiv la. Fo Euoan oions on an ass aying a yil, w hav: [ ] Gamma Γ masus h la snsiiviy o hangs in h unlying ass an i is sn mahmaially as son ivaiv o oion i o unlying i o is o ivaiv o h la o. Gamma is inial o boh all an h u oion an an b osiiv o ngaiv. Fo a Euoan oion on a non-ivinaying sok, i an b shown ha: wh, x x Fo a Euoan all o u oion on an ass aying a oninous ivin a a : * Gomi, gamma is h slo o h gah Δ = o onvxiy oion i o unlying i hangs, was us o masu isk ovag, so h isk osiion la nual. Gamma sns valus los o zo whn h oion is ou o h mony o in h mony an h maximum valu whn h oion is a h mony, sially whn h im o mauiy is u. ha Θ masus h oion i hang ov im, whil h oh aams a onsan. o min h ha o a Euoan oion on a nonivin-aying sok, h a h ollowing omulas: using

4 40 Finan Challngs o h Fuu X X Fo a Euoan all o u oion on an ass aying a oninous ivin a a : X X ha is in mos ass a ngaiv aam, baus as hy aoah mauiy, oion valu ns o as. Exion o his obsvaion a Euoan u oions on shas ha hav a vy song osiion in h mony, o all oions in h mony on unis ha hav a vy high ins a. Vga K. Blak-hols omula has bn monsa in h onsan volailiy unlying assumion o h im h oion i is alula. In ai, volailiy is a vaiabl aam, whih mins h oion i hangs. hs hangs a alula using vga whih sns h is ivaiv o an oion valu o volailiy. Fo a Euoan oion, a all o a u, having an unlying ass ha gnas no gain, oiin is givn by h ollowing omula: K K Fo a Euoan all o u oion on an ass oviing a ivin yil a a : K K Vga is always a osiiv oiin an a high valu inias a high snsiiviy o oion volailiy hangs. I vga has low volailiy hangs, i will hav a lil ima on oion i. ong ou o in h mony oions hav a low vga, an h on a h mony high, sially whn mauiy is lay. Rho Ρ masus h snsiiviy o h oion valu o ins a an i is alula as h is ivaiv o oion i o ins a. Fo a Euoan oion, a all o a u, having an unlying ass ha gnas no gain, h oiin is givn by h ollowing omula: X X h sam omula alis i h unlying Euoan oion gnas ivin.

5 Ya IX, o./00 4 Rho is always osiiv o a all oion, whil o a u oion h oiin is ngaiv. Examl no. : Consi a Euoan all oion an a Euoan u oion on a sok ha gna ivin an i has h ollowing haaisis: =0 RO, E= RO, =8.5%, σ=%, =3 monhs, =%. Alying h vious omulas, w obain: Call mium= Pu mium= Dminaion o snsiiviy oiins o an oion Call Pu Dla Gamm a Vga ha Rho ou: Own alulaions using DivaGm By h valus o la, i is no ha an inas by on uni in so i mins an inas by RO in h all mium an a uion by RO in h u mium. Gamma aks h valu , whih mans ha an inas in h sha i by RO om 0 o will inas h oion valu wih RO. Vga is , whih mans ha i h unlying volailiy inas by on nag oin om % o 3%, hn boh all an u mium will inas by ha inias h a o hang o h oion mium wih s o h assag o im. h uion o im o mauiy by on ay las o a uion by in all mium an by in u mium. Rho shows ha i h ins a inas by on nag oin, hn all mium inas by , an u mium as by Consiing h oiins la, gamma an ha in abov, uaion an b win wih h Blak- hols aial inial aoing o hm. hus, h laionshi bwn la, gamma an ha o a Euoan oion, in h Blak-hols mol assumions is: In aiion, i is no ha hs iniaos ina an may no b ga as saa niis. A high volailiy inass h la o ou o h mony an a h mony oions an i bings own in h mony oions, oy suling om la o masu h obabiliy o xising h oion. Changing h gamma o h inas volailiy is mo onoun o a h mony oions an low o hos ou o h mony. 3. A on-s binomial Consi a all oion on a nonivin-aying sok. h assumions o h mol a h sam o h Blak- hols mol, ha h mak i iin, h a no ansaion oss an no ax, suiis a ly ivisibl, sho slling is allow, vnus gna by a suiis a muna a h isk a,, whih is onsan, volailiy mains onsan houghou h li o h oion. A o his a ha h i o h unlying ass ollows a binomial oss in a im, so his is h only hyohsis on h voluion o h unlying ass i. o, i a h im 0 h sok i is, his an mov u in by u ims wih obabiliy o o mov own wih obabiliy -. h oss sib is all h binomial muliliaiv oss.

6 4 Finan Challngs o h Fuu h binomial mol is bas on builing a isk a oolio wih a sho osiion in a all oion an a long osiion in Δ shas. A on-s binomial Examl no. : W oos o valua an Euoan all oion wih a mauiy o h monhs an h xis i o RO. A sok i is unly 0 RO. W suos ha a h n o h monhs h sok i will b ih RO o 9 RO. I h sok i uns ou o b RO, h valu o h oion will b RO. I h sok i uns ou o b 9 RO, h valu o h oion will b zo. Consi a oolio onsising on a sho osiion in a all oion an a long osiion in Δ shas. W alula h valu o Δ ha maks h oolio isklss. I h sok i movs u om 0 o RO, h valu o h oion is RO, so ha h oal valu o h oolio is Δ-. I h sok i movs own o 9 RO, h valu o h oolio will b 9Δ. h oolio is isklss i h valu o Δ is hosn so ha h inal valu o h oolio is h sam o boh alnaivs. his mans Δ-=9Δ Δ=0,333. h isklss oolio onains 33 shas an on oion. Whh h sok i movs u o own, h valu o h oolio is always 3 RO *0,333-=9*0, In h absn o abiag oouniis, iskls oolios mus an h isk- a o ins. uos ha h isk- a is 6% annum. I u ollows ha h sn valu o h oolio is: 3-0,06*3/ =,955 h valu o h sok i oay is 0 RO. I w no h oion i by, hn: 0x0,333-=,955 =0,378 In onlusion, in h absn o abiag oouniis, h un valu o h oion mus b 0,378 RO. I h valu o h oion w mo han 0,378 RO, h oolio woul os lss han,955 RO an woul an mo han h isk- a. W an gnaliz h agumn jus sn by onsiing a sok whos un i is. W no wih h mauiy o h oion an w suos ha uing h li o h oion h sok i an ih movs own o o movs u o u, wh u> an <. h ooional inas in h sok i whn is an u movmn is u-, an h ooional as whn h is a own movmn is -. I h sok i movs u o u,w suos ha h ayo om h oion will b u an i h sok i movs own o, w suos ha h ayo om h oion will b. W will alula h valu o Δ ha maks h oolio isklss. I ollows ha Δ an b hosn so ha h inal valu o h oolio b h sam whh h i o h unlying ass inass o ass uing. uδ- u =Δ- Δ= u - u - In his as h oolio is isklss an mus an h isk- ins a. h vious uaion shows ha Δ is h aio o h hang in h oion i o h hang in h sok i as w mov bwn h nos. I is h isk- a, h sn valu o h oolios: uδ- u - h os o sing u h oolio is Δ-.

7 Ya IX, o./00 43 I ollows ha: Δ- =uδ- u - = Δ-uΔ- u - ubsiuing o Δ an simliying, w obain: = - [ u +- ] wh = u In h numial xaml onsi viously, u=., =0.9, =0.06, =0.5, u =, =0. I ollows ha: 0.06x = = = -0.06x0.5 [0.3837x x0]= Gnaliz Mol h on-s binomial an b xn o a numb o n ios, onsiing all ossibl sas o h sok i, wih i u movmns an n-i own movmns. h sok i is iniially, h isk- a is, an h lngh o h im s is δ yas. Givn h vious suls w obain: u = -δ [ uu +- u ] u = -δ [ u +- ] = -δ [ u +- ] ubsiuing h is wo uaions ino h las, w hav: = -δ [ uu +- u +- ] Examl no. 3: W onsi h all oion wih h sam haaisis as in h vious xaml. A B C D E 0 8. F 0 Ou objiv is o alula h oion i a h iniial no o h. W bgin by sing h oion i a h inal nos. A no D h sok i is 4.4, an h oion i is 4.4- =3.4. A nos E an F, h oion is ou o h mony an is valu is zo. A no C, h oion i is zo baus no C las o ih no E o no F an a boh nos h oion i is zo. W will alula h oion i a no B. W know ha u=., =0.9, =0.06, =0.5, so = I ollows ha h valu o h oion a no B is -0.06x0.5 [0.3837x x0]=.85 I mains o us o alula h oion i a no A. hus h valu o h oion is -0.06x0.5 [0.3837x x0]=0.486 h omula an b xn o n ios using h sam mhanism wih i u movmns an n-i own movmns o h sok i, o an Euoan all oion bing: n n! = -n δ i - n-i maxu i n-i -X,0 i 0 i! n i! In a simila mann, i an alula o an Euoan u oion: n n! = -n δ i - n-i maxx-u i n-i,0 i 0 i! n i! Pvious omula sib som wily us algoihms sially in h valuaion o h oions whih o xis oouniis bo h mauiy o whos unlying asss gna ivins. Essnially, a hoosing h numb o ios o iviing h oion li usually 30 o mo ss- i is buil h binomial nwok o h unlying ass, ollowing ha h oion i o b min lil by lil saing wih h inal nos o h nwok. Examl no. 4: W onsi an Euoan all oion wih a mauiy o h monhs an h xis i o RO. A sok i is unly 0 RO, h isk -a is 8% annum, h volailiy is % annum. h igu blow illusas h binomial wih 0 ss.

8 44 Finan Challngs o h Fuu wih 0 ss o h unlying ass i an an Euoan all oion Dminaion o snsiiviy oiins wih binomial mol an b on by h ansiion o iniions in i om, onsiing h inval δ. hus, i obains o a all oion: la Fo h is s, h oiin an b xss as: gamma As w an s gamma an b alula wih a lay o wo ios δ an o h is wo ios is valu an b aoxima: u u ha 0 In h vn ha his io h unlying ass i, o h oh aams main onsan, h oiin an b aoxima o h is wo nos: 0 K vga wh * is h valu o h oion o h volailiy σ+δσ. ho P Esimaion o snsiiviy oiins in h mann sib abov is usul o ollowing u hging ansaions, whos oninuoully sun is no ossibl. Examl no. 5: W onsi an Euoan all oion wih a mauiy o h monhs an h xis i o RO. A sok i is unly 0 RO,

9 Ya IX, o./00 45 h isk -a is 8% annum, h volailiy is % annum. nsiiviy oiins hav h valus blow in Blak-hols vsion an in binomial vsion wih, 0, 0, 30, 50, 00 ss. Calulaion o snsiiviy oiins. Blak-hols Mol vs. Mol Blak- hols Mol Mol n= Mol n=0 Mol n=0 Mol n=30 Mol n=50 Mol n=00 Dla Gamma Vga ha Rho ou: Own alulaions using DivaGm I an b sn ha h h ins a boming small as h numb o ss inass. imilaly, snsiiviy oiins an b alaula o a u oion. 5. Conlusions Us boh o hging isks an o sulaion, h oions a oun in h oolios o vaious insiuions - om hg uns an inanial insiuions, ooaions o iniviual invsos. Oions hav monsa sussully hi imoan ol in h inanial maks. In an ial sing, ivaivs iing hoy ovis a amwok in whih h isks inhn o an oion s osiion an b minimiz o limina via a ynami hging sagy. In ai, howv, h ivnss o suh sagy an b limi u o h lak o availabl hging insumns an mak miosuu issus suh as ansaion oss an mak illiuiiy. In aiion o h gnally high lvag o snsiiviy o ivaivs valu o h hang o unlying ass valu, a uniu an y vy imoan isk o oions is h soall mol isk ha aiss whnv ivaivs iing an/o hging sagis a bas on a miss-sii mol. Oions is, as hos asss ha onsiu hi suo, a a by sval aos. Knowing hs aos an how hy a h valu o oions is ssnial o us as a ool o inanial isk managmn. nsiiviy oiins o h oions mium masu h sons o hi i o ah o h aos inluning i, oviing an imag o h isk o a osiion on an oion. h vious xamls hav shown ha h in in h alulaion by h wo mols o oions i an hi snsiiviy oiins isaa as h numb o binomial ss inass.

10 46 Finan Challngs o h Fuu REFERECE Blak, F., hols, h iing o oions an ooa liabiliis, Jounal o Poliial M Eonomy, Volum 8, Issu 3, , 973; Boai, M., Amian oions wih sohasi ivins an volailiy: A non Dml, J., aami invsigaion, Jounal o Eonomis, 94: 53 9, 000; Ghysls, E., & os, O. Ca, P., Maan D. owas a hoy o volailiy aing, Volailiy, Risk Publiaions, R. Jaow, E.: 47 47, 000; Chan, D., Kuma, Euoan oion iing wih is sohasi ivin, Jounal o R., & Rih, D. Divaivs: 39 45, 00; Cox, J.C., Ross & Oion Piing: A imlii Aoah, Jounal o Finanial.A. Rubnsin, M. Eonomis, 7, 9-63, 979; El Kaoui,., Robusnss o h Blak an hols omula, Mahmaial Finan, Janblan, M., Vol. 8, Issu : 93-6, 998; hv,. Elam, E., Dixon, Examining h valiiy o uus mak iiny, h Jounal o B.L. Fuus Maks, Vol. 8: , 988; Hull, J. C. Oions, Fuus an Oh Divaivs, sixh iion, Pni Hall, w Jsy, 006; Mon, R. C. hoy o aional oion iing, h Bll Jounal o Eonomis an Managmn in, o. 4: 4-83, 973; Milăuş, P.G., Luu, Piaţa insumnlo inania iva, E. Eonomiă, Buuşi, R. 008; Roman, M., aisiă inania-banaă şi busiă, Eiua AE, Buuşi, 00; Panu,. ayaji D. Masuing Oion Pi nsiiviy h Gk Alhab o Risk, Risk Managmn an Finanial Divaivs, MGaw-Hill, 998; Wilmo, P. Paul Wilmo on Quaniaiv Finan, John Whily & ons, 000; *****