Exotic Options: Pricing Path-Dependent single Barrier Option contracts

Transcription

1 Exoic Opions: Picing Pah-Dpnn singl Bai Opion conacs Abuka M Ali Mahmaics an aisics Dpamn Bikbck Univsiy of Lonon YilCuv.com Pag of 8

2 Aac his pap iscusss h basic popis of bai opions an an analyical soluion fo picing such conacs. h significanc of monioing is consi fo xampl h iffnc bwn coninuous monioing an isc monioing. Pifalls aising fom a naïv applicaion of sana opion valuaion chniqus o bai opions a poin ou. W also iscuss h pacical issus la o bai opions such as h avanags hy povi o h buy as wll as o h wi an consi pacical issus bhin valuaion. y wos an phass: Bai Opions nock-ou Opions nock-in Opions Rba Pah-pnan Payoff Black & chols sic nsiy Rflcion Pincipal. YilCuv.com Pag of 8

3 . Inoucion Bai opions a a class of xoic opions which w fis pic by Mon 973. h mos common appoachs us o pic hs yp of ivaivs a h xpcaions mhos an h iffnial quaion mhos. h xpcaions mho has bn wok ou in ail by Rubinsin an Rin 99 an also Rich 994. h xpcaions picing mho quis h minaion of h isknual nsiis of h unlying pic as i bachs h bai fom abov an blow. If bas apply hn h fis xi im nsiis hough h bai a also qui. Bai opion pics a hn obain in h usual way by ingaing h iscoun bai opion pay-off funcion ov h calcula nsiis. I is consi ifficul o wok ou hs nsiis whn using h xpcaion appoach i is howv makabl ha clos fom soluions fo all yps of bai opions a in fac obain. A bif iscussion of h iffnial quaion mho can b foun in Wilmo 993. h basic ia of his appoach is ha all bai opions saisfy h Black- chols paial iffnial quaion bu wih iffn omains xpiy coniions an bounay coniions. In pincipal hs paial iffnial quaions PDE's can b ansfom o h iffusion quaion an solv. Onc again h analysis is complx an also quis h valuaion of ingals bu h sam clos fom soluions a obain. h soluion fom h PDE mho is of cous la o h soluions fom h xpcaion appoach. Richkn 995 has invsiga compuaional aspcs of bai opion picing using binomial an inomial laic. In his pap h PDE mho will b aop o show ha a ic an simpl analysis las o h clos fom soluions. h mho mploys symmy popis of h Black-chols B& PDE an quis lil mo han h wllknown basic Euopan vanilla opion soluions.. Picing of simpl coningn claims. Ass Pic Dynamics an Io Pocss h ynamics of sock pic a psn by h following Io pocss wih a if a of µ an vaianc a of : µ X. ivi boh sis by o obain h following sochasic iffnial quaion DE: / µ X. his pocss of sock pics known also as h gomic Bownian moion can b win in isc im sing as / µ.3 YilCuv.com Pag 3 of 8

4 Wh is a anom sampl fom isibuion wih zo man an a uni sana viaion. If w s 0 h m involving X in quaion. woul op ou an w a lf wih oinay iffnial quaion ODE / µ o / µ Wh µ is consan his can b solv xacly o giv xponnial gowh in h valu of h ass i.. 0 xp[ µ 0] h anom m X fom quaion. is known as a Win pocss which has h popis fin blow. h mol has h sock pic gowing a a consan a µ wih anom flucuaions supimpos. hs flucuaions a popoional o h sana viaion of h ass pic an a pnan on sana nomal anom vaiabl. his yp of pocss is known as Win pocss. Dfiniion. A sochasic pocss X is call Win pocss if h following hol.. W0 0. h pocss W has inpnn incmns i.. if < s < u hn Wu - W an Ws -W a inpnn sochasic vaiabls. 3. Fo s < h sochasic vaiabl W - Ws has a Gaussian isibuion 0 s. 4. W has coninuous ajcois W can wi X as X φ wh φ is a anom vaiabl wih givn by ~ 0 an a pobabiliy nsiy funcion φ fo < φ <.4 π Dfin h xpcaion opao ξ by ξ[ F.] φ F φ φ π.5 Fo any funcion F hn ξ [ φ] 0 an ξ [ φ ] Mos auhos woul us h l W o associa i wih h Win pocss. YilCuv.com Pag 4 of 8

5 I follows ha fom quaion. h xpcaion an vaianc of can b win as. Io's Lmma ξ[ ] ξ[ µ X ] µ a[ ] ξ[ ] ξ[ ] ξ[ X ] Io's Lmma is an impoan sul abou h manipulaion of anom vaiabls. Whil aylo's hom allows on o manipula funcions of minisic vaiabls Io's Lmma can b appli o manipula funcions of anom vaiabls. I las h small chang in a funcion of anom vaiabl o h small chang of in h anom vaiabl islf. W will us h following Io's muliplicaion abl X.X 0 0 If f is a smooh funcion of an w vay by small amoun hn h funcion f will also vay amoun small amoun f. Using h aylo sis xpansion w can wi h chang of h funcion f as: f f f....6 inc is givn by quaion. w squa i o fin ha µ X X µ X µ.7 h fis m is h lags fo small an ominas h oh wo ms. W a hn lf wih X... inc X. Bjok 998 fo chnical ails of hs suls. YilCuv.com Pag 5 of 8

6 W now suiu his sul an quaion. ino.6 such ha w fin: f f f µ X f f f µ X f f f f µ f X.8 his is Io's Lmma laing h small chang in a funcion of anom vaiabl o h small chang in h vaiabl islf. h fis componn of h igh han si quaion is minisic componn of h chang in h funcion f an is popoional o. h scon componn is a anom componn an is popoional o. h sul.8 can b xn o a funcion of wo vaiabl which nails h us of paial ivaivs sinc h a wo inpnn anom vaiabls i.. an. W can xpan f in a aylo sis abou o g f f f f... f f f f µ f X.9 3. h Black -chols Fomulaion of Opion Picing W illusa how o us h isklss hging pincipl o iv h govning paial iffnial quaion fo h pic of Euopan call. h ivaion follows h appoach us by Black an chols in hi sminal pap 973. hy ma h following assumpion in h financial mak: i ii iii iv v vi aing aks plac coninuously in im; h isklss ins is known an consan ov im; h ass pays no ivin; h a no ansacion coss in buying o slling h ass o h opion an no axs; h a no isklss abiag oppouniis; sho slling is pmi an h asss a ivisibl. L b h valu of an opion whos valu pns on boh an. Using Io's Lmma quaion.9 h anom walk follow by can b win YilCuv.com Pag 6 of 8

7 X µ 3. If w now consuc a pofolio of consising a long posiion of h opion an a sho posiion of h unlying ass h valu of h pofolio is Π 3. h chang in h pofolio in on-im sp is Π L; Puing. 3. an 3. ogh h chang in h pofolio can b win as X Π µ X µ X X µ µ inc X X Π µ µ by simply aanging h abov quaion w hav X X Π µ µ h fis wo ms fom h igh han si of h quaion cancls ach oh an h anom componn in h quaion is limina. his suls in a pofolio whos incmns is wholly minisic: Π 3.3 YilCuv.com Pag 7 of 8

8 o h uncainy u o X is cancll ou an u h pmium fo isk un on is also cancll ou. o only ha h m Π has no uncainy i is also pfnc f an no pnan on u a paam conoll by invsos isk avsion. If h pofolio valu is fully hg hn no abiag implis ha i mys an only isk f a of un. W hn hav Π 3.4 using quaion 3. sinc iviing boh sis by an aanging h quaion w hav finally his is h Black-chols paial ivaiv quaion an any ivaiv scuiy whos pic pns only on h cun valu of an on an which is pai fo up-fon mus saisfy h Black-chols quaion. h mos fqun yp of paial iffnial quaion in financial poblms is h paabolic quaion. Equaion 3.5 is call backwa paabolic sinc h quaion is lina an h signs of hs paicula ivaivs a h sam. YilCuv.com Pag 8 of 8

9 h pic of paicula ivaiv scuiy is obain by solving Equaion 3.5 subjc o h appopia auxiliay coniions minal payoff fo h cosponing ivaiv scuiy. h soluion of h Black-chols quaion wih iffn auxiliay coniions can hn povi valuaion fomulas fo iffn yps of ivaiv scuiis. h m X isappa fom h PDE which mans h is no uncainy. Whil h sock pic volvs in an uncain mann whn w valu ivaivs wih spc o sock pic his uncainy no long xis in h picing fomula fo his ivaiv. h m u which is h xpc a of un on h sock also isappa fom h PDE. h xpc un is affc by isk pfnc. h mo isk avs h invso h small h xpc un. Givn ha h xpc un os no appa in h picing fomula fo ivaivs valuaion of ivaivs in his famwok is pfnc f. h soluion o h iffnial quaion is hfo h sam in a isk-f wol as i is in h al wol. nc his yp of valuaion mho a ofn call isk nual valuaion laionship RR. Applicaion of R ss h xpc gowh a of sock qual o isk f ins a hn iscoun xpc payoff of opion a isk f a. h a many soluions o 3.5 ha cospon o iffn ivaivs f wih unlying ass. In oh wos wihou fuh consains h PDE in 3.5 os no hav a uniqu soluion. h paicula scuiy bing valu is min by is bounay coniions of h iffnial quaion. In h cas of a Euopan call h valu a xpiy c E0 svs as h final coniion fo h Black-chols PDE. 3. Lognomal popy an sock pic pocss Black-chols 973 assum ha h a wo funamnal asss: a bon wih a pic B. an a sock wih a pic.. h pic of h bon an h sock a assum o gow as follows fo any 0 : B xp an 0 xp µ w wh u an a consans an w is a sana Bownian moion 3. h aio of o can b win: 3 aison 985 fo mahmaical ails of Bownian moion. Bownian moion is nam af h boanis Rob Bown. Bown noic in 87 ha polln xhibis anom moion whn suspn in wa. h mahmaics of his "Bownian moion" i no com unil Bachli 900 an Einsin 905. YilCuv.com Pag 9 of 8

10 0 xp 0 xp w w µ µ xp w w µ 3.6 aking log of boh sis of h abov quaion w g ln w w µ h incmn w - w is isibu nomal 0 - so i follow ha µ ~ ln 3.7 Fom quaion 3.6 i can also b sn ha ln ln w w µ an hfo ha µ ~ ln fom quaion 3.6 h minal sock pic may b win as follows: xp w w µ W µ xp 3.8 wh is isibu as nomal w w W 0 un h usual pobabiliy masu. YilCuv.com Pag 0 of 8

11 As mnion abov h pic of Euopan opion a im can b foun by iscouning h xpc payoff of h call opion Wh E nos xpcaion akn un h isk-nual pobabiliy masu. h xpcaion is akn coniional on infomaion a im [ha is coniional on ]: [ ] 0 max E c 3.9 ow suiuing quaion 3.8 ino 3.9:. w w f c W w W µ 3.9a w f W is h pobabiliy nsiy funcion pf of w. Wih 0 ~ W i follows ha h pobabiliy nsiy funcion of W is givn by w W w f π uiu his fo in quaion 3.9a o g h call opion valu: w f W w W c µ x. w w π o simplify l w so ha w an h call opion valu bcoms π µ c l 0 b such ha 0 hn 0 ln YilCuv.com Pag of 8

12 h fomula fo c simplifis slighly bcaus h ingan 4 is inically zo whn 0 < π µ c 0 W may spli h ingan an hnc h ingal ino wo componns: π c 0 0 π Collc ms an simplify: 0 π c 0 π h xponn in h ingan of h fis m is scal pfc squa saisfying ' wh '. ow suiu his ino h fis ingal o simply h xpssion fo c: ' 0 ' ' π π k c 4 h ingan is ha pa ha of h ingal ha falls bwn h an h YilCuv.com Pag of 8

13 h ingans a nomal sana pf's. hfo h ingals involv nomal sana nomal cf's. h opion valu c may now b win in ms of h cumulaiv sana nomal funcion. as follows: [ ] [ ]. 0 0 c Fom h popy of h cumulaiv sana omal funcion: ] [ z z his may b us o simplify c: 0 0 c By algbaic manipulaions on can show ha if ln 0 hn ln 0 an ln 0 If w labl h la wo ms an an spcivly you g h Black-chols fomula fo h pic of a sana Euopan call on a non ivin-paying sock: 3.9. wh ln an 3.9.a 3.9.b YilCuv.com Pag 3 of 8

14 h abov call pic fomula can b inp using h languag of pobabiliy. Fis is sn as h pobabiliy of h call bing in-h-mony a xpiy an so can b inp as h isk nual xpcaion of h paymn ma by h hol of h call opion a xpiy on xcising h opion. con is h isk nual xpcaion of h ass pic a xpiy coniional on h call bing in-h-mony. nc h xpcaion of h call valu a xpiy is which is hn iscoun by h faco psn valu of h call. in h isk nual wol o giv h 4. Bai Opions Opions wih h bai fau commonly call bai opions a consi o b on of h simpls yps of pah-pnn opions. h uniqu fau is ha h payoff pns no only on h final pic of h unlying ass bu also on whh o no h unlying ass pic has ach on-ouch som bai lvl uing h lif of h opion. An ou-bai opion knock-ou opion is on wh h opion is nullifi pio o xpiaion if h unlying ass pic ouchs h bai. h opion hol may b compnsa by a ba paymn fo h cancllaion of h opion. An in-bai opion knock-in opion is bai opion yp which coms ino play if h asss pic his o cosss h pfin bai lvl. Whn h bai is appoach fom blow h bai opion is call an up-opion; ohwis i is call own-opion. On can inify igh yp of Euopan bai opions such as own-in calls up-in calls own-ou calls up-ou calls. An simila fou yps of opions fo h Euopan bai pu opions. All hs opions a call sana o vanilla bai opions. h aacivnss of bai opions is ha hy a chap han hi cosponing vanilla opions as h sum of h pmiums of a knock-in an is cosponing knock-ou is always h sam as h pmium of hi cosponing vanilla opion if h a no bas. 4. anilla bai opions Anoh nam fo bai yp opions is also a igg opion. his is bcaus h payoff pns ciically on whh a p-spcifi bai o a igg is ouch uing h lif of h opion. If h bai is bach uing his im h hol is nil o civ a Euopan opion. Ohwis h hol gs a ba a h mauiy of h opion. his kin of bai is known as knock-in bai opion o simply knock-in. Givn h unlying ass pic h bai lvl can b plac abov of blow i. If h bai is blow h unlying pic h knock-in opion is call a own knock-in opion DI - fo own an in opion. h payoff of a own knock-in opion PDI can b fomally givn as YilCuv.com Pag 4 of 8

15 [ ω ω0] PDI max{ > fo som } an < 4.a o PDI Rm if > an > fo all < 4.b an Wh a h cun an xpiaion im of h opion spcivly; is h knock-in bounay of h opion o h consan bai lvl. is h sik pic of h opion; ω is a binay opao fo a call an - fo a pu. Rm is h ba of h bai opion pai a mauiy if h bai is no ouch. Blow w also so fin h payoff fo maining vanilla bai yps such as up-anin PUI own-an-ou PDO. Fo up-an-ou PUO payoff s Zhang 998. h payoff of an up-an-in bai opion PUI is givn fomally as; o [ ω ω0] PUI max{ < an fo som } < 4.c PUI Rm if < an < fo all < h Payoff of a own an knock ou bai opion o simply own-an-ou is givn; o [ ω ω0] PDO max{ > an > fo som } < 4.c PDO R if > an fo all < R is in his cas also h ba funcion which is im pnan. R is mos ofn an incasing funcion of im saing fom zo o R'> 0 an R00. h ba fin in 4.c is call non-f ba implying ha h ba is pai as soon as h bai is ach. h ba can also b f ha is h ba paymn can b pospon unil mauiy. YilCuv.com Pag 5 of 8

16 4.. Pah Inpnnc an Pah Dpnnc A scuiy is pah-inpnn if is valu a a givn poin in im pns on h socall sa-of-h-wol a h im an now on how h wol volv o ha sa. Fo xampl h pmium of Euopan opion pns on h pic an h un volailiy of h unlying a a givn poin in im bu is inpnn of h acual pic hisoy ha anspi pio o ha im. Bai opions a pnan on pic hisoy fo mining if a bai has bn hi o no. Fo an ou-opion his yp of pnncy is hoically no iffn han h pah-pnncy inhn in h aly xcis of an Amican opion. A non-lina bai opion xis fo an Amican opion fin a im by h ciical pic a which h invso shoul xcis. In pacis his bai is subjciv o h xn ha h invso ns o spcify volailiy bfo h Amican bai can b inifi. Also h aional invso woul xcis bu is no qui o o so whas h baching of h bai iggs a conacual povision in a knock-ou o knock-in opion. Financial ngins a concn wih y anoh yp of pah pnnc - whh an how backwa cusion can b us fo picing. Backwa cusion fs o h mhos such as Cox-Ross-Rubinsin. his is of ins bcaus backwa cusion is flxibl an fficin whn compa o Mon Calo simulaions. In o o us backwa cusion on quis h scuiy coningn claim bing pic b pah-inpnn in a wak sns. Founaly bai opions a bah-inpnn in his sns 4.3 Rflcing bai A Bownian moion wih flcing bai is also call Bownian moion flc abou som paicula poin. A Bownian moion X flc abou h lin x b is givn as follows. ~ X X fo < b b X fo > b imula ass pic Figu. imula ass pics wih a fix bai lvl of 5. YilCuv.com Pag 6 of 8

17 h wll-known sul abu h flcing bai is h flcion pincipl which sas ha fo vy sampl pah wih X > b h a wo sampl pahs X ~ an X wih h sam pobabiliy of occunc. Bcaus of h symmy wih spc o b of a Bownian moion X saing a b h "pobabiliy" of oing his is h sam as h "pobabiliy" of avlling fom b o h poin b - X. h ason fo his is ha fo vy bah which cosss lvl b an is foun a im a a poin ~ blow b h is a shaow pah X obain fom h flcion abou h lvl b which xcs his lvl a im an hs wo pahs hav h sam pobabiliy. h acual pobabiliy fo h occunc of any paicula pah is zo. Wih h agumn sa abov w can wi h quaion of h flcion pincipl as follows: P < X < b P < X > b P X b b [ ] [ ] [ ] b b > Wh sans fo h im whn h flcing bai b is fis ouch an P is h pobabiliy. h flcion pincipl can b us o fin h fis passag im. h soluion of h nsiy funcions fo h Bownian moion wih a flcing bai can b foun in sval x books in financial mahmaics Unsic isibuion an aobing bai L g san fo h annual coninus ivin yil on h unlying ass. h sochasic pocss which govns h unlying ass pic movmn givn in. bcoms µ g z wh all h paams a h sam as quaion. xcp fo incopoaing h coninus ivin payou. h soluion o h abov DE is givn blow xp[ v w ] [ ] wh san fo cun im an xpiaion im of h opion spcivly v g / an w is sana Gauss-Win pocss no h ha w hav chang h noaion slighly. W know ha X ln[ / ] is h log-un of h unlying ass hn h nsiy funcion of X is nomally isibu wih man v an vaianc. is pf is hn givn by: 5 Fo xampl s: Yu-un wok Mahmaical Mols of Financial Divaivs 998 Also s: P Jams Opion hoy 003 YilCuv.com Pag 7 of 8

18 x v f x xp. π 4.. Blow w povi h sul fom Cox an Mill 965 fo h nsiy funcion of a Bownian pocss wih an aobing bai. An aobing bai is a bai which upon ouching all h paicls vanish. f x v x xp π. / x a xp v av fo x < a 4..a 4.4. Rsic Disibuions Fom h spcificaion of h payoff of a bi opion w know ha in o o pic i w cainly n anoh nsiy funcion coniion on whh h bai is ach uing h lif of h opion. Dfin: M [ ] max{ s s } 4.3a an m [ ] min{ s s } 4.3b wh X sans fo h s of al numbs saing fom an ning a. max an min psn h funcions giving h maximum an h minimum of a s of numbs spcivly. x sans fo ha x blongs o X; [ ] h wo vaiabls givn in 4.3a an 4.3b a h maximum an minimum of all unlying ass pics wihin h lif of h opion. W can xpss hs in ms of log-uns: Y ln M / an y ln M / 4.3c a l san fo h im h unlying ass pic fis achs an up bai U. h following always hol: YilCuv.com Pag 8 of 8

19 a Y P U M P P a < < > 4.4a an a Y P U M P P a 4.4b Equaion 4.4a shows ha h bai is nv hi wihin h lif of h opion sinc h fis im bai is bach is af h xpiaion im of h opion. his is quivaln o h fac ha h maximum valu of h unlying ass pic wihin h lif of h opion is always blow h bai in a pobabilisic sns. Equaion 4.4b is h complmn of 4.4a an implis ha h bai is ouch wihin h lif of h opion sinc h fis im h bai is hi is uing h lif of h opion. h join-cumulaiv isibuion bwn h log-un of h unlying ass an h ansf maximum givn in 4.3c is givn as follows [s aison 985] fo y x : : 0 y / v y x v x y Y x X F yv 4.5 wh. is h cumulaiv funcion of a sana nomal isibuion. h joincumulaiv funcion in 4.5 is quivaln o h following / < v y x v x y Y x X F yv 4.5a Equaion 4.4a an 4.5a ogh imply ha 4.5a is h cumulaiv funcion of h log-un of h unlying ass coniional on h fac ha h bai is nv ouch wihin h lif of h opion. Diffniaing 4.5a wih spc o x yil h nsiy funcion of h log-un of h unlying ass coniional on h fac ha h bai U is nv ouch wihin h lif of h opion.: 4.5b / a x f x f a Y x av < φ o / a x f U x f a Y x v < φ fo x < a 4.5c an 0 < a Y x φ fo x a wh fx is h unsic nsiy funcion of h log-un of h unlying ass givn in 4... h sic nsiy funcion givn in 4.5b o 4.5c is xacly h sam as h soluion o h Bownian moion wih an aobing bai a > 0 givn in 4..a. YilCuv.com Pag 9 of 8

20 h complmn of bing always blow h bai is no always bing abov o a h bai bcaus i is possibl ha h bai is ach an h pic ns up blow. h nsiy funcion ha h bai is ouch can b obain fom h following iniy. φ x Y a φ x Y < a f x 4.5 his quaion can b inp as h summaion of h pobabiliy whn h bai is ouch an h pobabiliy whn h bai is nv ouch wihin h liv of h opion an his is h sam as h unsic nsiy givn in 4..a Disibuion of h fis passag im h fis passag im o a paicula poin on h pah of h unlying ass pic is h fis im ha his paicula poin is fis ach. h join pobabiliy ha x y a > 0 fo an up-bai cab b obain using 4.4a an 4.5 P X a Y a P X a > a a v av / a v 4.6 if h if m v g / 0 h nsiy funcion of h fis passag im fom zo o h ansf bai poin a ln U / > 0 can b obain by iffniaing 4.6 wih spc o h im o mauiy. h a > 0 F X a Y a a a v xp 3 π 4.6a quaion 4.6a is h isibuion of h fis passag im. 5. Picing sana bai opions On of h ols bai opion yps such as own-an-ou call opions w fis ma availabl in h U.. mak fom 967. h cosponing valuaion fomula fo hs opions was ivn by Mon 973. A ca la Bgman 983 vlop a famwok fo picing pah-pnan claims such as bai opions an Cox an Rubinsin 985 us hi own-an-ou fomula o pic fix incom scuiis wih mb chaacs. Rubinsin an Rin 99 also conibu ail suls fo all bai opion yps wih h assumpion ha h unlying ass pic follows lognomal pocss. YilCuv.com Pag 0 of 8

21 h xpc payoffs of in an ou bai opions can b calcula in h sam way as in vanilla opions wih h only xcpion ha h sic nsiy funcion shown abov is us. Using a isk-nual valuaion laionship on can obain bai opion pics by iscouning h xpc payoffs a h isk-f a of un. h bai opion is howv also affc by h laiv magniu of h sik pic an h bai lvl. Fo a own-an-in call wih a sik pic ga han h bai lvl an wihou any ba h valu of h call can b foun by ingaing h payoff of a vanilla call opion wih h sic nsiy funcion fo all possibl unlying ass pic saing fom h sik pic o infiniy. If howv h sik pic is blow o low h bai h payoff of h own-anin call bai opion inclus wo pas: h ingaion of h payoff funcion of a vanilla call opion wih h sic nsiy funcions givn in 5.a fo all possibl unlying ass pics saing fom h bai L o infiniy an h ingaion of h sam payoff funcion wih h nsiy funcion givn in 5.b fo all saing fom h sik pic o h bai L. v / bv / L φ x Y b f x b f x a fo x > b 5.a an φ x Y b f x fo x b 5.b wh b lnl/ an L sans fo a own-bai L <. 5.a is h sic nsiy funcion of h unlying ass log-un un h coniion ha h ownbai is ouch wihin h opions lifim. 5. Down-an-in bai call opion h payoff of a own-an-in bai call opion can b ivi in wo pas; on pa incluing h payoff of h cosponing vanilla opion if h bai is ach any im wihin h lif of h opion an h ba if h bai is nv ach. Wihin h lif of h opion. Ls fis consi h cas wh h sik is ga han h bai lvl >. h valu of own-an-in call opion DIC wihou any ba if h bai is ach is aily obain by iscouning i is xpc payoff givn in quaion 4.a a h isk-f a of un: DIC v / g wh YilCuv.com Pag of 8

22 / ln / / ln v g a an xn vsion Black-chols paams w hav alay sn abov. W can also xn h Black-chols soluion fo h call pic an us his fomula o giv an alnaiv compac fomula fo h own-an-in bai call opion; [ ] [ ] C g ω ω ω ω h valu of own-an-in call bai opion can hn also b win in fom / C DIC v 5. h fomula 5. givs h valu of a own-an-in call opion wihou any ba whn h sik pic is ga han h bai. In h cas whn h sik is low han h bai h whol ingaion angs mus hn b ivi ino sub-angs. Fo xampl h ingaion ang ino an bcaus h cosponing nsiy funcions a iffn ino hs sub-angs. Fo h ang w can obain h valu of h opion in his up poion DUP / DUP g v [ / C v ] 5.3 Wh C is again h xn Black-chols fomula givn abov. inc h ang is quivaln o h iffnc of h wo angs - an - w can obain h valu of h own-an-in call fo h ang [ ] P P DIC B 5.4 wh YilCuv.com Pag of 8

23 P is h Black-chols fomula fo vanilla Euopan pu opion. h valu of h own-an-in opion wihou any ba is hfo h sum of h valus of h opions givn in 5.3 an 5.4. Fom h abov analysis h picing fomula of a own-an-in opion pns on whh > o <. In o o obain a gnal fomula o cov boh siuaions on can us an inicao B > which quals on whn > an zo if ohwis. Givn h abov inicao w can xpss h pic of own-an-in call opion DIC wihou ba: DIC v / C max [ max ] max { P [ ]} B P > 5.6 Wh max is h funcion which givs h lag of h wo numbs an an oh paams a h sam as in 5.3 an 5.4. Whn > h picing fomula givn is 3.6 bcoms h sam as 5. bcaus max an B > 0. W can also chck whn < max B > h picing fomula 3.6 is h sun of h wo picing fomulas givn in 5.3 an 5.4 an h sum psns h valu of h own-an-in call opion whn h is no ba. W now coninu o pic own-an-call opion by using numical xampl an wih h assumpion ha h own bai is ouch. his implis ha h ba is zo. o fin h fin h pic of own-an-in bai call opion consi h following paam; sik pic 98 spo pic 00 bai lvl of 95 ins a 8% h yil of h unlying ass g 3% an volailiy of h unlying ass 0% uiuing w g 0.03 an 0.5 ino 5.6 yils v g / / 0.03 / 95 / max max YilCuv.com Pag 3 of 8

24 ln [ / / ] v inc 98 > 95 h call opion pic B > 0. W can hn fin h ownan-in call pic fom 5.6 as follows: DIC v / 98 C v / g 0.95 x0.03 / x x0.5 [ ]. 73 whn h sik pic 9 max - max B > an all h ms in 5.6 an nonzo. uiuing 00 9 B > sigma g 0.03 an 0.50 ino 5.6 yils: max ln [ / / ] v P P max 0.5 g [ [ ] k [ ].580 [ ] [ ] g YilCuv.com Pag 4 of 8

25 C 95 max x x hus h valu of h own-an-in call opion is DIC x0.03 / x0. [ ] 0.08x0.05 [ ] h pic of h own-an-in bai call opion can also b incopoa in h cas whn h bai is nv ouch an h opion pays som ba. h psn valu of his ba is givn abov. Fo compac soluions fo all maining yps of bai opions can b foun un aug 998. h valu of h ba a h opion mauiy can b obain by ingaing h sic nsiy funcion blow fom which h own bai L o infiniy. v / L φ x Y > b f x f x b fo x > b { PUI } E < an < < < v / Rm [ ] [ ] 5.7 h psn valu of h ba is obain by iscouning 5.7 a h isk f a : RBDI v / Rm [ ] [ ] 5.8 RBDI is h psn valu of h ba fo own-an-in call opion. YilCuv.com Pag 5 of 8

26 h pic of own-an-in call opion PDIC can now b xpss using 5.6 an 5.8: PDIC DIC RBDI Wh RBDI an DIC a givn in quaions 5.6 an 5.8 spcivly. o fin h psn valu of h ba whn h ba is pai.5 a mauiy if h bai is no ouch wihin h liv of h call opion fom h numical xampl abov suiuing Rm sigma g 0.03 an 0.50 Ino quaion 5.8 yils ln / v ln00 / x ln / v ln95 / x RBDI x x0.03 / 0. [ ] [ 0.566] w can now fin h pic of h own-an-in call opion whn h ba is pai.5 a mauiy if h bai is no ouch wihin h liv of h opion h own-an-in call opion pic wih sik pic 9 DIC 9 RBDI An impoan issu of picing bai opions is whh h bai cossing is monio in coninus im. Mos mols assum coninus monioing of h bai. In oh wos in h mols a knock-in o knock-ou occus if h bai is ach a any insanc bfo h mauiy of h conac mainly bcaus his las o analyical soluions; s fo xampl Mon 973 ynn & a 994a994b an kuniomo & Ika 99 fo vaious fomula fo coninuously monio bai opions un h classical Bownian moion famwok; s ou & Wang 00 fo coninuously monio bai opions un a jump iffusion famwok. owv in pacic mos if no all bai opions a in maks a iscly monio. In oh wos hy spcify fix ims fo monioing of h bai ypically aily closings. YilCuv.com Pag 6 of 8

27 Bsis pacical implmnaion issus h a som lgal an financial asons why iscly monio bai opions a pf o coninuously monio bai opions. Fo xampl som iscussions in a's liau "Divaivs Wk" may 9 h 995 voic concn ha whn h monioing is coninuous xanous bai bach may occu in lss liqui maks whil h majo wsn maks a clos an may la o cain abiag oppouniis. Alhough iscly monio bai opions a popula an impoan picing hm is no as asy as ha of hi coninuous counpas fo h asons. h a ssnially no clos soluions xcp using m-imnsional nomal isibuion funcion m is h numb of monioing poins which can haly b compu asily if fo xampl m > 5; s Rin 000. Dic Mon Calo simulaion o sana binomial s may b ifficul an can ak hous o vn ays o pouc accua suls; s Boai Glassman an ou Alhough h Cnal Limi hom asss ha as m h iffnc bwn coninuously an iscly monio bai opions shoul b small i is wll known in h as liau ha numically h iffnc can b supisingly lag vn fo lag m. o giv a fl fo h accuacy of hs mols s abl.. his is pouc wih pmission fom Boi Glassman an ou 997. h numical suls shown in abl. suggs ha vn fo aily monio isc bai opions h can sill b big iffncs bwn h isc pics an h coninuous pics. abl. Up-an-ou Call opion pic suls wih m 50 aily monioing an following paams: p ya 0. an 0. ya which psns oughly 50 aing ays. Coninuous Coc Rlaiv o Bai Bai Bai u YilCuv.com Pag 7 of 8

28 Rfncs Boai M. Glassman P. an ou G. 997 A coninuous cocion fo isc bai opions Mah. Financ 7 pp Boai M. Glassman P. an ou G. 999 Conncing isc an coninuous pah-pnan opions Finan. ochasics 3 pp Espn G. aug. 997 h compl gui o Opion picing McGaw-ill ynn R. C. an a. M. 994b Paial bai opions Jounal of Financial Engining Jams P. 003 Opion hoy John Wily & ons L M.. Joshi. 003 h concps an pacic of mahmaical financ Cambig Univsiy Pss uniomo. an Ika M. 99 Picing opions wih cuv bounais Mah. Financ Mon R. C. 973 hoy of aional opion picing Bll J. Economic Managmn an cinc 4 pp Mon R. C. 974 On h picing of copoa b: h isk sucu of ins as Jounal of Financ 9 pp Rin E Convoluion Mhos fo Pah-Dpnan Opions Ppin UB Wabug Dillon Ra Richkn P. 995 On picing bai opions Jounal of Divaivs 3 pp. 9-8 Rubinsin M. Rin E. 99 Baking own h bai Risk 4 pp Wilmo P. owison Dwynn 993 h Mahmaics of Financial Divaivs Cambig Univsiy Pss YilCuv.com Pag 8 of 8