How To Get A Better Price On A Drug

Transcription

1 Biomial approximaio mhos for opio pricig Yasir hrwai U.U.D.M. Proc Rpor 7: Examsarb i mamaik poäg Halar och xamiaor: Joha ysk Ji 7 Dparm of Mahmaics Uppsala Uivrsiy

2

3 Biomial Approximaio Mhos for Opio Pricig Ackowlgm I h am of Allah h mos bfic h mos mrcifl. h sy was coc a Cr of Mahmaics a Iformaio chology MIC Dparm of Mahmaics Uppsala Uivrsiy Uppsala w. I am plas o r my hmbl grai for all h popl who wr wih m from h commcm of sy ill is rmiaio. I wol lik o pay my rspc o all hos achrs a collois who hlp m o compl his program. Firs of all I wol lik o haks h al-mighy Allah who gav m corag a abiliy o compl his Masr program i Mahmaical Fiac. I wol lik o xprss my haks for h Prof. Joha ysk who accp m as a Masr s a kily sprvis m rig h whol sy prio. I xprss my grai a high rgars for Prof. Lif Abrahamsso Dircor of Iraioal Masr of cic Programm for givig m a opporiy o paricipa i h programm. I giv my sicr haks o Prof. Joha ysk my sprvisor for hlpig a giig m o oly for h hsis work b also for achig m h Fiacial Mahmaics cors which boos my irs i his fil. I wol also lik o giv my sicr haks o Prof. Maci Klimk for achig m h cors Fiacial Drivaivs. I wol lik o haks all h Pakisai ss livig i Uppsala I rally oy yor compay rig my say i Uppsala I hav sp a lo of wor moms rig my say i Uppsala wih yo gys I wol lov o rcall hs moms. I am rally blss a fora o hav fris lik yo i my lif. I wish yo all h sccss i yor livs. A h my highs rgars for my family pariclarly my fahr for his fiacial sppor a coragm. I am i arh of vocablary o xprss my flig owars yo all. ii

4

5 Biomial Approximaio Mhos for Opio Pricig Cos. Irocio.. Irocio. Opio Pricig hory.. Opio... Eropa Opios... Amrica Opios..3. Brma Opios Asia Opios Urlyig Ass ock Opios Forig Exchag Opios Ix Opios Fr Opios Call Opios P Opios Biary Opios 7.. Arbirag Fr Pricig 9... Implmaio of Arbirag Fr Pricig 9.3. P Call Pariy.4. Biary P Call Pariy 3. Opio Pricig Mols 3.. h Black-chols Opio Pricig Mol 3... h Assmpio bhi Black-chols Eaio 3... Logormal Disribio Browia Moio Io s formla Rplicaio Porfolio h Black-chols Eaio Cosa Divi Yil 3.. h Biomial Mho 3... Biomial Ass Pric Procss 3... Mli-prio Biomial Mho Exampl 7 iii

6 Biomial Approximaio Mhos for Opio Pricig Approximaig Coios im Prics h Biomial Paramrs Drivig Black-chols Eaio sig Biomial Mho Cosa Divi Yil h Black-chols formla for Eropa Opios Exampl B Pricig Formla for Biary Ps a Calls B Pricig Formla for Cosa Divi Yil Compariso a Rsls h Biomial Mho for Eropa Opios h Rcrsiv Algorihm Assmpios Compariso a Rsls h Biomial Mho for Amrica Opios Amrica Opios Amrica call a P Compariso a Rsls mmary 6 7. Rfrcs 6 iv

7 Biomial Approximaio Mhos for Opio Pricig Chapr Irocio. Irocio Mor opio pricig chis ar of cosir amog h mos mahmaically complx of all appli aras of fiac. Fiacial aalys has rach a poi whr hy ar abl o calcla wih alarmig accracy h val of a opio. Bcas of is simpliciy a covrgc h biomial mho has arac h mos aio a has b moifi io a mbr of varias which rsl i improv accracy somims a h xps of sp. Compaioal mhos hav always xchag sp a accracy i.. maximizig sp a miimizig rrors. h hsis als wih biomial approximaio mhos for opio pricig o pric Eropa a Amrica opios. W cosir h logormal mol of ass pric yamics a h arbirag fr pricig cocp hrogh hs w ca ily rmi h pric of a opio giv h riskfr yil r h volailiy σ a h spo pric of h ass. I his hsis w will iscss wo ival ways of rmi h arbirag fr val of h opio. i.. Risk-ral xpcaio formla a h Black-chols parial iffrial aio. W will cosir biomial mol o riv h risk-ral xpcaio formla w will also iscss mli-prio biomial mol a approximas coios im prics wih iscr im mols w will riv h Black-chols aio sig h biomial mol giv h risk-ral xpcaio formla frhr w will riv Black-chols pricig formla for Eropa opios a s rsls for iffr xpiris. Fially w will pric Eropa a Amrica opios sig biomial mols. W will pric h opios by sig rcrsiv algorihm compar hir accracy a sabiliy. h mai rfrc i his hsis is [] i h chapr for h mai rfrcs is [6] a [].

8 Biomial Approximaio Mhos for Opio Pricig Chapr Opio Pricig hory I his chapr w will iscss som basic cocps abo opio hory a sy h Pricipal of No-Arbirag.. Opio A opio givs h holr h righ o ra by or sll a spcifi aiy of a rlyig ass a a fix pric also call h srik pric a ay im o or bfor a giv a or xpiraio a. ic i is a righ a o a obligaio h owr of h opio ca choos o o xrcis h opio a allow h opio o xpir. I ay opio corac hr ar wo paris ivolv. A ivsor ha bys a opio ha is a opio holr a a ivsor ha slls a opio ha is a opio wrir. h opio s holr is sai o ak a log posiio whil h opio s wrir is sai o ak a shor posiio... Eropa Opios h Eropa opio ar opios which ar oly xrcisabl a h xpiry a of h opio a ca b val sig Black-chols Opio Pricig formla. hr ar oly fiv ips o h classic Black-chol Mol: spo pric srik pric im il xpiry irs ra a volailiy. As sch Eropa opios ar ypically h simpls opio o val. h ivi or yil of h rlyig ass ca also b a ip o h mol. h rm Eropa is cofi o scribig h xrcis far of h opio i.. xrcisabl oly o h xpiry a a os o scrib h gographic rgio of h rlyig ass. For xampl a Eropa Opio ca b iss o a sock of a compay lis o a Asia xchag... Amrica Opios A Amrica opio is a opio which ca b xrcis a ay im p o a iclig h xpiry a of h opio. his a flxibiliy ovr Eropa opios rsls i Amrica opios havig a val of a las al o ha of a iical Eropa opio alhogh i may cass h vals ar vry similar as h opimal xrcis a is of h xpiry a. h arly xrcis far of hs opios complicas h valaio procss as h saar Black-chols coios im mol cao b s. h mos commo mol

9 Biomial Approximaio Mhos for Opio Pricig for valaig Amrica Opios is h biomial mol. h biomial mol is simpl o implm b is slowr a lss accra ha 'clos-form' mols sch as Black chols...3 Brma Opios Brma opios ar similar i syl o Amrica syl opios i ha hr is a possibiliy of arly xrcis b isa of a sigl xrcis a hr ar prrmi iscr xrcis as. hy ar commoly s i irs ra a FX marks b w gralis hm i his cas for ay yp of opios. h mai ifficly i rmiig siabl valaio for Brma opios coms i h form of h boary problm. Bcas of mlipl xrcis as rmiig h boary coiio i orr o solv h pricig problm ca b ifficl. By rmiig h opimal xrcis sragy a h rspciv boary coiio o ca grally s simlaio mhos o rmi siabl prics for Brmas...4 Asia Opios A Asia opio is bas o h avrag pric of h rlyig ass ovr h lif of h opio a o s a srik pric. Asia opios ar of s as hy mor closly rplica h rirms of firms xpos o pric movms o h rlyig ass. For xampl a airli migh prchas a o yar Asia call opio o fl o hg is fl coss. h airli will b charg h mark ra for fl hrogho h lif of h opio a so a Asia opio bas o h avrag ra rig h prio wol b prfrabl o a opio bas o a sigl srik pric...5 Urlyig Asss Opios ca b ra o a wi rag of commoiis a fiacial asss. h commoiis icl wool cor wha sgar i prolm gol c. a h fiacial asss icl socks crrcis a rasry bos. Exchag ra opios ar crrly acivly ra o socks sock iics forig crrcis a fr coracs...5. ock Opios om xchags raig sock opios icl OMX wish-fiish Fiacial Compay i opras six sock xchags i h Noric a Balic coris FE Loo sock xchag CBOE Chicago boar opio xchag amog ohrs. Opios ar ra o mros socks mor ha 5 iffr socks i fac. h opios ar saariz i sch a way ha a opio corac cosiss of shars. Exampl is IBM Jly 5 call. his is a opio corac ha givs a righ o by IBM s shars a a srik pric of $5 ach i hr mohs im. 3

10 Biomial Approximaio Mhos for Opio Pricig..5. Forig Exchag Opios his corac givs h holr h righ o by or sll a spcific forig crrcy a a fix fr im a a fix pric. h siz of his corac grally ps o h forig crrcy i sio. Forig crrcy opios ar impora ools i hgig risk a hy ca b ra as ihr a Amrica or a Eropa opio Ix Opios his has h sam saariz siz as a sock opio. O corac is o by or sll ims h ix a spcifi srik pric. A Amrica of his yp of opio corac is a call corac o h &P wih a srik pric of 98. If h opio is xrcis h sm of moy ival o h payoff of h opio is giv o h holr of h opio Frs Opio I a fr opio h rlyig ass is a fr corac. h fr corac ormally mars shorly afr h xpiraio of h opio. Wh a p call fr opio is xrcis h holr acirs a shor log posiio i h rlyig fr corac is aiio o h iffrc bw h srik pric a h fr pric. 4

11 Biomial Approximaio Mhos for Opio Pricig..6 Call opio A call opio givs h holr of h opio h righ o by h rlyig ass by a crai pric o a crai a blow is or iscssio for Eropa opios. h payoff fcio for a Eropa call opio ps o h pric of h rlyig ass.g. a sock a xpiry a h srik pric K. h Eropa call opio payoff ca b xprss as: C K Payoff K E K Figr.: Payoff fcio of h call opio If a xpiraio h val of h ass is lss ha h srik pric h opio is o xrcis a xpirs worhlss. O h ohr ha if h val of h ass is grar ha h srik pric h opio is xrcis h byr of h opio bys h ass a h xrcis pric. A h iffrc bw h ass val a h xrcis pric compriss h gross profi of h opio ivsm. If h ass val is lss ha h srik pric yo los wha yo pai for h call. N payoff o call opio rik Pric Pric of rlyig ass Figr.: Ngaiv Payoff fcio of h call opio 5

12 Biomial Approximaio Mhos for Opio Pricig..7 P Opio A p opio givs h holr of h opio h righ o sll h rlyig ass by a crai pric o a crai a blow is or iscssio for Eropa opios. h payoff fcio for a Eropa p opio ps o h pric of h rlyig ass.g. a sock a xpiry a h srik pric K. h Eropa p payoff P ca b xprss as: P K Payoff K E K Figr.3: Payoff fcio of h p opio If h pric of h rlyig ass is grar ha h srik pric h opio will o b xrcis a will xpir worhlss. O h ohr ha if h pric of h rlyig ass is lss ha h srik pric h holr of h p opio will xrcis h opio a sll h sock a srik pric. h iffrc bw h srik pric a h mark val of h ass as gross profi. N payoff o call opio If h ass val is grar ha h srik pric yo los wha yo pai for p. rik Pric Pric of rlyig ass Figr.4: Ngaiv Payoff fcio of h p opio 6

13 Biomial Approximaio Mhos for Opio Pricig..8 Biary Opio Biary opios bhav similarly o saar opios b h payo is bas o whhr h opio is o h moy o by how mch i is i h moy. For his raso hy ar also call cash-or-ohig opios. As wih a saar Eropa syl opio h payoff is bas o h pric of h rlyig ass o h xpiraio a. Ulik wih saar opios h payoff is fix a h wriig of h corac. Payoff K K Figr.5: Payoff fcio of h biary call opio Payoff K K Figr.6: Payoff fcio of h biary p opio 7

14 Biomial Approximaio Mhos for Opio Pricig h biary p opio has a payoff P B if if < K K a h biary call opio has a payoff C B if if K > K. h biary opios ar xampl of opios wih a iscoios payoff. 8

15 Biomial Approximaio Mhos for Opio Pricig. Arbirag-fr Pricig Mol h absc of arbirag mas ha o ivsor i h mark is abl o mak risklss profi by sllig a byig scriis... Implmaio of Arbirag-Fr Pricig mol h basic assmpio i applyig h pricipl of o arbirag is xisc of risk-fr scriy a fal fr ass. I his hsis w will assm ha hr is a risk-fr scriy ha has a cosa coiosly compo yil r. h rars a ivsors ca borrow a l a his ra. ppos w hav a porfolio which is collcio of scriis. A porfolio may coai log a shor posiios of varios asss. A posiio is a im i h porfolio sch as a sock or cash i a moy mark acco. Posiios ar frhr classifi as log posiio a shor posiio which ar fi blow: Dfiiio: A log posiio wh liia cras a posiiv cash flow a is hs a ass o s rasry bo sock or a opio ha w prchas ar xampls of log posiio. Dfiiio: A shor posiio cras a poial gaiv cash flow wh liia so i is a liabiliy borrow sock or a wri opio ar xampls of shor posiio. ppos w hav wo risk-lss porfolios PA a PB wih rspciv rmiisic yils a a b ovr som prio of im. hir crr prics ar A a B rspcivly. W will assm ha A B if a > b. Now cosir a porfolio PC wih a log posiio o porfolio PA a a shor posiio o porfolio PB. h prs val C will b: C A B C 9

16 Biomial Approximaio Mhos for Opio Pricig whil a im h val of porfolio PC will b: P C A B a b P C A B ab [ ] b P C A >. ab As w assm ha A B a P C > i shows ha porfolio PC os o cos ayhig a hc ivolvs o risk. W hav hs a arbirag opporiy. I h abov xampl h xcss ma of h high yilig porfolio PA a xcss spply of h low yilig porfolio PB wol riv h pric of porfolio A p a rag h pric of porfolio B ow. Agai assm ha A B if a < b. Porfolio PB wih a log posiio o porfolio PC a porfolio PA wih a shor posiio o porfolio PC. I or his cas h pric of porfolio PB wol go p a h pric of porfolio PA wol go ow. As h pric of h porfolios chags h ilibrim pric is obai wh rspciv yils a b i mas ha h yils of ay wo risk-lss porfolios ms b ival r a b.

17 Biomial Approximaio Mhos for Opio Pricig.3 P Call Pariy By sig h pricipl of o arbirag w ca riv a impora rlaioship bw Eropa p a call prics wih h sam srik pric cosir h followig porfolios: r Porfolio A: o Eropa call opio pls a amo of cash al o X. Porfolio B: o Eropa p opio pls o shar. Boh ar worh max X a xpiraio of h opios. Hc h porfolios ms hav h sam val a h prs im. hrfor: C X r P whr C is h val of h Eropa call opio a P is h val of h Eropa p opio. Hc calclaig h val of h call opio also givs s h val of h p opio wih h sam srik pric..4 Biary P Call Pariy By sig h pricipl of o arbirag w ca riv h impora rlaioship bw biary p a call prics. Biary p a call hav a simpl rlaio. A porfolio cosisig of o biary p ca o biary call yils rgarlss of h pric of h rlyig ass a xpiry. h porfolio is risk-lss. By h o arbirag argm h porfolio ms grow a h risk fr ra r. h prs val of h combiaio porfolio r is : P B C r B whr is h rmaiig im il xpiry.

18 Biomial Approximaio Mhos for Opio Pricig Chapr 3 h Opio Pricig Mols I his chapr w will iscss wo famos Opio Pricig Mols: h Black-chols mol a h Biomial mol. W also show ha how o ca obai h Black-chols aio by approximaio wih biomial mol. 3. h Black-chols Opio Pricig Mol 3.. Assmpios bhi h Black-chols Eaio. hr ar svral assmpios ivolv i h rivaio of h Black-chols aio.. h sock pric follows a log-ormal raom walk.. h mark is assm o b lii hav pric coiiy b fair a provi all h ivsors wih al accss o availabl iformaio. his implis ha zro rasacio coss ar assm i h Black-chols aalysis. 3. I is assm ha h rlyig scriy is prfcly ivisibl a ha shor sllig wih fll s of procs is possibl. 4. h pricipl of o-arbirag is assm o b saisfi. 5. W assm ha hr xiss a risk-fr scriy which rrs $ a im wh r $ is ivs a im. 6. Borrowig a lig a h risk-fr ra i possibl.

19 Biomial Approximaio Mhos for Opio Pricig 3.. Logormal Disribio Cosir a Eropa opio wih xpiry a im. W will assm h prs im is zro. h val of h opio ps boh o h crr val of h rlyig ass a h im o xpiry. L sppos ha h pric of h ass a som poi of im by. h spo pric is h crr pric of h ass. I h Black-chols opio pricig mol h mai assmpio is ha h rlaiv chag of h ass ovr a prio of im is ormally isrib h ma a variac ar μ σ rspcivly. h ra of rr ovr som im irval is giv blow: h ra of rr ca b xprss as:. μ σ Z 3. whr Z is saar ormal raom variabl wih ma a variac rspcivly. h abov aio 3. lls s ha im passs by a amo of h ass pric chags by μ a also mps p or ow by a raom amoσ Z. W will o h raom compo of aio 3. by a sigl variabl. L s wri: W Z. If w mak im irvals smallr a smallr h i mas h limi h raom procss bcoms a coios raom procss; w call his a sochasic procss a s iffrials o scrib ifii chargs. W ca wri h aio as μ σw 3. whr Lig W is a sochasic variabl. X 3

20 Biomial Approximaio Mhos for Opio Pricig aio 3. bcoms: X μ σ 3.3 W whr h variabl µ is call h rif ra. Usig h fac ha a log X w ca wri as X. 3.4 his mas ha h logarihm of is ormally isrib hc w say ha h isribio of is logormal. 4

21 Biomial Approximaio Mhos for Opio Pricig 3..3 Browia Moio h variabl W apparig i h sochasic iffrial aio 3. fis a spcial ki of a raom walk call Browia moio or Wir Procss. I is a ormally isrib raom variabl havig ma a variac. Eaio 3. lls s ha h logarihm of givs s a way o scrib h chags is a Browia moio wih rif μ i h ass pric as im passs. a Figr 3.: Graphical rprsaio of Browia moio 5

22 Biomial Approximaio Mhos for Opio Pricig 3..4 Io s formla W ow cosir h fcio f pig o h ass pric a im. If h ass pric wr a rmiisic variabl w wol simply xpa f a f i aylor sris: f f... f h Io calcls is sochasic procss ival o Nwoia iffrial calcls. I h limi rms of of highr orr ha as i h oriary iffrial calcls ar cosir small a ca b omi. I cas of logormal raom walk w ca wri aio 3. as bcas h cosir μ σ Z ps o i cas of raom procss. μ σ Z sic Z is saar ormal Z is isrib wih gamma isribio wih ma. hrfor I h limi E [ σ Z σ ] 3 [ ] O σ E Z σ O. 3 σ. hrfor if f is a fcio p o i is also a sochasic procss f sch ha f f σ 3.5 6

23 Biomial Approximaio Mhos for Opio Pricig or wriig o w obai Io s formla for h opio giv h rlyig ass is a sochasic procss wih μ σ w : f f σ w μ σ. 3.6 h sochasic variabl W prs i h formla; his mas ha h opio pric f also movs raomly Rplicaig Porfolio A rplicaig porfolio limias h raomss of h opio a maks h opio ival o a risklss porfolio. Oc w hav sch a porfolio or ls a arbirag opporiy will occr. W hs assm h absc of arbirag o rmi h fair pric of h opio. A risk of a porfolio is h variac of h rr o h ivsm. I or logormal mol his isσ. W fi h volailiy of h porfolio byσ. A risk-lss porfolio hs has σ μ r a o raom compo. Fiig a way o limia h raom compo i h followig Io s aio: f f σ w μ σ. o fi h pric of a opio is hs h ky o fiig h fair pric for h opio. h rsl will b h Black-chols aio which ms hol i h absc of arbirag. hs if w ca limia h raomss by a rplicaig porfolio w ca fi h pric of h opio. 7

24 Biomial Approximaio Mhos for Opio Pricig 3..6 h Black-chols Eaio Assm h Io s formla f f σ w μ σ. W hav a sochasic procss f pig o aohr procss. Cosrc a porfolio cosisig of o opio a a shor posiio G is o h sock. h val of his porfolio is π f G afr o im-sp h porfolio will charg by π f G. 3.7 Apply Eaio 3.5 o Eaio 3.7 f f σ w μ σ. W will g h followig aio f π σ G f π G σ. o limia h raom compo w choos G a π σ f f π σ f π σ. 8

25 Biomial Approximaio Mhos for Opio Pricig I his porfolio hr ar o raom compos i i. o i mas ha his porfolio is rmiisic a hs risklss. By h o-arbirag argm h yil o his porfolio is as h sam as ha of a risklss scriy. If w assm ha h yil of a risklss scriy is r a porfolio val π a ivr i his scriy yils rπ rig im sp. h rplica opio o rmiisic porfolio o h ohr ha yils π. hir yils ms b al so rπ π. P h val of π ic w kow ha i h abov aio f π σ f rπ σ. 3.8 π f G P h vals of π i aio 3.8 π f. f r f σ f r f σ. his is h Black-chols parial iffrial aio which ms hol for all Eropa opios. ic h poi of im was arbirary a his form h aio is sally wri i f σ r rf 3.9 9

26 Biomial Approximaio Mhos for Opio Pricig or ivally f σ r rf. h rif ra µ is o prs i h aio. h pric of h rivaiv scriy is govr by aio 3.9 giv ha rlyig ass follows logormal yamics i h absc of arbirag. I hols for mor gral choics of σ b i his hsis w will cosir cosa volailiis oly a h im of xpiraio f als h payoff fcio Cosa Divi Yil W cosir h cas wh h rlyig ass pays a coios ivi a som fix ra D wiho loss of graliy. L a h spo pric of h ass b. Afr a ifiisimal imsp h holr of h ass gais D i ivis. Howvr h ass pric ms fall by h sam amo or ls hr is a arbirag opporiy: byig h ass a a a sllig i immialy a afr rcivig h ivi wol yil a risk-fr profi of D. hs w ms hav his aio σ W μ D. 3. o limia h ffc of h ivi i h pric of h opio h Black-chols aio chags o h aio giv blow: f r D σ rf. 3.

27 Biomial Approximaio Mhos for Opio Pricig 3. h Biomial Mol h biomial mol prss aohr way o scrib h raom ass pric yamics. L s ak wo possibl ass prics pr im-sp by icrasig h mbr of imsps i h limi w will vally arriv a h corrc pric of h opio a fi a alraiv way o rprs h val of h opio amly h risk-ral xpcaio formla. 3.. Biomial Ass Pric Procss h biomial mol sars o wih a xrmly simpl wo sa mark mol show i h figr.a: Figr 3.: O prio biomial mol If is h spo pric of a risky ass a im afr som im prio i ca oly assm wo isic vals: a whr a ar ral mbrs sch ha >. Morovr w will assm h xisc of a risk-lss ass wih a cosa yil r. r hs w ca say ha a ivsm of ollars a im yils ollars a im. h o arbirag argm is also vali hr w ms rir ha or < r < < r <. 3.

28 Biomial Approximaio Mhos for Opio Pricig If i is r ha i mas ha a risk-lss ivsm ca b br wors or v as wll as a risky ivsm. If his is o r h h risky ass is o risky a all. If r < < o wol vr prfr h risk-fr ass o h risky ass; borrowig φ is of moy a h risk-lss ra r a byig h risky ass wol yil a profi of a las φ r > a im. If r sch a mark posiio log ass shor bo wol yil a posiiv val. If r h mark posiio will b rvrs log bo shor ass yil a posiiv val. ppos ow ha h opio yils f a rspcivly. f if h rlyig ass gos p or ow Cosir a porfolio cosisig of is of h risky ass.g. a sock a ψ is of risk-lss ass.g. a moy mark acco forms a rplicaig porfolio r ψ f 3.3 a r ψ f. 3.3 b his is sysm of wo aio wih wo kows ψ hr is a i solio xis if a oly if f f r f f ψ. ic h opio payoff a is al o ha of his porfolio h val of h porfolio ms b al o ha of h opio. L say h prs val of h opio is p h vals of a Ψ w will g Ψ f f f f r r f f f f

29 Biomial Approximaio Mhos for Opio Pricig f f r f f. hs f f r f f. W iroc a w variabl r. h val of h opio a ca b xprss as [ f f ]. 3.4 r h o arbirag argm garas ha < <. hs h val of h opio rcs o a crai ki of xpcaio formla [ f ] r E 3.5 whr h xpcaio is ak r h probabiliy masr giv by. his masr has h spcial propry ha if is h val of opio a r [ ] ψ E r [ ] E. his probabiliy masr is call risk-ral probabiliy masr. 3

30 Biomial Approximaio Mhos for Opio Pricig Figr 3.3: -Prio Biomial mol wih cosa ivi yil σ. r. N 8 / 365 a D /N h graph is gra hrogh malab. 4

31 Biomial Approximaio Mhos for Opio Pricig 3.. Mli-Prio Biomial Mol Mli-prio biomial mols appli o h sam oal prio of im N as h mbr of prios icras im sp h isribio of log approachs o ormal isribio. I mli-prio mol h xpiry of h opio is ivi io wo al im-sps a risky ass movs pwar by a facor of a owwar by a facor of. 3 3 Figr 3.4a: wo prio biomial mol Figr 3.4b: hr prio biomial mol his rcombiig biomial r has h ass vals a im. ppos ow h opio payoff fcio is f. L h hr opio possibl vals a b h f f f m f f f. Assmig h o-arbirag a risk-ral w ca apply h followig formla [ f f ] r o ach of h iivial brachs i his r o obai a val for h opio sp by sp. A im w will g ihr of h wo vals of h opio. or [ f f ] r m [ f f ]. r 5

32 Biomial Approximaio Mhos for Opio Pricig 6 Applyig h formla oc agai [ ] r v v. 3.6 Isrig h vals of v v w ca wri his as f f f r or r f. For h N-Prio mol whr N w obai N N N r f J N. 3.7 h payoffs a ach o i h N-Prio mol ca b xprss as fcio of h payoffs i a N prio mol: N N r N f f f. 3.8 Rplacig 3.8 io 3.7 N N N r N N r f N N f N N r f as w kow N N N. h abov aio bcoms: N N N N r f N. Which cofirms ha formla is also vali for N - prio mol. By h pricipl of icio a formla prs i aio 3.7 is r for N-prios os.

33 Biomial Approximaio Mhos for Opio Pricig 3... Exampl Figr 3.5: -prio biomial mol wih cosa ivi yil σ. r. N 8/365 a D /N h graph is gra o malab. Cosir a hr yar Eropa p opio o a o-ivi-payig sock wh h sock pric is 9 EK h srik pric is EK h risk-fr irs ra is 6% pr am a h volailiy is.3. ppos ha w ivi h lif of h opio io for irvals of lgh.75 yars. hs w hav ha a 9 K r.6 3 σ σ σ

34 Biomial Approximaio Mhos for Opio Pricig r.6* * Figr blow shows h biomial r for his xampl. Each o has a formla lik 4 a a mbr lik h formla is h sock pric a ha o a h mbr is h val of h opio a ha o. h val of h opio a im is calcla by h formla max N K. For xampl i h cas of o A i N 4 i figr 3.5 w hav ha h val of h opio is: f 4 max K N 4 f max 9 *.97 *.77 4 max max Figr 3.6: r s o val a sock opio 8

35 Biomial Approximaio Mhos for Opio Pricig For all h ohr os i.. all h os xcp h fial os h val of h opio is calcla sig h followig aio: [ f f ]. i N i r f i i i For xampl i h cas of o B i 3 i figr w hav ha h val of h opio is f [ f 3 3.] [.53 f. f ] r 3 f f.6* * 3.956[.53.7] 4. [.477 * ] f f f.956* f.69. imilarly for h cas of o Ci w hav ha h val of h opio is f [ f. ] [.53 f. f ] r f f.6* * [ ]. [.477 *.438 ] f f f.956*.557 f.488. his is a mrical sima for h opio s crr val of.473 EK. 9

36 Biomial Approximaio Mhos for Opio Pricig 3..3 Approximaig Coios im Prics wih Discr im Mols h N-prio mol riv i h prvios scio is for iscr mol whil h Black-chols mol riv by ochasic iffrial aio was coios wha wol happ if N or? Cosir h val of h rlyig ass h sock afr prios has pass. hr has b a raom mbr say X p-mps a X ow mps h val of ass is h X X. h logormal procss iroc i h prvios scio ivolvs a sock X pric whr X is giv by is iffrial aio i is a sochasic procss wih rif μ a variacσ. L h val of will b followig μ σ μ σ. I is clar ha < if w assm ha arbirag rirm ass pric afr prios will b: < r < saisfis h h μ σ X μ σ N X μ X σ. If h limi h μ X σ X? if i os h biomial mol ca b s for approximaig h logormal procss. By sig h vals of w ca wri as r r μ σ μ σ μ σ. 3

37 Biomial Approximaio Mhos for Opio Pricig 3 Expaig h omiaor a mraor i aylor ris 3 3 O O r σ σ μ σ σ μ σ σ μ 3 3 O O r σ σ σ O r σ σ μ. h raom compo of is X σ X σ X σ. L X Y ow w will fi h ma a variac of Y. X ca b scrib as sm of ip Brolli raom variabls i.. raom variabls oig h mbr of has i flips of a coi hr h coi is risk ral coi wih probabiliy of has al o. h ma of X is a h variac of X is -.

38 Biomial Approximaio Mhos for Opio Pricig h ma of Y is h [ ] E Y [ X ] E. P h val of i h abov aio a w will g h ma of Y : h variac of Y is r μ σ E σ var [ Y ] O X 4 var X 4 4. [ Y ] var P h val of i h abov aio a w will g h variac of Y var [ ] O Y. I h limi N h Y σ s o a sochasic procss Y isrib ormally wih ma r μ σ a h variac σ. W ca wri his as Y r μ σ σw. 3

39 Biomial Approximaio Mhos for Opio Pricig whr W is a Browia moio w hav show ha h iscr raom procss μ X σ covr o coios sochasic procss μ Y. L X μ Y h w obai X r σ σw. 3.9 W ca hs approxima logormal ass pric procss wih biomial mol by sig r σ σ r σ σ L s ow fi X apply Io calcls o abov aio a 3. b r. 3. c X X X r σ σ σ r σ σ σ r σ. W W W his implis ha i h risk ral worl h sochasic iffrial aio is X μ σ has rif μ r. W I ohr wors or E [ X ] E[ μ σ ] E W r X [ ] E[ ] E r [ ]. E 3. h pric of h sock is xpc o grow a h risk-ral ra r. 33

40 Biomial Approximaio Mhos for Opio Pricig Figr 3.7: N-Prio Biomial r Mol Figr 3.7 simas h biomial mol for a s of N. chck accracy for compaio also ivsigas how log i aks o vala r. 34

41 Biomial Approximaio Mhos for Opio Pricig 3..4 h Biomial Paramrs I his scio w will show h paramrs of biomial mol for coios im prics sig h logormal pric procss cosir h biomial paramrs which ar fi i aio 3.3 a r σ σ r σ σ r which ar o h oly possibl ways o cosrc a risk ral biomial r. h logormal mol is flly spcifi by h ma a variac of h raom variabl or whr X is X r σ σw. X h variac of is: X X var [ ] [ ] X [ ] X X E E E[ ] [ ] X X E X whr has ma Io calcls: r X as show i aio 3.4. o fi h ma of w will apply X X r σ σ r σ σ. W W 35

42 Biomial Approximaio Mhos for Opio Pricig o fi I mas X X has ma X X X X X rσ r σ σ σ r σ 4σ σ r σ σ σ W W W r σ σ W r σ σ. W a h variac of X is giv blow: var X [ ] [ ] r σ X E as w kow X hs w ca wri ma a variac as E ar r r σ 3. a E. 3. b W will apply ma a variac o o-prio biomial mol wih.. I his mol h ma a variac ar giv blow: a limi E 3.3 a ar Comparig aio 3. a a 3.3 a E E E. 3.3 b 36

43 Biomial Approximaio Mhos for Opio Pricig r hr r. Agai comparig aio 3. b a 3.3 b ar E ar r σ rσ E hr agai r σ r σ. r 3.4 a r σ 3.4 b Now w hav wo aios wih hr kows variabls o variabl ca b chos for xampl. P h val of i aio 3.4 a r r r r. 37

44 Biomial Approximaio Mhos for Opio Pricig 38 Agai p h val of i aio 3.4 b r σ r σ r σ r σ. Now agai w hav wo aios r 3.5 a r σ. 3.5 b From hs wo aios w ca g h val of which ar giv blow: r σ r σ. his provs ha h biomial mol approximas h logormal pric procss. o cocl h cisio abo h ivalc of wo mols.

45 Biomial Approximaio Mhos for Opio Pricig Drivig h Black-chols Eaio sig h Biomial Mol I his scio w will show ha giv h risk ral biomial procss w ca riv h Black-chols aio from h risk-ral xpcaio formla giv blow: [ ] f E r. Cosir h sigl prio biomial mol. L h crr spo pric b. h risk ral xpcaio is giv blow: [ ] E. 3.6 Expa i aylor sris: 3 O. Expa i aylor ris: 3 O. P h vals of a i aio 3.6 [ ] 3 O E 3 O

46 Biomial Approximaio Mhos for Opio Pricig 4 [ ] 3 O. W will s h aliy r a by risk ral argm his amo b al o O r r δ [ ] 3 O r 3 O r σ. By h risk-ral argm his ms b al o O r r. Rarragig 3 O r r σ wh h limi w will fially g h Black-chols parial iffrial aio r r f σ.

47 Biomial Approximaio Mhos for Opio Pricig 3..6 Cosa Divi Yil ppos w hav a ass wih cosa ivi yil D byig φ is of h ass a D D h crr spo pric worh φ if h pric gos p a φ if h pric gos ow. D φ φ D φ Figr 3.8: Biomial mol of a ass wih cosa ivi yil byigφ is of h ass a. D If w mliply h r by h w will g h biomial r wih a ach D o. B h iiial val chags o. his is illsra blow: φ φ D φ Figr 3.9: Biomial mol wih cosa ivi yil byigφ is isa maks h os val as i h oivi mol. D 4

48 Biomial Approximaio Mhos for Opio Pricig 4 P h vals of D i aio 3.3a b w will g r D D r D r. his mas ha i h rivaio of a w will s r D isa of r i h. W will g followig vals of a D r σ σ 3.7 a D r σ σ 3.7 b D r. 3.7 c Figr 3.: If w isco h prs val of h ass by is ivi yil D w ca apply h o ivi mo wih risk-fr ra r - D isa of r. D

49 Biomial Approximaio Mhos for Opio Pricig 3..7 h Black-chols Pricig Formla for Eropa Opios h Eropa opio ar val sig Black-chols Opio Pricig formla. h Black- chols aio solv Eropa call a p opios a Eropa biary opio. h mos sraighforwar way o riv h formla is by risk-ral xpcaio formla which is giv blow: [ ] X r E. h formla will b s a rfrc wh sig h compaioal mhos. ak h Eropa p opio P. I payoff a xpiry is K so by h val of h opio a ca b xprss as: P [ K ] r E r X E K [ ]. Do h xpcaio val rsric o his omai by E so by h liariy of h xpcaio opraor w ca wri i as: X [ ] E [ ] r P KE. 3.8 For saar ormal raom variabl Z h xpcaio val [ f Z] giv by: If E z f zxp [ f Z ] π z E for Z < z is z. X is h sochasic procss fi i aio 3.9 also giv blow: for o has X X r σ σ W r σ σ W. h h saar ormal raom variabl ca b fi as: X r σ Z 3.9 σ 43

50 Biomial Approximaio Mhos for Opio Pricig h X is rsric o < X < log K / By h variabl chag 3.9 w obai K log x r E [] σ xp x. σ π σ z E [] xp z z φ z 3.3 π whr φ. is h cmlaiv ormal isribio fcio fi by h impropr igral o h lf ha si a z is obai by h variabl chag i 3.9: z K log r σ. 3.3 σ imilarly w will comp E K log x r σ σ X [ ] xp x x σ π W will obai h followig by chagig h variabl a complig h sar whr E r z X [ r ] xp z z φ z π z z z σ K log r σ. 3.3 σ. 44

51 Biomial Approximaio Mhos for Opio Pricig Applyig h xpcaio formlas o aio 3.8 yils h Black-chols p opio formla: r P K φ z φ z h Black-chols call opio formla is riv asily by sig h p-call pariy formla which is giv blow: r C K P r C P K. Pig h vals of p opio i abov aio w will g: r C K φ z φ z K r r C φ z K K φ C z r φ z K φ z r which is ival o Exampl C z r φ z K φ Cosir h siaio whr h sock pric six mohs from h xpiraio of a opio 4 EK h xrcis pric of h opio is 4 EK h risk fr irs ra is % pr am a h volailiy is. pr am. Wha is h pric of Eropa call a p opio? W hav ha 4 K r..5 σ. z K log r σ. σ P h vals w will g 4 log...5 z

52 Biomial Approximaio Mhos for Opio Pricig a z K log r σ z σ 4 log...5 z 4..5 z K r 4.* K r r K h Black-chols p opio formla is: h Black-chols call opio formla is: P C z r K φ z φ 38.49φ.678 4φ z r φ z K φ 4φ φ Black-chols Pricig Formla for Biary ps a calls h biary p opio val is simply [ ] r PB E. By aio 3.3 w ca wri abov aio as P B z r φ By h biary p call pariy formla w obai h biary call opio formla which is giv blow: C B z r φ

53 Biomial Approximaio Mhos for Opio Pricig Black-chols Pricig Formla for Cosa ivi yil If h rlyig ass yils a cosa ivi yil D. h formlas w riv abov simply moifi by rplacig h risk-fr ra r wih r D. alraivly w ca rplac D h spo pric wih. his also affcs h variabls. 3.3 Compariso a Rsls Figr 3.: h Black-chols solio for Eropa p opio. Figr 3. shows h Black-chols solio for Eropa p opios sig Black- chols pricig formla sa i aio 3.33 sig hs paramrs r.5 σ.3 h hick coios crv has xpiry. a h hi coios crv has xpiry 7. r h payoff fcio is show as a o li h crv ochs a K. 47

54 Biomial Approximaio Mhos for Opio Pricig Figr 3.: h Black-chols solio for Eropa p opio wih cosa ivi yil D. Figr 3. shows h Black-chols solio for Eropa p opios wih cosa ivi yil sig Black-chols pricig formla sa i aio 3.33 sig hs paramrs r.5 σ.3 D. h hick coios crv has xpiry. a h hi coios crv has xpiry 7. h payoff fcio is show as a o li h r crv ochs a K. 48

55 Biomial Approximaio Mhos for Opio Pricig Figr 3.3: h Black-chols solio for Eropa call opio. Figr 3.3 shows h Black-chols solio for Eropa call opios sig Black- chols pricig formla sa i aio 3.34 sig hs paramrs r.5 σ.3 h hick coios crv has xpiry. a h hi coios crv has xpiry 7. h payoff fcio is show as a o li as h val of h call approachs r o K i im h val K r approachs o zro so h val of h call obviosly approachs ha of h ass islf. 49

56 Biomial Approximaio Mhos for Opio Pricig Figr 3.4: h Black-chols solio for Eropa call opio wih cosa ivi yil D. Figr 3.4 shows h Black-chols solio for Eropa call opios wih cosa ivi yil sig Black-chols pricig formla sa i aio 3.34 sig hs paramrs r.5 σ.3 D. h hick coios crv has xpiry. a h hi coios crv has xpiry 7. h payoff fcio is show as a o li D r as h val of h call approachs o K i im h val K r D a boh rms approachs o zro val of h call graally loss is val. 5

57 Biomial Approximaio Mhos for Opio Pricig Figr 3.5: h Black-chols solio for Eropa biary p opio. Figr 3.5 shows h Black-chols solio for Eropa biary p opios sig Black-chols pricig formla sa i aio 3.35 sig hs paramrs r.5 σ.3 h hick coios crv has xpiry. a h hi coios crv has xpiry 7. h payoff fcio is show as a o li as h crv ochs h a r. Figr 3.6: h Black-chols solio for Eropa biary call opio. Figr 3.6 shows h Black-chols solio for Eropa biary call opios sig Black-chols pricig formla sa i aio 3.35 sig hs paramrs r.5 σ.3 h hick coios crv has xpiry. a h hi coios crv has xpiry r 7. h payoff fcio is show as a o li as > K h crv is ar. 5

58 Biomial Approximaio Mhos for Opio Pricig Chapr 4 h Biomial Mol for Eropa Opio A Eropa opio givs is holr h righ b o h obligaio o sll or by a prscrib ass for prscrib pric a som prscrib im i h fr. h Black- chols parial iffrial aio r som assmpios ca b s o rmi h val of h opio. h biomial mho is a simpl mrical chi for approximaig h pric of h Eropa opio. 4. Rcrsiv Algorihm W ca approxima h val of a Eropa opio by sig h rcrsiv algorihm. ppos ha h opio is ak o a im a ha h holr may xrcis h opio. i.. sll h ass a a im call h xpiry im. 4.. Assmpios. A ky assmpio is ha bw sccssiv im lvls h ass pric ihr movs p by a facor > or movs ow by a facor <. a pwar movm occrs wih probabiliy p a a owwar movm occrs wih probabiliy p.. h iiial ass pric is kow. Hc a im ar a. h possibl ass prics I gral a im blow: i : i hr ar i possibl ass prics which ar giv i i i hc a h xpiry im i for M hr ar M possibl ass prics. h vals i a M form a rcombiig biary r. Which is giv blow: 5

59 Biomial Approximaio Mhos for Opio Pricig Figr 4.: Rcombiig biary r for ass prics L K b h xrcis pric of h opio. his mas ha holr of h p opio may xrcis h opio by sllig h ass pric a pric K a im. Clarly h holr of h opio will xrcis h opio whvr hr is a profi o b ma ha is whvr h acal pric of h ass rs o o b lss ha K. Hc if h ass has pric a im M h h val of h opio will b max K. L i o h val of h opio a im i M M max K. W will calcla i h opio val a im a for h lar im lvl. h followig rcrsiv formla w will s o calcla h vals: i r i i i M whr h cosa r is risk-fr irs ra. h vals of a ar giv blow: M M 53

60 Biomial Approximaio Mhos for Opio Pricig r r σ Whr A A A / r. 4. Compariso a Rsls a cosa σ rprss h volailiy of h ass. Figr 4.: Eropa p opio calcla by 64-Prio Biomial mho. Figr 4. shows Eropa p opio calcla by h Biomial mho sig hs paramrs K.. σ.3 a r.5 whr N 64 h biomial mho is appli o svral spo prics. Each o shows a spo pric whil h coios hi crv ovr os is Black-chols solio of Eropa p opio. 54

61 Biomial Approximaio Mhos for Opio Pricig Figr 4.3: Eropa call opio calcla by 64-Prio Biomial mho. Figr 4.3 shows Eropa call opio calcla by h Biomial mho sig hs paramrs K.. σ.3 a r.5 whr N 64 h biomial mho is appli o svral spo prics. Each o shows a spo pric whil h coios hi crv ovr os is Black-chols solio of Eropa call opio. 55

62 Biomial Approximaio Mhos for Opio Pricig Figr 4.4: Rlaiv rror ε / i Eropa call a p opio sig 64-Prio Biomial mho. Figr 4.4 lls s abo h rlaiv rror occrr whil pricig Eropa p opio a w hav almos h similar rsls for Eropa call opio. W fi rror as h iffrc bw h Biomial approximaio a h val comp by Black chols aio. 56

63 Biomial Approximaio Mhos for Opio Pricig Chapr 5 h Biomial Mol for Amrica Opio 5. Amrica Opios Mos of h opios ra a xchags ar of Amrica yp; hrfor a accra a ffici valaio of Amrica opios is impora. Bcas a Amrica opio ca b xrcis a ay im is val ca vr b lss ha h payoff. Ohrwis i wol b xrcis immialy P If h siaio < P occrr h opio wol b xrcis immialy a is payoff is val wol b P. Frhrmor bcas h arly xrcis is a aiioal righ o h xrcis a xpiry a Amrica opio is a las as valabl as a Eropa opio. h rcrsiv formla ca b s almos chag o val Amrica opios. Oly o aio has o b moifi i h followig way: whr f is h payoff of h opio. 5.. Amrica Call a P r max [ f ] A Amrica call opio s payoff is τ K if i is xrcis a imτ similarly a Amrica p opio s payoff is K τ if i is xrcis a imτ. h val of h Amrica call opio a im ca b wri as: rτ [ ] C max E * K τ 5. whr h max is ovr all soppig ims τ. For xac sificaio of aio 5. as h appropria pricig formla s h rfrc [3] a [4]. 57

64 Biomial Approximaio Mhos for Opio Pricig 5. Compariso a Rsls Figr 5.: Amrica p opio calcla by 64-Prio Biomial mho h figr 5. shows Amrica p opio calcla by h biomial mho sig hs paramrs K.. σ.3 a r.5 whr N 64 h val of h corrspoig Eropa p opio is show as a smooh crv parly ovrlapp by h os h payoff fcio is show as o crv. 58

65 Biomial Approximaio Mhos for Opio Pricig Figr 5.: Amrica call opio calcla by 64-Prio Biomial mho. h figr 5. shows Amrica call opio calcla by h biomial mho sig hs paramrs K.. σ.3 a r.5 whr N 64 h val of h corrspoig Eropa call opio is as sam as h Amrica call opio. h val of h corrspoig Eropa call opio is show as a smooh crv. 59

66 Biomial Approximaio Mhos for Opio Pricig Chapr 6 mmary I his hsis w sy biomial approximaio mhos for Eropa as wll as Amrica opios. W sy opios o socks wih as wll as wiho ivis. W also icl a chapr o how o riv Black-chols aio from a biomial mol. Or sy shows how vrsail h biomial mho is boh from a horical a a pracical poi of viw. 6

67 Biomial Approximaio Mhos for Opio Pricig Rfrcs [] Krma Joi. Nmrical Mhos for Opio Pricig: Biomial a Fiiiffrc Approximaios Nw York Uivrsiy. [] Börk omas. Arbirag hory i Coios im Oxfor Uivrsiy Prss [3] Hag A. Robr. Mor Ivsm hory 3 r iio Pric Hall. [4] Ogbil Olfmi Olsola. Biomial Opio Pricig Mol Africa Isi of Mahmaical cics. [5] Elk Kor a Ralf Kor. Opio Pricig Maagm Mahmaics for Eropa chools MaMaEch. [6] Higham J. Dsmo. Ni Ways o Implm h Biomial Mho for Opio alaio i Malab ociy for Isrial a Appli Mahmaics. [7] Khyami Aiasai. Nish. Opio Pricig Java Programmig a Mo Carlo imlaio. [8] Chao Jg Ch. Pricig Eropa a Amrica Opios wih Exrapolaio Naioal aiwa Uivrsiy. [9] Wiicks Aricopolos. Nwo a Dck. O h hac Covrgc of aar Laic mhos for opio pricig Uivrsiy of Machsor. [] ia. Yisog. A Flxibl Biomial Opio Pricig Mol York Uivrsiy. [] J Ngi. A Approxima formla for Pricig Amrica Opios Joral of Drivaivs Wir 999. [] Cox J.C a.a. Ross. Opio Pricig a implifi Approach Joral of Fiacial Ecoomics 979. [3] Dai Mi a Y.K. Kwok. Kock-I Amrica Opios h oral of Fr Marks 4. [4] karamaa Aaa. Amrica pra Opio Pricig Uivrsiy of Raig 5. [5] Aiahlia F. a Carr P Amrica Opios: A Compariso of Nmrical Mhos. 6

68 Biomial Approximaio Mhos for Opio Pricig [6] Barcci Emilio a Lai Loaro Compaioal Mhos i Fiac: Opio Pricig. [7] hompso arah imlaio of Opio Pricig. [8] Biga imo a Wir Zvi h Biomial Opio Pricig Mol. [9] Fischr Black a Myro chols. h Pricig of Opios a Corpora Liabiliis Joral of Poliical Ecoomy 973. [] Pal Wilmo am Howiso a Jff Dwy. h Mahmaics of Fiacial Drivaivs Cambrig Uivrsiy Prss 995. [] Joh C. Hll. Opios Frs a Ohr Drivaivs Pric-Hall 997. [] Pal Wilmo. Drivaivs: h hory a Pracic of Fiacial Egirig Joh Wily a os L 998. [3] Bsossa A. O h hory of Opio Pricig Appli Mahmaics 984. [4] Karazas I. O h Pricig of Amrica Opio Appli Mahmaics 998. [5] hp:// [6] hp:// 6