UNIVERSITÉ PARIS I PANTHÉON-SORBONNE MASTER MMMEF

Transcription

1 UNIVRIÉ PARI I PANHÉON-ORBONN MAR MMM Pacou inanc Ramzi MAALOU PARICULARII of h COMMODII MARK Réumé C appo d ag pén un vu global du maché d maiè pmiè L impoanc d c maché n a pa cé d augmn c dniè anné n ff, l anné 8 éai un anné è inéan, duan laqull l maché a éé émoin d un nvolé d pix du péol, d l o, du cuiv, d podui agicol La pmiè pai du documn inodui l maché m l accn u la coub fowad nui, ll concn u la «convninc yild», un noion paiculiè au mond d commo La duxièm pai élabo l uj d opion volailié «aly xpiy», qui aui un concp pécifiqu au mond d commo La oiièm pai déci un méhod uilié pou xai la coélaion implici n dux ou-jacn à pai d un opion bak La dniè pai oulign l impoanc d opion binai dan l maché d commo l impac qu a l mil d volailié u l pix d c opion Rappo d ag péné l LUNDI 5 PMBR a 6H45 Juy Guillaum OUCHR BICH Philipp Naixi Univié Pai Commodiy Divaiv Liu du ag Naixi 47, uai d'auliz 753 Pai anc

2 Paiculaii of h Commodii Mak abl of Conn uu conac in commodiy mak 3 Diffn commodiy a cla 3 Populaiy of fuu conac in commodiy mak 3 3 Convninc yild 4 4 Compaing fuu and fowad 5 5 hap of h fowad cuv 6 5 Conago 7 5 Backwadaion 8 53 Impac of h hap of fowad cuv on opion pic 9 aly xpiy Opion Implid Volailii aly xpiy opion: poblm dcipion 3 Mhodology 4 4 Numical xampl 5 3 Opion on Bak 7 3 Picing opion on bak 7 3 Momn maching 8 33 Dmining implid colaion 4 Digial opion 4 Populaiy of digial opion 4 Volailiy impac on h pic of a digial opion 3 43 mil impac on h pic of a digial opion 5 5 Rfnc 7

3 Paiculaii of h Commodii Mak uu conac in commodiy mak Diffn commodiy a cla h diffn commodiy a cla a h pciou mal gold, ilv, plainum and palladium, h ba mal copp, nickl, and aluminum, h ngy commodii cud oil, naual ga, h agiculual poduc, alo calld of commodii, wha, oyban, coff, h CO miion pmi and cdi, and la bu no la, lciciy Populaiy of fuu conac in commodiy mak h main play in commodiy mak a: - poduc and conum looking a hdging h commodiy pic ik - abiag looking fo a ikl and pofiabl ading agy in commodiy mak 3- invo and pculao looking fo an xpou o commodiy pic mov In cona o h quiy mak, h ypical undlying fo commodiy divaiv poduc i no h po o phyical commodiy, bu ah h fuu conac on h did commodiy o xampl, in quiy mak, a ypical call opion on a Micoof ha ha h ha ilf a undlying In commodiy mak on h oh hand, a ypical call opion on Bn ha h Bn fuu conac a undlying hank o h abiag h nd cla of play in commodiy mak, h pic of h fuu conac convg o h po pic a h fuu mauiy hu, h payoff of a uopan divaiv poduc on a po commodiy i quivaln o a imila poduc on h commodiy fuu conac Mahmaically, W dno : f :h pic a of h fuu conac mauing a :h po pic a of h commodiy Auming No Abiag Oppounii f NOA : o, compaing a uopan po call opion o a uopan fuu call opion wih boh opion having a mauiy d K f K K wh i h ik nual pobabiliy, w hav : d f K 3

4 Paiculaii of h Commodii Mak o, hoically, boh call opion a woh h am hi can b gnalizd o all yp of uopan payoff hu, givn h pmium a idnical almo idnical in pacic, popl nd o choo h mo liquid a a undlying Conqunly, in quiy mak, ock a pfd In commodiy mak howv, fuu a pfd fo h following aon: - hy avoid all ik aociad wih po ading, namly anpoaion, oag, and counpay ik - hi pic a adily availabl on xchang, unlik po pic which can only b known by conacing a numb of dal 3- fuu and fuu opion a gnally add on h am xchang, faciliaing hdging and pculaion and ducing abiag 3 Convninc yild In quiy mak, h abnc of abiag dfin a ic laion bwn fowad and po pic No ha I hav pupofully ud h m fowad inad of fuu bcau I do no wan o conid daily magin call If w dno : B, :h pic a of h zo coupon bond mauing a hn hi i no u fo commodiy fowad conac du o h following aon: - h difficuly of ho lling om po commodii - h illiquidiy of om po commodii 3- h anpoaion co 4- h oag co 5- h bnfi obaind by h ownhip of h phyical commodiy Wha i u howv i ha : :h pic a of h fowad conac mauing a NAO giv : B, c B, wh c i h oag co of of h po pic h commodiy, xpd a a popoion h oh inqualiy do no hold bcau of h aon mniond abov, namly poin and 5 hfo, o g h qualiy, w inoduc a nw paam b, which pn h bnfi fom holding h phyical commodiy ah han i B, cb y B, coponding fuu o fowad conac 4

5 Paiculaii of h Commodii Mak 5 h convninc yild y, combind boh h oag co and h phyical bnfi of holding h commodiy No ha fo impliciy, h convninc yild i akn o b conan A mo gnal ca will follow 4 Compaing fuu and fowad In h financ wold, fuu and fowad a ud inchangably hoically howv, hy a no quivaln o compa h wo, w fi look a h po diffnial quaion, af w inoducd h convninc yild paam: { { w hav : pobabiliy, h fowad nual maingal und, a inc h fowad pic i On h oh hand, maingal an xponnial ha of inc h nd xpxaion i : w g h bownian moion filaion, Auming h ochaic in a i indpndn of { { w hav : pobabiliy, h ik nual maingal und, inc h fuu pic i a yild fo impliciy conan convninc auming a d d y dw d d y dw d d y dw d d y f f f dw d y

6 Paiculaii of h Commodii Mak 6,,,, { w hav : pobabiliy, chang of By a B B B B y dw d y dw d y d, o o fowad duc o compaing compaing fuu o, B d a quivaln mau and h wo pobabiliy bcau und dminiic in a, no a upiing ul, hi i Conqunly,, w hav : In h ca wh h in a a dminiic, W conclud ha, w hav : x, convx funcion of /x i a Uing Jnn' inqualiy and h fac ha fx d d d d f B f B inc in h commodii mak, h in a a no h main ouc of uncainy, hy a conidd o b dminiic hu, fuu and fowad a ud inchangably hi i wha I will b auming fo h of h documn 5 hap of h fowad cuv W can ha in cona o quiy mak wh h ni fowad cuv can b dmind fom h po pic, a commodiy fowad cuv i dicly ad fom h mak Diffn fuu conac fo h am commodiy a conidd a diinc hough colad undlying a

7 Paiculaii of h Commodii Mak Conqunly, in pacic, w wok backwad fom h fowad cuv o dduc h implid convninc yild om fowad cuv may b complx nough quiing a non-conan convninc yild o xplain hi hap h gaph how h pic p bal on h y-axi and h mauiy of h fuu conac on h x-axi o, a dminiic funcion of im will do h ick h gnalizd fomula bcom: y B, d Onc h convninc yild i dducd fom h availabl mak quo, i can b ud o dmin h pic of fuu ha a no quod on h xchang h mo common hap of fowad cuv in h commodii mak a h nx ubcion 5 Conago If w aum fo impliciy ha h in a and convninc yild a conan, Conago i h ca wh y > Uually, i i h ca wh h cun wold invnoi fo h commodiy in quion a laxd, and h common fling in h mak i h fa of a poibl upply cunch in h fuu, imulaing h buying of long dad fuu W hfo h long dad nd of h cuv high han h ho dad on 7

8 Paiculaii of h Commodii Mak Blow i an xampl of conago: h gaph how h pic p bal on h y-axi and h mauiy of h fuu conac on h x-axi 5 Backwadaion h fowad cuv i in backwadaion whn -y < Uually, hi i h ca wh h cun wold invnoi a igh, and conum pf o buying now wha hy will b uing in h na fuu, faing ha invnoi will kp on dcaing W hfo h ho dad nd of h cuv high han h long dad on Blow i an xampl of backwadaion: h gaph how h pic p bal on h y-axi and h mauiy of h fuu conac on h x-axi 8

9 Paiculaii of h Commodii Mak I i woh noing ha alhough i occu vy aly, a fowad cuv may chang fom backwadaion o conago o vic va hi wa h ca fo h Bn and WI cud oil fuu duing May and Jun 8 Indd, conum fad ha oil fild and v will oon b dpld bcau of h maiv dmand of mging couni lik China and India hi puhd h fowad cuv fom conago o backwadaion 53 Impac of h hap of fowad cuv on opion pic pculao, conum and poduc uually buy a h mony opion o hm, h impoan fnc i h po commodiy pic Howv, hy uually buy opion on h fuu conac wih mauiy clo o ha of h opion h a calld andad opion Mo on andad opion la B& fomula fo a call opion on a fuu giv u: C B, { wih ln d f f K N d bak K N d d d and bing h fuu volailiy Moov, bcau of h abnc of abiag, h po pic i gnally clo o h fuu pic wih h ho mauiy, calld h fi naby hi xplain h ul hown in h abl blow hap of fowad Cuv Conango Backwadaion abl - uu Compad o ik AM man fi Naby K f f o, h opion i acually IM AM man fi Naby K f f o, h opion i acually OM Call pmium xpniv Chap Obviouly, h oppoi i u fo a pu opion A a concluion, backwadaion i uiabl whn invo hav a bullih viw, and conango i uiabl whn hy hav a baih viw, inc in h mniond ca, h pmium hy pay i chap 9

10 Paiculaii of h Commodii Mak aly xpiy Opion Implid Volailii inc h pobabiliy diibuion of undlying a i no lognomal a aumd in h B& wold, a volailiy mil o kw xi in h mak fo a givn opion mauiy hi giv h implid volailiy fo ach ik h implid volailiy i h volailiy ha h mak u o pic opion fo givn ik hy a hu obaind by aking h opion pmium fom h mak and inving h B& fomula numically Blow i an xampl of a volailiy mil: 35% 3% 5% % 5% % 5% % h gaph how h volailiy mil of WI cud oil opion wih ya o mauiy On h x-axi, w hav h ik On h y-axi, w hav h implid volailii If w conid val mauii, w obain wha i calld a volailiy ufac In od o hav conin mil, w hould nomaliz h ik by dividing hm by h fuu pic W obain wha i calld a monyn aio An xampl of a volailiy ufac i hown blow:

11 Paiculaii of h Commodii Mak 5% 4% 3% Implid Vol % % % 45% 6% 75% 9% Monyn 5% % 35% Mauiy h gaph how h volailiy ufac of Bn Cud Oil h mauiy i in day h volailiy maix coponding o h abov ufac i hown blow: Mauiy in day Monyn 45% 6% 75% 9% 5% % 35% % 475% 445% 393% 349% 3335% 348% 7 48% 4563% 4369% 397% 3479% 337% 336% % 45% 44% 383% 343% 368% 376% % 4383% 453% 3735% 3384% 36% 39% % 49% 468% 3667% 3335% 38% 364% 5 44% 4% 3999% 366% 395% 335% 35% % 4% 384% 349% 348% 338% 37% 39 49% 399% 3758% 3447% 37% 396% 369% % 3785% 368% 337% 38% 385% 356% % 375% 3584% 3347% 359% 366% 335% % 387% 3594% 38% 33% 886% 85% % 368% 3453% 344% 95% 756% 689% abl -

12 Paiculaii of h Commodii Mak aly xpiy opion: poblm dcipion A mniond in cion 43, mak paicipan nd o buy o ll andad opion A andad opion i an opion who undlying i h fuu conac which mau lighly af h xpiy of h opion o claify ma, w dno: fuu n h mauiy of h nh fuu conac lid on h xchang opion n h mauiy of h andad opion on h nh fuu conac dcibd abov No ha h andad opion a lid on h am xchang a hi coponding fuu conac, and hy obviouly mau a fw day ali han h fuu conac mauiy opion n < fuu n h aon fo ha i o avoid phyical dlivy of h commodiy if h opion i xcid a mauiy inc fo h nh fuu, andad opion wih val ik a lid on h xchang and a gnally liquid poduc, a volailiy mil can b implid fo ha fuu W call hi mil h andad volailiy mil of fuu n o, fo vy fuu conac diinguihd fom oh conac by i mauiy, a andad volailiy mil can b implid fom h mak h ul i a andad volailiy maix fo h commodiy in quion andad volailiy maix uu uu uu uu 3 uu 4 uu 5 uu 6 uu 7 uu Mauiy fuu fuu fuu fuu fuu fuu fuu 7 Opion mauiy in day Monyn 8% 9% AM % % opion = 6 49% 438% 3784% 347% 363% opion = % 438% 3756% 358% 359% opion 3 = % 46% 379% 3493% 349% opion 4 = % 393% 36% 349% 3335% opion 5 = 7 445% 397% 3597% 3398% 337% opion 6 = 6 49% 383% 3534% 3356% 368% opion 7 = 88 43% 3735% 3483% 334% 36% abl - omim, howv, paicipan may b ind in buying o lling opion on fuu conac wih long dad mauii o b mo pcific, hy a ind in opion

13 Paiculaii of h Commodii Mak xpiing much ali han hi undlying fuu conac, hnc h nam aly xpiy opion A an xampl, h WI cud oil opion wih mauii byond ya a only lid fo h mauiy monh of Jun and Dcmb o, if h cun monh i July 8, and a conum wan o hdg again an inca in pic of oil in pmb 9, h/h will no find h appopia call opion in h xchang H/h mu buy an Ov h Coun aly mauiy opion In od o pic ha opion, w nd o know h aly xpiy implid volailiy o, h poblm a hand i h following: Givn a andad volailiy maix in monyn foma of fuu conac on a cain commodiy, w nd o gna an aay of aly volailiy maic on aly volailiy maix p fuu Now h quion i: Givn h andad volailiy maix hown in h abl abov, w nd o find h volailiy of opion h opion wih mauiy on h fuu n a undlying, fo all k mall han n In k oh wod, h poblm i h on of gnaing, fo ach fuu, h following aly volailiy maix: aly volailiy maix of fuu 5 uu uu Mauiy Opion mauiy in day Monyn 8% 9% AM % % uu 5 fuu 5 opion = 6????? = 35????? opion = 66????? opion 3 = 97????? opion 4 opion 5 = 7 445% 397% 3597% 3398% 337% abl -3 3

14 Paiculaii of h Commodii Mak 3 Mhodology o ackl hi poblm, w ak wo main aumpion: Aumpion : o a givn commodiy, AM inananou volailiy i aionay By aionay, w man ha h inananou volailiy do no dpnd on im bu ah on h dianc o mauiy L If w dno : h inananou implid volailiy of opion wih mauiy a im, [, ] h inananou implid volailiy of opion wih mauiy a im, [, ] hn aumpion giv: { { and { { If w dno : n i, j h avag AM volailiy on h piod [i, j] of mauiy i n h opion who andad hn aumpion giv:,,, { { d d, { d { d In wod, hi man ha n fom oday, h volailiy duing h la monh of h lif of an opion i h am a h volailiy of an opion ha xpi in on monh Obviouly, hi inpaion appli o all duaion oh han on monh Viually, 4

15 Paiculaii of h Commodii Mak am avag AM volailiy fo h pcifid duaion am avag AM volailiy fo h pcifid duaion 3 Aumpion : o a givn fuu conac, h andad volailiy mil i plicad in i aly volailiy mil h aon fo ha i ha ad bliv i i h b ima fo h aly volailiy mil hy aum mak paicipan will kp h am kwd pobabiliy diibuion fo h undlying, indpndn of h opion mauiy hfo, h mhodology o gna an aay of aly volailiy maic i h following: o fuu n, w fi a by compuing h AM aly volailii bad on aumpion hn, w compu aly IM and OM volailii bad on h pviouly compud aly AM volailii and aumpion 4 Numical xampl h following xampl b illua h algoihm uppo ou andad volailiy maix i h on hown in abl - uppo, moov, ha w wan o fill h aly volailiy maix of fuu 5, hown in abl -3 No ha h la ow i xacly h 5 h ow of andad volailii givn in h andad volailiy maix h maining volailii a o b calculad W how in dail h calculaion of h hadd volailii W fi calcula h 66day aly AM volailiy Accoding o h noaion dcibd in h pviou cion, h fi volailiy in quion i: 7, 66 W know ha: ,7,66 66,7 5

16 Paiculaii of h Commodii Mak By aumpion, w hav: 7 66,7 = 6, omula hu bcom: ,7,66,6 Moov, fom h andad volailiy maix, w can dicly ad h following volailii: 7,7 3597% 66, % 35, % hu, by lina inpolaion bwn h coupl 66, w find : 6, 6 376% 66 35, 66 and 35,, 35 o, 7, ,7 7, % 66 andad volailiy givn by h mak 66 7 aly Volailiy Unknown Volailiy dmind by inpolaing andad volailii givn by h mak Af auming AM volailii a aionay igu - Illuaing h aumpion of aionay AM volailii W now calcula h 66day aly 8% volailiy h cond volailiy in quion, dnod by 8% aly 8% aly 8% By aumpion, w hav: andad AM AM aly andad 6

17 Paiculaii of h Commodii Mak h igh hand id of h quaion can dicly b ad fom h andad volailiy maix AM, on h oh hand, i calculad in h pviou p aly W hfo hav: 445 8% 3483 % 4% aly 3597 In h am fahion, h of h aly volailiy maix i compud h ul i hown in h abl blow aly volailiy maix of fuu 5 uu uu Mauiy Opion mauiy in day Monyn 8% 9% AM % % uu 5 fuu 5 opion = 6 484% 385% 3545% 3349% 37% = % 384% 35% 336% 347% opion = 66 4% 3783% 3483% 39% 3% opion 3 = % 385% 3545% 3348% 369% opion 4 opion 5 = 7 445% 397% 3597% 3398% 337% abl -4 3 Opion on Bak 3 Picing opion on bak Opion on a bak of undlying a a vy common in commodiy mak W can imagin a conum who u boh naual ga Na Ga and cud oil WI fo h manufacuing of hi good hi conum would b ind in buying a bak call opion o hdg an inca in pic ah han buying paa call opion, inc i i chap o do o Cainly, hi ducion in co ul fom a wak pocion, inc h bak call opion i only pofiabl whn h avag pic xcd h ik Mahmaically, if w dno,, :h pic a of h fuu conac mauing a :h pic a of h fuu conac mauing a :po pic a of undlying :po pic a of undlying 7

18 Paiculaii of h Commodii Mak No ha if h po pic i no adily availabl o accibl in h mak, i i placd by h fi naby ha i, w plac i by lik cud oil and naual ga i, fi naby in all wha follow hi i h ca fo a lo of commodii, L u dfin h poc :,, Y A uopan call opion on an qually wighd bak of undlying aand ha h following payoff a mauiy : Payoff Y K o an AM call, w K % Uing Jnn' inqualiy and h fac ha h pic of a call opion i a convx funcion of h po pic, w can ha : Call Y Call, Call, h wo mo common way of picing bak opion a: Mon Calo Momn Maching I will no add h fi mhod inc i i py aigh fowad h nx cion will dcib h nd mhod 3 Momn maching W know ha und pcific condiion, h poduc of lognomal ochaic poc i a lognomal ochaic poc hi um howv i no mpiical udi hav hown ha i can b appoximad a uch wihou looing a lo of accuacy whn calculaing h pic of vanilla opion h advanag i h pd of calculaion compad o Mon Calo o, if w aum Y i a lognomal ochaic poc, hn B& can b applid o dmin h valu of a call on an qually wighd bak 8

19 Paiculaii of h Commodii Mak 9 In fac, B& giv: maingal un bcau h fuu pic i a hi i ] [ : L ] and [ h wo unknown in h abov quaion a o, % ] [ ln ] [ {,,, Y Y M Y d d K Y d wih d N K d N Y B C bak bak bak bak bak Now ha w found h fi unknown, l u y o find h cond, bak : o do o, w hav o pa by an inmdia p: d dw d dw d Y M,,,,,,,, { w hav : dminiic, i and auming maingal und, an xponnial Bu inc h fuu pic i { { { 4 ] [ L : d dw d dw d,,,, {

20 Paiculaii of h Commodii Mak { 4 ] [ : w g i conan, w fuh aum fo impliciy ha If {,,,,,,,,,,,,,, d d d d d d W d d Y M Now ha w found M, w hav o find a laion bwn M and bak o any xponnial maingal, and fo Y in paicula, w hav: ln { M M M Y M Y Y bak W W bak bak bak bak bak Now ha w found all h paam w nd, B& fomula can adily b applid 33 Dmining implid colaion omim h poblm facd by ad i invd hi man, hy obv in h mak namly bok, inc bak opion a no add on xchang, pic of poduc calld dipion A dipion poduc bwn WI and Na Ga i h diffnc bwn h pic of h paa call and h pic of a call on a bak In oh wod: Dipion = 5 x CallWI + CallNa Ga Callbak h ad can aily pic h paa call o, givn h dipion pic, h pic of h call bak i dmind Onc h call bak pmium i dmind, h B& fomula can b numically invd o find h implid bak volailiy, bak

21 Paiculaii of h Commodii Mak,, bak bak hn, fom quaion abov, w find M M Af dmining M, w hav vyhing w nd o dmin h implid colaion fom quaion Raaging quaion giv u : ln{ M [ 4,,,, ] In mo ca, h implid colaion un ou o b diffn han h hioical colaion, ju lik implid and hioical volailii a diffn Implid colaion i inpd a h colaion h mak i uing o pic bak opion om quaion and, w can ha M i an incaing funcion of, and bak i an incaing funcion of M W alo know ha h call valu i an incaing funcion of volailiy hi how ha h pic of a call bak i monoonically incaing wih o, if h mak i picing a high implid colaion, i man h i a common nimn ha boh undlying a a going o inca ogh hi ca a high dmand on call bak fo hdging pupo, puhing hi pic high Onc hi implid colaion i dmind, i can ih b ud o pic oh bak opion, o i can b ud o build a colaion maix, which i in un ud o pic bak opion on mo han undlying a 4 Digial opion 4 Populaiy of digial opion A uopan digial o binay opion wih mauiy and ik K on an undlying fuu conac, i h opion who payoff a i K Digial o binay opion a vy common in h commodii mak hy a ud in h picing of popula payoff, lik uopan bai opion knock in and knock ou opion h aon fo ha populaiy i ha h opion ucu digad h highly impobabl vn wih vy lag payoff, ducing h pmium fo h clin An xampl of uch a payoff i hown blow:

22 Paiculaii of h Commodii Mak Payoff Pcnag Monyn W can ha h invo i pocd again h i in h pic of h undlying up o h lvl of 75%, af which a ba of 5% of h nominal i paid inc hi vn i conidd a highly unlikly, giving away hi payoff i no much of a poblm fo h invo, and will duc h ovall co of h ucud poduc On h oh hand, if h pic dop, h clin i no xpod o h downid lo unl h dop i bigg han 5% In h la ca, h invo xpinc a % paicipaion lo, which i cappd a 5% of h nominal h abov payoff i aily ucud uing vanilla and digial call and pu In fac, if w dno N CK PK DCK DPK Nominal amoun invd in h ucu Vanilla call opion wih ik K Vanilla pu opion wih ik K Digial call opion wih ik K Digial pu opion wih ik K Payoff = P5% P75% 5 x N x DP75% + C% C75% 5 x N x DC75%

23 Paiculaii of h Commodii Mak 4 Volailiy impac on h pic of a digial opion Unlik vanilla call opion, h pic of a digial call i no an incaing funcion of volailiy hi can b xplaind inuiivly In fac, wo ca a o b conidd Ca : h digial call i in h mony IM In hi ca, h ininic valu of h call i, bu h im valu i ngaiv, inc wih im, w ik o fall back blow h ik o, whn h opion i IM, w wan h volailiy o b a low a poibl, idally, o main abov h ik ad ay hy a ho volailiy Ca : h digial call i ou of h mony OM In hi ca, h anw i no vy obviou I dpnd on how clo w a o h ik pic If w a vy fa blow, w a long volailiy If w a vy clo blow, w a ho volailiy om calculaion hav o b mad o dmin h xac lvl a which digial call Vga chang ign Bifly paking, h digial call opion bhav diffnly han vanilla call opion whn i com o volailiy h aon i ha h Digial upid ponial i limid L udy h abov ul mahmaically d DC K By changing o h fowad nual mau, w g : DC B, K B, K K wh z ~ N, W K W K ln z ln K d o, DC B, z d B, N d Now, w will udy h fi divaiv of d wih pc o volailiy, which will dicly giv u h bhavio of h pic inc Nx i an incaing funcion of x d ln K 3

24 Paiculaii of h Commodii Mak Ca : h digial call i in h mony IM d ln K o, h pic of h digial call i alway dcaing wih volailiy diplay a -v Vga IM Digial Call Opion Monyn = 5% % 7% 3% 9% 5% 3% 37% Digial Call Pic 43% 49% 55% 6% 67% 73% 79% 85% 9% 97% 3% 9% 5% % 7% 33% 39% 45% Volailii Ca : h digial call i ou of h mony OM ln K d d whn ohwi ln K o, h pic of h digial call i incaing wih volailiy +v Vga up o a cain poin, wh mo volailiy will acually hind h call valu 4

25 Paiculaii of h Commodii Mak OM Digial Call Opion Monyn = 5% 4 Digial Call Pic % 9% 7% 5% 33% 4% 49% 57% 65% 73% 8% 89% 97% 5% 3% % 9% 37% 45% 53% 6% 69% 77% 85% 93% Volailii 43 mil impac on h pic of a digial opion ad hdg digial call opion by lling h plicaing pofolio h mak andad i o appoximaly plica a digial call opion wih ik K by a vanilla call pad, cnd a K wih a vy mall diffnc bwn h ik K and K Payoff K K K po Pic 5

26 Paiculaii of h Commodii Mak In pacic, hi plicaing pofolio aly ha h am valu a h fomula fo h pic of a digial call divd in h pviou cion hi diffnc i claly n whn h volailiy mil i no fla Inuiivly, whn h volailiy kw i upwad loping, h plicaing pofolio coni of buying a lag numb of call wih ik K and lling h am numb of call wih ik K inc CK i picd wih a volailiy V a bi mall han V which i ud o pic CK, h plicaing pofolio i woh l han h ca wh V = V whn h volailiy mil i fla h oppoi can b aid whn h volailiy i downwad loping o, givn h am paam Mauiy, ik, Volailiy, and in a, and h diffn volailiy mil hown blow, Volailii mil 3 mil mil ik W can ay, DCmil > DCmil > DCmil 3 o how hi ul mahmaically: DC, K, K,, C lim, K, K,, C, K, K, C, K,,, K K B, N d, K C, K,,, K Vga of Vanilla Call lop d K dk of volailiy mil i, w obv ha whn h mil i fla B& wold, h lop i zo, and h quaion duc o h fomula obaind in h pviou cion cond, inc h Vga of a vanilla call opion i alway poiiv, h pic i only affcd by h ign of h lop paam In h ca i i poiiv, h pic dca and vic va 6

27 Paiculaii of h Commodii Mak 5 Rfnc - Commodii and Commodiy Divaiv: Modling and Picing fo Agiculual, Mal, and ngy by Hly,Pof Gman - Opion, uu and Oh Divaiv, by John C Hull 3- Commodiy ad dfiniion and picing, CDA Analyic, Naixi 4- In Ra Modl hoy and Pacic, by Damiano Bigo abio Mcuio 7