Derivations and Applications of Greek Letters Review and
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1 Rvi //008 Chap 0 Divaion an Applicaion of Gk L Rviw an Ingaion By Hong-Yi Chn, Rug Univiy, USA Chng-Fw L, Rug Univiy, USA Wikang Shih, Rug Univiy, USA Abac In hi chap, w inouc h finiion of Gk l. W alo povi h ivaion of Gk l fo call an pu opion on boh ivin-paying ock an non-ivin ock. Thn w icu om applicaion of Gk l. Finally, w how h laionhip bwn Gk l, on of h xampl can b n fom h Black-Schol paial iffnial quaion. Ky wo Gk l, Dla, Tha, Gamma, Vga, Rho, Black-Schol opion picing mol, Black-Schol paial iffnial quaion 0. Inoucion Gk l a fin a h niivii of h opion pic o a ingl-uni chang in h valu of ih a a vaiabl o a paam. Such niivii can pn h iffn imnion o h ik in an opion. Financial iniuion who ll opion o hi clin can manag hi ik by Gk l analyi. In hi chap, w will icu h finiion an ivaion of Gk l. W alo pcifically iv Gk l fo call (pu) opion on non-ivin ock an ivin-paying ock. Som xampl a povi o xplain h applicaion of Gk l. Finally, w will cib h laionhip bwn Gk l an h implicaion in la nual pofolio.
2 Rvi // Dla ( ) Th la of an opion,, i fin a h a of chang of h opion pic pc o h a of chang of unlying a pic: S wh i h opion pic an S i unlying a pic. W nx how h ivaion of la fo vaiou kin of ock opion. 0.. Divaion of Dla fo Diffn Kin of Sock Opion Fom Black-Schol opion picing mol, w know h pic of call opion on a non-ivin ock can b win a: C N S N (0.) an h pic of pu opion on a non-ivin ock can b win a: P N S N (0.) wh S ln S ln T N i h cumulaiv niy funcion of nomal iibuion.
3 Rvi //008 u u u u f N Fi, w calcula N N (0.) S S N N ln (0.4) Eq. (0.) an Eq. (0.) will b u piivly in mining following Gk l whn h unlying a i a non-ivin paying ock. Fo a Euopan call opion on a non-ivin ock, la can b hown a N( ) (0.5) Th ivaion of Eq. (0.5) i in h following:
4 Rvi //008 C N N S S S S N S N S N N N S S S S S N S S S S N Fo a Euopan pu opion on a non-ivin ock, la can b hown a Th ivaion of Eq. (0.6) i N( ) (0.6) P N N N S S S S ( N ) ( N ) ( N ) S S S S ( N ) S S S S N S S S N If h unlying a i a ivin-paying ock poviing a ivin yil a a q, Black-Schol fomula fo h pic of a Euopan call opion on a ivin-paying ock an a Euopan pu opion on a ivin-paying ock a C S N N (0.7) q an 4
5 Rvi //008 q P N S N (0.8) wh S ln q S ln q To mak h following ivaion mo aily, w calcula Eq. (0.9) an Eq. (0.0) in avanc. N N ( ) N N ( ) S ln q S (q) (0.9) (0.0) Fo a Euopan call opion on a ivin-paying ock, la can b hown a 5
6 Rvi //008 (0.) q N( ) Th ivaion of (0.) i N C N N S S S S q q N N N S q q S S q q S (q) N S S S q q q N S S S S q N Fo a Euopan call opion on a ivin-paying ock, la can b hown a q N( ) (0.) Th ivaion of (0.) i P N N N S S S S q q ( N ) ( N ) ( N ) S q q S S S (q) q q q ( N ) S S S S S q (N ) q q (N ) S S 0.. Applicaion of Dla Figu 0. how h laionhip bwn h pic of a call opion an h pic of i unlying a. Th la of hi call opion i h lop of h lin a h poin of A 6
7 Rvi //008 coponing o cun pic of h unlying a. Figu 0. By calculaing la aio, a financial iniuion ha ll opion o a clin can mak a la nual poiion o hg h ik of chang of h unlying a pic. Suppo ha h cun ock pic i $00, h call opion pic on ock i $0, an h cun la of h call opion i 0.4. A financial iniuion ol 0 call opion o i clin, o ha h clin ha igh o buy,000 ha a im o mauiy. To conuc a la hg poiion, h financial iniuion houl buy 0.4 x,000 = 400 ha of ock. If h ock pic go up o $, h opion pic will go up by $0.4. In hi iuaion, h financial iniuion ha a $400 ($ x 400 ha) gain in i ock poiion, an a $400 ($0.4 x,000 ha) lo in i opion poiion. Th oal payoff of h financial iniuion i zo. On h oh han, if h ock pic go own by $, h opion pic will go own by $0.4. Th oal payoff of h financial iniuion i alo zo. Howv, h laionhip bwn opion pic an ock pic i no lina, o la chang ov iffn ock pic. If an invo wan o main hi pofolio in la nual, h houl aju hi hg aio pioically. Th mo fqunly ajumn h o, h b la-hging h g. Figu 0. xhibi h chang in la affc h la-hg. If h unlying ock ha a pic qual o $0, hn h invo who u only la a ik mau will coni ha hi pofolio ha no ik. Howv, a h unlying ock pic chang, ih up o own, h la chang a wll an hu h will hav o u iffn la 7
8 Rvi //008 hging. Dla mau can b combin wih oh ik mau o yil b ik maumn. W will icu i fuh in h following cion. Figu Tha ( ) Th ha of an opion,, i fin a h a of chang of h opion pic pc o h paag of im: wh i h opion pic an i h paag of im. If T, ha ( ) can alo b fin a minu on iming h a of chang of h opion pic pc o im o mauiy. Th ivaion of uch anfomaion i ay an aigh fowa: ( ) wh T i im o mauiy. Fo h ivaion of ha fo vaiou kin of ock opion, w u h finiion of ngaiv iffnial on im o mauiy. 8
9 Rvi // Divaion of Tha fo Diffn Kin of Sock Opion Fo a Euopan call opion on a non-ivin ock, ha can b win a: S N ( ) N( ) (0.) Th ivaion of (0.) i C N( ) N( ) S N( ) N( ) N( ) S N( ) ln S N( ) ln S ln S N( ) ln S S S N ( ) N( ) N( ) 9
10 Rvi //008 Fo a Euopan pu opion on a non-ivin ock, ha can b hown a S N ( ) N( ) (0.4) Th ivaion of (0.4) i P N( ) N( ) N( ) S ( N( )) ( N( )) ( ) ( N( )) S ln S ( ) ( N( )) ln S ln ( N( )) S ln S ( N( )) S S ( N( )) N ( ) S N( ) N ( ) 0
11 Rvi //008 Fo a Euopan call opion on a ivin-paying ock, ha can b hown a q q S qs N( ) N ( ) N( ) (0.5) Th ivaion of (0.5) i C q q N( ) N( ) q S N( ) S N( ) N( ) N( ) q S N( ) S N( ) q q ln q q q q q S N( ) S N( ) ln q S (q) q q q ln q q q S N( ) S N( ) ln q S q S N( ) S N( ) q q S q S N( ) N ( ) N( ) Fo a Euopan call opion on a ivin-paying ock, ha can b hown a q q S N( ) qs N( ) N ( ) (0.6)
12 Rvi //008 Th ivaion of (0.6) i P N( ) q q N( ) N( ) ( q)s N( ) S ( N( )) ( N( )) ( N( )) qs N( ) S q q ln S q (q) q ( N( )) q ln q q q qs N( ) S ln q q q ( N( )) S q ln q q q q ( N( )) qs qs N( ) S q q ( N( )) qs N( ) S q S N( ) N ( ) q q S N( ) qs N( ) N ( ) 0.. Applicaion of Tha ( ) Th valu of opion i h combinaion of im valu an ock valu. Whn im pa, h im valu of h opion ca. Thu, h a of chang of h opion pic wih pciv o h paag of im, ha, i uually ngaiv. Bcau h paag of im on an opion i no uncain, w o no n o mak a ha
13 Rvi //008 hg pofolio again h ffc of h paag of im. Howv, w ill ga ha a a uful paam, bcau i i a poxy of gamma in h la nual pofolio. Fo h pcific ail, w will icu in h following cion. 0.4 Gamma ( ) Th gamma of an opion,, i fin a h a of chang of la pc o h a of chang of unlying a pic:: S S wh i h opion pic an S i h unlying a pic. Bcau h opion i no linaly pnn on i unlying a, la-nual hg agy i uful only whn h movmn of unlying a pic i mall. Onc h unlying a pic mov wi, gamma-nual hg i ncay. W nx how h ivaion of gamma fo vaiou kin of ock opion Divaion of Gamma fo Diffn Kin of Sock Opion Fo a Euopan call opion on a non-ivin ock, gamma can b hown a Th ivaion of (0.7) i N (0.7) S
14 Rvi //008 C C S S S N N S S N S Fo a Euopan pu opion on a non-ivin ock, gamma can b hown a Th ivaion of (0.8) i N (0.8) S P P S S S (N ) S N S N S Fo a Euopan call opion on a ivin-paying ock, gamma can b hown a q N (0.9) S Th ivaion of (0.9) i 4
15 Rvi //008 C C S S S q N( ) q q N S N S q N S S Fo a Euopan call opion on a ivin-paying ock, gamma can b hown a q N (0.0) S Th ivaion of (0.0) i P P S S S q (N ) q q N S (N ) S q N S S 5
16 Rvi // Applicaion of Gamma ( ) On can u la an gamma ogh o calcula h chang of h opion u o chang in h unlying ock pic. Thi chang can b appoxima by h following laion. chang in opion valu chang ic in ock p (chang in ock pic) Fom h abov laion, on can obv ha h gamma mak h cocion fo h fac ha h opion valu i no a lina funcion of unlying ock pic. Thi appoximaion com fom h Taylo i xpanion na h iniial ock pic. If w l V b opion valu, S b ock pic, an S 0 b iniial ock pic, hn h Taylo i xpanion aoun S 0 yil h following. n V ( S0) V ( S0) V ( S0) V ( S) V ( S0) ( S S0) ( S S 0) ( S S0) n S! S! S V ( S0) V ( S0) V ( S0) ( S S0) ( S S 0) o( S) S! S If w only coni h fi h m, h appoximaion i hn, n V ( S ) V ( S ) V ( S) V ( S ) ( S S ) ( S S ) S! S. ( S S0) ( S S0) Fo xampl, if a pofolio of opion ha a la qual o $0000 an a gamma qual o $5000, h chang in h pofolio valu if h ock pic op o $4 fom $5 i appoximaly, chang in pofolio valu ($0000) ($4 $5 $7500 ) ($5000) ($4 $5) Th abov analyi can alo b appli o mau h pic niiviy of in a la a o pofolio o in a chang. H w inouc Moifi Duaion an Convxiy a ik mau coponing o h abov la an gamma. Moifi uaion mau h pcnag chang in a o pofolio valu uling fom a pcnag chang in in a. 6
17 Rvi //008 Chang in pic Moifi Duaion Pic Chang in in a / P Uing h moifi uaion, Chang in Pofolio Valu Chang a in in ( Duaion P) Chang in in a w can calcula h valu chang of h pofolio. Th abov laion copon o h pviou icuion of la mau. W wan o know how h pic of h pofolio chang givn a chang in in a. Simila o la, moifi uaion only how h fi o appoximaion of h chang in valu. In o o accoun fo h nonlina laion bwn h in a an pofolio valu, w n a con o appoximaion imila o h gamma mau bfo, hi i hn h convxiy mau. Convxiy i h in a gamma ivi by pic, Convxiy /P an hi mau capu h nonlina pa of h pic chang u o in a chang. Uing h moifi uaion an convxiy ogh allow u o vlop fi a wll a con o appoximaion of h pic chang imila o pviou icuion. Chang in Pofolio Duaion Valu P ) (chang in a Convxiy P (chang in a) A a ul, (-uaion x P) an (convxiy x P) ac lik h la an gamma mau pcivly in h pviou icuion. Thi how ha h Gk can alo b appli in mauing ik in in a la a o pofolio. Nx w icu how o mak a pofolio gamma nual. Suppo h gamma of a la-nual pofolio i, h gamma of h opion in hi pofolio i, an i h numb of opion a o h la-nual pofolio. Thn, h gamma of hi nw pofolio i o o o o To mak a gamma-nual pofolio, w houl a * o / opion. Bcau h o poiion of opion chang, h nw pofolio i no in h la-nual. W houl chang 7
18 Rvi //008 h poiion of h unlying a o mainain la-nual. Fo xampl, h la an gamma of a paicula call opion a 0.7 an.. A la-nual pofolio ha a gamma of -,400. To mak a la-nual an gamma-nual pofolio, w houl a a long poiion of,400/.=,000 ha an a ho poiion of,000 x 0.7=,400 ha in h oiginal pofolio. 0.5 Vga ( ) Th vga of an opion,, i fin a h a of chang of h opion pic pc o h volailiy of h unlying a: wh i h opion pic an i volailiy of h ock pic. W nx how h ivaion of vga fo vaiou kin of ock opion Divaion of Vga fo Diffn Kin of Sock Opion Fo a Euopan call opion on a non-ivin ock, vga can b hown a S N (0.) Th ivaion of (0.) i 8
19 Rvi //008 C N( ) N( ) S N( ) N( ) S S ln S S S ln S S ln ln S S S S N Fo a Euopan pu opion on a non-ivin ock, vga can b hown a S N (0.) Th ivaion of (0.) i 9
20 Rvi //008 P N( ) N( ) S ( N( )) ( N( )) S S ln S S S ln S S ln ln S S S S N Fo a Euopan call opion on a ivin-paying ock, vga can b hown a q S N (0.) Th ivaion of (0.) i 0
21 Rvi //008 C N( ) N( ) S q S q N( ) N( ) S ln q q S S (q) S ln q S S ln q ln q q q S S S S N q Fo a Euopan call opion on a ivin-paying ock, vga can b hown a q S N (0.4) Th ivaion of (0.4) i
22 Rvi //008 P N( ) N( ) S q ( N( )) ( N( )) S q q S ln q S (q) S q S ln q S S ln q ln q q q S S S S N q 0.5. Applicaion of Vga ( ) Suppo a la-nual an gamma-nual pofolio ha a vga qual o an h vga of a paicula opion i. Simila o gamma, w can a a poiion of / in o opion o mak a vga-nual pofolio. To mainain la-nual, w houl chang h unlying a poiion. Howv, whn w chang h opion poiion, h nw pofolio i no gamma-nual. Gnally, a pofolio wih on opion canno mainain i gamma-nual an vga-nual a h am im. If w wan a pofolio o b boh gamma-nual an vga-nual, w houl inclu a la wo kin of opion on h am unlying a in ou pofolio. Fo xampl, a la-nual an gamma-nual pofolio conain opion A, opion B, an unlying a. Th gamma an vga of hi pofolio a -,00 an -,500, pcivly. Opion A ha a la of 0., gamma of., an vga of.5. Opion B ha a la of 0.4, gamma of.6 an vga of 0.8. Th nw pofolio will b boh gamma-nual an vga-nual whn aing of opion A an of opion B ino h oiginal A B o
23 Rvi //008 pofolio. Gamma Nual: Vga Nual: Fom wo quaion hown abov, w can g h oluion ha A A B B A =000 an B = 50. Th la of nw pofolio i 000 x x 0.4 = 800. To mainain la-nual, w n o ho 800 ha of h unlying a. 0.6 Rho ( ) Th ho of an opion i fin a h a of chang of h opion pic pc o h in a: ho wh i h opion pic an i in a. Th ho fo an oinay ock call opion houl b poiiv bcau high in a uc h pn valu of h ik pic which in un inca h valu of h call opion. Similaly, h ho of an oinay pu opion houl b ngaiv by h am aoning. W nx how h ivaion of ho fo vaiou kin of ock opion Divaion of Rho fo Diffn Kin of ock opion Fo a Euopan call opion on a non-ivin ock, ho can b hown a ho N( ) (0.5) Th ivaion of (0.5) i
24 Rvi //008 C N( ) N( ) ho S N( ) N( ) N( ) S N( ) S S N( ) S N( ) N( ) S Fo a Euopan pu opion on a non-ivin ock, ho can b hown a ho N( ) (0.6) Th ivaion of (0.6) i P N( ) N( ) ho N( ) S ( N( )) ( N( )) ( N( )) S S ( N( )) S ( N( )) S S N( ) Fo a Euopan call opion on a ivin-paying ock, ho can b hown a ho N( ) (0.7) Th ivaion of (0.7) i 4
25 Rvi //008 C q N( ) N( ) ho S N( ) N( ) N( ) S N( ) q q S (q) S N( ) S q q N( ) S N( ) Fo a Euopan pu opion on a ivin-paying ock, ho can b hown a ho N( ) (0.8) Th ivaion of (0.8) i P N( ) q N( ) ho N( ) S ( N( )) ( N( )) ( N( )) S q S (q) q ( N( )) S ( N( )) S S q q N( ) 0.6. Applicaion of Rho ( ) Aum ha an invo woul lik o how in a chang affc h valu of a -monh Euopan pu opion h hol wih h following infomaion. Th cun ock pic i $65 an h ik pic i $58. Th in a an h volailiy of h ock i 5% an 0% p annum pcivly. Th ho of hi Euopan pu can b calcula a following. 5
26 Rvi //008 6 (0.05)(0.5) pu ln(65 58) [0.05 (0.) ](0.5) Rho N( ) ($58)(0.5) N( ).68 (0.) 0.5 Thi calculaion inica ha givn % chang inca in in a, ay fom 5% o 6%, h valu of hi Euopan call opion will ca (0.0 x.68). Thi impl xampl can b fuh appli o ock ha pay ivin uing h ivaion ul hown pviouly. 0.7 Divaion of Sniiviy fo Sock Opion Rpciv wih Exci Pic Fo a Euopan call opion on a non-ivin ock, h niiviy can b hown a C N( ) (0.9) Th ivaion of (0.9) i ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( N S N S S N S N N N S N N N S C Fo a Euopan pu opion on a non-ivin ock, h niiviy can b hown a P N( ) (0.0) Th ivaion of (0.0) i
27 Rvi //008 P N( ) N( ) N( ) S ( N( )) ( N( )) ( N( )) S S ( N( )) S ( N( N( ) S S )) Fo a Euopan call opion on a ivin-paying ock, h niiviy can b hown a C N( ) (0.) Th ivaion of (0.) i C q N( ) N( ) S N( ) N( ) N( ) S N( ) q q S (q) S N( ) N( ) q S q S N( ) Fo a Euopan pu opion on a ivin-paying ock, h niiviy can b hown a P N( ) (0.) Th ivaion of (0.) i 7
28 Rvi //008 P N( ) q N( ) N( ) S ( N( )) ( N( )) ( N( )) S q S ( q) q ( N( )) S q q S S ( N( )) N( ) 0.8 Rlaionhip bwn Dla, Tha, an Gamma So fa, h icuion ha inouc h ivaion an applicaion of ach iniviual Gk an how hy can b appli in pofolio managmn. In pacic, h inacion o a-off bwn h paam i of concn a wll. Fo xampl, call h paial iffnial quaion fo h Black-Schol fomula wih non-ivin paying ock can b win a S S S S Wh i h valu of h ivaiv cuiy coningn on ock pic, S i h pic of ock, i h ik f a, an i h volailiy of h ock pic, an i h im o xpiaion of h ivaiv. Givn h ali ivaion, w can wi h Black-Schol PDE a S S Thi laion giv u h a-off bwn la, gamma, an ha. Fo xampl, uppo h a wo la nual ( 0) pofolio, on wih poiiv gamma ( 0) an h oh on wih ngaiv gamma ( 0) an hy boh hav valu of $ ( ). Th a-off can b win a S Fo h fi pofolio, if gamma i poiiv an lag, hn ha i ngaiv an lag. 8
29 Rvi //008 Whn gamma i poiiv, chang in ock pic ul in high valu of h opion. Thi man ha whn h i no chang in ock pic, h valu of h opion clin a w appoach h xpiaion a. A a ul, h ha i ngaiv. On h oh han, whn gamma i ngaiv an lag, chang in ock pic ul in low opion valu. Thi man ha whn h i no ock pic chang, h valu of h opion inca a w appoach h xpiaion an ha i poiiv. Thi giv u a a-off bwn gamma an ha an hy can b u a poxy fo ach oh in a la nual pofolio. 0.9 Concluion In hi chap w hav hown h ivaion of h niivii of h opion pic o h chang in h valu of a vaiabl o paam. Th fi Gk i la ( ) which i h a of chang of opion pic o chang in pic of unlying a. Onc h la i calcula, h nx p i h a of chang of la wih pc o unlying a pic which giv u gamma ( ). Anoh wo ik mau a ha ( ) an ho ( ), hy mau h chang in opion valu wih pc o paing im an in a pcivly. Finally, on can alo mau h chang in opion valu wih pc o h volailiy of h unlying a an hi giv u h vga ( v ). Th laionhip bwn h ik mau a hown, on of h xampl can b n fom h Black-Schol paial iffnial quaion. Fuhmo, h applicaion of h Gk l in h pofolio managmn hav alo bn icu. Rik managmn i on of h impoan opic in financ oay, boh fo acamic an paciion. Givn h cn ci cii, on can obv ha i i cucial o poply mau h ik la o h v mo complica financial a. Rfnc Bjok, T., Abiag Thoy in Coninuou Tim, Oxfo Univiy P, 998. Boyl, P. P., & Emanul, D. (980). Dicly aju opion hg. Jounal of Financial Economic, 8(), Duffi, D., Dynamic A Picing Thoy, Pincon Univiy P, 00. 9
30 Rvi //008 Fabozzi, F.J., Fix Incom Analyi, n Eiion, Wily, 007. Figlwki, S. (989). Opion abiag in impfc mak. Jounal of Financ, 44(5), 89-. Galai, D. (98). Th componn of h un fom hging opion again ock. Jounal of Buin, 56(), Hull, J., Opion, Fuu, an Oh Divaiv 6 h Eiion, Paon, 006. Hull, J., & Whi, A. (987). Hging h ik fom wiing foign cuncy opion. Jounal of Innaional Mony an Financ, 6(), -5. Kaaza, I. an Shv, S.E., Bownian Moion an Sochaic Calculu, Sping, 000. Klban, F.C., Inoucion o Sochaic Calculu wih Applicaion, Impial Collg P, 005. McDonla, R.L., Divaiv Mak n Eiion, Aion-Wly, 005. Shv, S.E., Sochaic Calculu fo Financ II: Coninuou Tim Mol, Sping, 004. Tuckman, B., Fix Incom Scuii: Tool fo Toay' Mak, n Eiion, Wily, 00. 0
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