NGHIEN Ctfu & TRAO DOI
w
ifNG DUNG MO HINH CAY NHI PHAN
• I
DE DjNH GIA QUYEN CHON VA HQlP DONG GIAO SAU
TS. BUI PHUC TRUNG*
Thi trddng chdng khodn phdi sinh (TTCKPS) VN dd dinh se ra ddi cud'i ndm 2014 (theo UBCKNN); vi vdy, ngay td bdy gid phdi nghien cdu cdc phiidng phdp luan cUng nhd cdng cu di dinh gid CKPS la rdt quan trgng. Ky thudt rdt pho Men di dinh gid quyen chgn chdng khodn la ky thugt xdy ddng cdu true cdy nhi phdn. Ddy la do thi biiu diin cdc hiidng phdt triin khdc nhau cd the cd cua gid chdng khodn trong suot thdi ky tin tgi cua quyen chgn. Bdi viit ndy gidi thifu md hinh cdy nhi phdn vd mgt so nguyen ly tdi chinh quan trgng di dinh gid quyen chgn vd hdp dong giao sau.
1. Dfng toin hpc cda md hinh ciy nh[ phin Chiing ta dd bie't cde eay nhi phdn mdt bUde va hai bUde dd'i vdi ehffng khodn khdng trd co tffe dl dinh gid ede quyin ehpn k i l u chdu Au vd k i l u My. Tuy nhidn, cde cdy nhi phdn ndy Id nhffng md hinh it phu hdp vdi thffc te'. Md h i n h thffc td' htfn la md hinh ma gia dinh r a n g nhflng b i l n dpng gia chffng khodn dffpfe chia t h a n h n h i l u bid'n ddng nhi phdn nhd. Ddy Id gid dinh ctf sd eua thu tue sd' hpc rd't quen thupe, dffpfe d l xud't d i u tidn bdi Cox, Ross, va Rubinstein.
S,u
1-/7 S,d
Hinh 1. Cdc bidn d$ng gid chffng khodn trong thdi
\ gian 5t theo mo hinh nhj phan
Xem xdt cdch dinh gid mdt quyen ehpn trdn chfing khodn khdng t r a co tffe. Chffng ta chia thdi k^ t i n tai eua q u y i n chpn t h d n h n h i l u khoang thdi gian nho cd dp ddi la 5t. Chffng ta gia dinh r&ng trong moi k h o a n g t h d i gian thi gid ehffng khodn dich chuyen tff gid tri ban dau SQ eua nd
dd'n mdt trong hai gid tri mdi la SQU va Sod. Md hinh ndy dUtfe minh hpa trong Hinh 1. T i n g qudt, u>l va d < l . Do dd, bie'n ddng tff SQ dd'n SQU la mdt bid'n ddng tdng, vd bid'n ddng tff Sg d i n Spd Id mdt bid'n ddng gidm. Xde sud't eua mdt bid'n ddng tdng se dfftfc ky hidu Id p. Xac sud't eua mdt bid'n ddng giam se dffpfe ky hidu la 1-p.
1.1 Ly ludn ve nguyen tdc dinh gid rdi ro trung hda cd bdn
Nguyen tdc dinh gid rui ro trung hda phdt bilu rang: Mdt quyin ehpn (bode phdi sinh khde) ed t h i dfftfc dinh gia dffa trdn gid dinh rdng mdi trffdng cd rui ro trung hda. Dilu ndy cd nghia Id, khi tinh toan thi ehung ta cd t h i gia dinh nhff sau:
Lai sud't ky vpng tff td't cd cdc chffng khodn dfftfc giao dich Id lai suat phi rui ro.
- Cde ddng t i l n m a t trong tUtfng lai cd t h i dUtfc dinh gid bang each chie't khau cac gid tri ky vpng eua ehung tai mffe ldi sud't phi rui ro.
Chung ta t a n dung k i t qua ndy khi sff dung ede cdy nhi phdn. Cdy nhi phdn md chung ta xdy dffng eho mdt ehffng khoan khdng trd c l tffe se t h i hidn nhffng bie'n ddng gid chffng khodn trong mdt mdi trffdng rui ro trung hda.
1.2 Cdch xdc dinh p, u, vd d
Cde t h a m s i p, u, vd d phai cd cdc gid tri phu hdp vdi trung binh vd phfftfng sai cua nhffng thay
NGHIEN Ctfu & TRAU u u i
ECOWOMC DBaOPMENTBEVgW
d l i gia ehffng khodn trong sud't khoang thdi gian cd dp ddi 8t. Bdi vi chffng t a dang lam viec trong mdi trffdng rui rd trung hda, cho n e n lai suat ky vpng tff ehffng k h o a n se b a n g lai suat phi rui ro r.
Vi t h e , gia tri ky vpng cua gid ehffng k h o a n t a i thdi diem ke't thue k h o a n g t h d i gian 5t la Se''**, trong dd S la gid chffng khodn tai t h d i d i l m b a t d i u khoang thdi gian nay. Tff bieu thffc :
Se'-'^ = pSu + (1-p)Sd (1) hay
e'^ = pu + (1 -p)d (2^
Trong qua t r i n h ngdu n h i e n , t h i phfftfng sai cua p h i n t r a m t h a y d l i gia chffng khodn trong mpt khoang thdi gian nhd 5t la cr^St. Bdi vi phfftfng sai eua mpt bie'n Q dUtfc dinh nghia la E(Q^)- [E(Q)]^, cho n d n suy ra:
pu2 -f- (l-p)d2 [pu -I- (l-p)d]2 = cr'dt
Thay the' p tff phUtfng t r i n h (2) vao d l rfft gpn phfftfng t r i n h nay, chung t a dUtfe:
e''"(u-i-d) - ud - e' ,2r5t _ a^St (3) PhUtfng trinh (2) vd (3) dp ddt 2 d i l u kidn ldn p, u, vd d. D i l u kidn thff 3 dfftfc Cox, Ross, vd Rubinstein dffa ra la:
a
Ba dilu kien nay dua de'n:
a — d
« = £ • " * 1 „-a./Tt
trong dd
(4) (5) (6)
a = e--*' (7) va cde so h a n g ed bae eao htfn 5t dffpc bd qua.
Bid'n a ddi khi dfftfc xem Id h e so' phdt trien.
1.3 Md hinh cdy nhi phdn gom cdc mdc gid chdng khodn
Hinh 2 minh h p a cay n h i p h a n h o a n ehinh gom cac mffe gid ehffng khodn dfftfc xet den khi sff dung md h i n h nhi phdn. Tai thdi d i l m 0, gid ehffng khoan SQ dfftfc bie't trffde. Tai thdi d i l m 5t, ed 2 mffe gia chffng khodn ed the la Sgu vd Sgd; tai thdi d i l m 25t, ed 3 mffe gia chffng khodn ed t h i la SQU^, SQ, vd Sgd^; V.V.. Tong qudt, t a i thdi diem iSt, ed i-i-1 mffe gid ehffng khodn dfftfc xem xet. Nhffng mffe gid ndy la:
Souid% j=0, 1,..., i
Chff y r a n g md'i quan hd u=l/d dffpfe sff dung de t i n h gia ehffng k h o a n t a i moi nut cua cdy nhi phdn d H i n h 2. Vi du nhff SQUM = SQU. Chu y rang eay n h i p h a n se k e t htfp lai ed nghia la mdt biln dpng giam theo sau mdt bien dpng t a n g se di den eung m d t mffe gia nhff k h i mpt bien dpng tdng theo sau mpt bid'n ddng giam.
Hinh 2. Cay nhj phdn dUdc sff dung de d[nh gid mpt quyin chon chffng khodn
1.4 Qud trinh tinh ngUdc tren cdy nhi phdn Cae q u y i n chpn dfftfc dinh gid bang edeh bat d i u tai thdi d i l m k i t thuc cua cdy (thdi dilm T) va t i n h ngfftfc len. Gia t r i cua quyen chpn tai thdi diem T dffpfe bie't trffde. Vi du, mdt quyen ehpn bdn cd gid tri b a n g max(K ST, 0) va mdt quyen ehpn mua ed gia tri b a n g max(Sx - K, 0), trong dd ST la gia ehffng k h o a n t a i t h d i d i l m T vd K la gid thifc hidn. Bdi vi gia dinh r a n g mdi trUdng ed rui ro trung hda, eho n e n gid t r i d m§i nut tai thdi dilm T - 5t cd the dUpfe t i n h b a n g vdi gid tri ky vpng tai thdi d i l m T dUtfe chie't k h a u tai mffe lai sud't r trong khoang t h d i gian 8t. Tfftfng tff, gid tri d moi n u t t a i thdi d i l m T 28t ed t h e dffpfe tinh bang vdi gia t r i ky vpng t a i t h d i diem T - 6t dfftfc chiit khdu t a i mffe lai s u i t r trong khoang thdi gian 5t v.v.. Ne'u quyen chpn n a y theo k i l u My, thi cin p h a i k i e m t r a t a i mdi nfft de xem xet lidu vide thffc hidn sdm quyen chpn cd td't hdn vide giff quyen chpn t h e m mot khoang thdi gian 5t nffa hay khdng. Cud'i eung, sau k h i t i n h ngUde qua td't ca ede nut, t h i chffng t a ed t h e tim dffpfe gid tri cua quyen chpn n a y t a i t h d i diem 0.
Vi du 1: Xet mpt quyen chpn b a n 5 thdng kilu My t r e n mpt chffng k h o a n khdng t r a eo tffe khi gid chffng khodn hien tai la 50 USD, gia thffc M$n
Phat trien kinh te - Thang Nam nam 2011
INGHIEN Ctfu & TRAO D O I
la 50 USD, ldi s u i t phi rui ro la 10%/ndm, vd dp blp benh Id 40%/ndm. Theo ky hieu thdng thudng cua ehung ta, thi dieu nay cd nghia la SQ = 50, K
= 50, r = 0.10, (J = 0.40 vd T = 0.4167. Gia sff r a n g chung ta chia t h d i ky t o n t a i eua q u y i n ehpn thdnh 5 khoang t h d i gian ed dp ddi 1 t h d n g (=
0.0833 ndm) d l xdy dffng cdy nhi phdn. Sau dd, va sff dung phfftfng t r i n h (4) de'n (7), thi:
u = e'"^ =l.l22'i, d = e-""^ = 0.S909 a — d
a = e'"= 1.0084, 1 - p = 0.4927
P = u 0.5073
Hinh 3 t h i hien cdy nhi p h a n dfftfc xdy dffng bdfi phan m i m DerivaGem. Tai mdi nut d i u cd hai
; con s i . S i bdn trdn Id gid ehffng khodn; so ben
; dudi la gid tri cua quyin chpn. Xde suat eua mdt biln ddng tdng ludn bang 0.5073; xde s u i t eua mdt : biln dpng gidm ludn bang 0.4927.
i
Gid chffng khodn t a i nut thff j (j=0, 1,..., i) tai thdi dilm i5t (i=0, 1,..., 5) dffdc tinh Id SoUJd'^ Vi I du, gid chffng khodn tai n u t A (i=4, j = l ) (nghia la nut thff 2 d t r e n tai thdi d i l m kd't thffc khoang thdi gian thff 4) la 50 x 1.1224 x 0.8909^ = 39.69 USD.
Cdc mffe gid quyin chpn tai cdc nut cud'i eung dupc tinh Id max(K - ST, 0). Vi du, gid quyin chpn tai niit G Id 50.00 - 35.36 = 14.64. Gid quyin ehpn tai niit ke cud'i dffpfe tinh tff gid quyin ehpn tai nut cuoi ciing. Trffde tidn, ehffng t a gia dinh quyen chpn khdng dfftfc thffc hidn tai cdc nut. Dieu nay CO nghia Id gid quyin chpn dffpfe t i n h bang gid tri hidn hanh cua gid quyin chpn ky vpng sau mdt bifdc thdi gian. Vi du, tai nut E, gid quyin chpn dUtfc tinh Id:
(0.5073 x 0 + 0.4927 x 5.45)e-'' i'"""'^^^ = 2.66 Trong khi tai nut A, gid quyen ebon dUde tinh Id:
(0.5073 X 5.45 -i- 0.4927 x I4.64)e-'' i'''*''*'^ = 9.90 Sau dd, ehung ta k i l m t r a xem vide thife hien quyen chpn thi ed ldi htfn vide ehd dtfi hay khdng.
Tai niit E, vi§e thife hidn sdm quyin ehpn se eho gid tri quyin chpn b a n g 0, bdi vi ca gid ehffng khodn vd gid thffc hidn deu b a n g 50 USD. Rd rdng, tot n h i t la chung t a ndn chd dpfi. Vi thd', gid quyin chpn chinh xdc cua q u y i n ehpn tai nut E b k g 2.66 USD. Tai n u t A t h i khde h a n . Nd'u quyin chpn dfftfc thffc hi^n, nd se cd gid la
50.00USD 39.69USD, hay 10.31 USD. Gid tri nay thi ldn htfn 9.90 USD. Do dd, n l u tid'n dd'n diem A, thi quyen ehpn se dffdc thffc hidn vd gid tri chinh xde eua quyin chpn tai nut A la 10.31 USD.
Tgi moi nut:
Gia tri tren = Gid tai san ca sd Gid tfi dtrdi = Gid quy^n chpn
NhOng chT so in d$m Id t^i do quyen chQn di/?c \lnifc hi^n Gia thi/c hi$n - 50
H$ so chiit khiu IrSr mbi H$ so phat IriSn Ir§nm6i D$ \6n cua biSn d^ng tan D$ I6n cOa biln dpng gid
D 50.0(
4.4S
y
\
56. i ; 2.ie
44.5!
6.9e
y
>
S
but)c \i = 0.9!
33 nam, 30.42 3ti6c, a = 1.00
p = 0.5073 g = 1.1224 m, d = 0.8909
F 6 2 . 9 !
0 . 6 '
C
50.0C 3.77
B
3 9 . 6 ! 10.3f
y N.
^ N
N.
17 ngay 84
7 0 . 7 ( O.OC
56.12 1.3C
4 4 . 5 ! 6 . 3 !
35.36 14.6^
^
N y
S
7 9 . 3 ! O.OC
6 2 . 9 ! O.OC
E 50.0C
2.66
A 39.6S 10.31
31.5C 18.5C
y
>
\
>
N»
^
8 9 . 0 : O.OC
70.7C O.OC
56.i;
O.OC
44.5!
5.4!
G 35.3!
14.6'
28.01 21.9:
Nut thcri gian:
0.0000 0.0833 70.70
0.1667 70.70
0.2500 70,70
0.3333 70.70
0.4167
Hinh 3. Cay nhj phan dff^c xay dffng tff phan mem DerivaGem cho quyen chon bdn Idlu IVIy trdn chffng
Ichoan Ichdng trd cd tffe (Vi dv 1)
Gid q u y i n chpn tai cde nut trffde dfftfc tinh tfftfng tff. Chff y rang khdng phdi luc ndo vide thife hidn sdm mpt quyen chpn cung tot n h a t khi quyen ehpn dd dang d trong vung hdi ra t i l n . Hdy xem xet nut B. Nd'u quyin ehpn dfftfc thffc hidn, thi nd cd gid tri Id 50.00USD 39.69USD, hay 10.31 USD. Tuy nhidn, nd'u giff lai quyen chpn ndy, thi nd ed gid tri Id:
(0.5073 X 6.73 -i- 0.4927 x I4.64)e*i<«'<'83^ ^ 10.36 Vi t h i , quyin ehpn khdng ndn dfftfc thffc hien tai d i l m ndy, vd gid tri quyin chpn ehinh xdc tai nut ndy Id 10.36 USD.
Khi tinh ngffpc trd lai dau cdy, thi gid tri quyin ehpn tai nut dau tidn Id 4.49 USD. Ddy la con sd ffde Iffdng cua chffng t a cho gid tri hidn h a n h cua quyin chpn. Trong thffc t l , mpt gid tri nhd htfn , va nMeu nut htfn se dfftfc sff dung. Phan m i m DerivaGem da t i n h dffpfe rang qua 30, 50, 100 va 500 bffdc thdi gian, gid tri quyen ehpn se hkng tfftfng ffng Id 4.263, 4.272, 4.278, vd 4.283.
NGHIEN Ctfu & TRAcrrft?
KMWMCOftBOPKBflHfvtw
1.5 Bieu dien phddng phdp cdy nhi phdn theo dai so'
I iGia tri ciJa quyen chpn 5.00
4.80- • 4.60-•
4,40 4 20 4.00- 3 80 3 60
25 30 35 40 45 50
S 6 bvfcfc
Hinh 4. SU hpi tu cua gid quyen chpn trong Vf du 1 dUtfc tinh tff cac ham sd trong Application Builder
cua DerivaGem
Gia sff rang thdi ky ton tai eua mpt quyen chpn bdn kilu My trdn mpt ehffng khodn khdng trd eo tffe dffpfe chia thdnh N khodng thdi gian nhd ed dp ddi 8t. Chung ta se xem nfft thff j tai thdi dilm idt Id nut (i, j), trong dd 0 < I < N vd 0 < j < 1. Dinh nghia fy la gia tri eua quyin ehpn tai nfft (i, j).
Gia chffng khoan tai nut (i, j) la SoU'd'^ Bdi vi gid tri cua mpt quyen ehpn bdn kilu My tai thdi dilm ddo ban bang max(K - ST, 0), eho ndn ehung ta bie't rang:
fNj = max(K - SouidN-J, 0), j=0, 1,..., N
Cd xac suat p d l dich chuyin tff nut (i, j) tai thdi dilm iSt dd'n nut (i-i-1, j-i-1) tai thdi diem (i-i-l)5t, va xdc sua't 1-p d l dich ehuyin tff nut (i, j) tai thdi dilm idt dd'n nfft (i-nl, j) tai thdi diem (i+l)5t. Nd'u ehung ta gia dinh khdng thffc hien sdm quyen ehpn, thi nguydn tae dinh gia rui ro trung hda se cho:
fij = e-"[pfi,ij^i + (l-p)fi^ij]
vdi 0 < i < N-1 va 0 < j < 1.
Khi ed thffc hidn sdm quyin ehpn, thi gid tri fjj nay phai dfftfc so sanh vdi gid tri thffc chd't cua quyin ehpn, vd ehung ta dfftfc:
fy = maxlK SoU*dH, e-" [pfi^i^^i + (l-p)fi+ij] 1 Chu y rdng, bdi vi cac kd't qua tinh toan bat dau tai thdi diem T vd tinh ngfftfc lai, cho ndn gid tri tai thdi dilm idt khdng chi anh hffdng dd'n nhffng kha ndng thffc hien sdm quyen chpn tai thdi diem idt ma cdn anh hUdng den vide thifc
Men sdm quyin ehpn tai nhffng thdi dilm tilp theo.
Vdi gidi ban khi 5t t i l n dl^n 0, thi ehung ta se tim dUtfe gid tri chinh xde eua quyIn ehpn bdn kieu My. Trong thUe te, N=30 thudng dem lai nhffng ke't qua hpfp ly. Hinh 4 cho thay sif hdi tu eua gid quyin chpn trong vi du ehffng ta dang xet.
Do thi nay dffdc tinh todn bang each sff dung cdc ham cffa p h i n m i m DerivaGem.
1.6 lfdc Iddng Delta vd cdc tham so dif phdng khdc
Delta cua mpt quyen ehpn la ty Id giffa thay dli gid quyen chpn theo thay dli gia cua tai sdn ctf sd. Nd ed t h i dfftfc ffdc Ifftfng nhu sau:
^ SS
trong dd 5S la lUpfng thay doi nhd eua gia ehffng khodn vd 5f la Iffdng thay doi nhd tfftfng ffng cua gid quyen chpn. Tai thdi dilm 5t, ehung ta cd ffdc Iffpfng f^^ cua gia quyIn ehpn khi gid ehffng khoan la SQU, va ffdc Iffpfng f^o eua gid quyin ehpn khi gia ehffng khodn la Sod. Dilu ndy cd nghia la, khi 8S = SQU Sgd, tM gid tri eua 5f Id fii - fio- Vi the, Udc lUpfng eua A tai thdi dilm 5t Id:
fn —fio
A = (8)
SoU — Sod
Dl xde dinh r ehff y rang ehffng ta ed hai Udc lUtfng cua A tai thdi diem 25t. Khi S=V&(Sou2-i- SQ) (nam giffa nfft thff 2 va thff 3), thi A Id (£22 - f2i)/(Sou2 -1- So); khi S=i/2(So -1- Sod^) (nam giffa mit thff 1 vd thff 2), thi A la (fgi f2o)/(So -i- Sod^) Chenh lech giffa hai gia tri eua S la h, trong d6:
h = 0.5(Sou2-Sod2)
r tinh dffpfe bang A cMa eho h:
P _ [ U -Ji,)/{Sou' - 5,)j-!(/. -i,)/(5o - Sod'^ (9) Nhffng thu tue nay dem lai nhffng Udc lUtfng eua A tai thdi diem 8t va Ude lUtfng eua F tai thdi dilm 28t. Trong thifc te, cdc Udc lUpfng nay cung thudng dffdc sff dung dffdi dang cdc ffde Ifftfng cua A vd r tai thdi diem 0.
Mpt tham sd bao hd khac ma cd t h i tim trifc tie'p tff cdy Id ©. Day la ty Id giffa thay dli gia quyin ehpn theo thdi gian khi td't ed cdc ylu to khde Id khdng doi. Nd'u cdy bat diu tai thdi dilm 0, tM Udc lupfng cua 0 la:
Phat trien kinh te - Thang Nam nam 2011
MGHIEN Ctfu & TRAO DOI
(10)
v cd the dffpfe tinh b a n g each ldm eho dp bap bdnh thay doi nhd vd xdy difng mpt cdy mdi d l tim ra gid tri mdi eua quyen chpn. (Vdi cung budc thdi gian Id St.) Udc lUpfng cua v la:
trong dd f vd r lan lUpft Id cdc Ude lUtfng eua gid quyin chpn tff cdy ban dau vd eay mdi. Rho ed the dfftfc tinh tfftfng tff.
Vi du 2: Xet lai Vi d u l . Tff H i n h 3 , fio = 6.96 va fl 1 = 2.16. Phfftfng t r i n h (8) cho ffdc Iffpfng cua A bang:
2.16-6.96 _ Q^^
5 6 . 1 2 - 4 4 . 5 5 " "'^^
Tff phfftfng trinh (9), ehung ta cd t h i tim dfftfc Udc Iffpfng v l r cua quyin chpn tff cde gid tri tai cdc nut B, C, vd F bang:
((0.64,- 3.77)/( 62.99 - 50.00)] - [{3.77 - 10.36)/( 50.00 - 39.69)] _ „ » ,
11.65 ""•"^" -0.41
Tff phfftfng trinh (10), chffng ta ed t h i tim dfftfc Udc Ifftfng v l 0 eua quyin chpn tff cdc gid tri tai cdc nut D, vd C bang:
3.77 - 4.49 4 - 5 ^ 5 ; „ a ^ 0.1667 = - 4 - 3 moi ndm
hay bang -0.012 moi ngdy. Dfftfng nhidn, ddy chi la nhffng ffdc Iffdng stf bp. Chffng se d i n dan chinh xdc htfn khi chung ta tdng so bffdc thdi gian tren cdy. Sff dung 50 bffdc thdi gian, thi De- rivaGem tinh dffpfe ede ffde Ifftfng cua A, r , vd 0 lin lUtft Id -0.415, 0.034, vd -0.0117.
2. 0ng dyng mfi hinh ciy nhf phin cho cac qiqrin chpn trin chi sflT, tiln t$, vi h^p ding giao sau
i> Nhu dd trinh bay, phUtfng phdp cay nhi phdn dl dinh gid ede quyen ehpn t r e n ehffng khodn khdng trd e l tffe ed the dffpfe sffa d l i de dinh gid I cdc quyen chpn mua vd b a n kieu My trdn ehffng
khodn trd hoa Ipfi e l tffe la q.
Bdi vi ede co tffe mang lai mffe ltfi suat la q, cho ndn thdng thffdng gid ehffng khodn trong mdi tnfdng riii ro trung hda phai mang lai mffe Ipfi sud't Id r - q. Vi vdy, phfftfng t r t n h (1) trd thdnh:
Se'^-'i»'=pSu-H(l-p)Sd
sao eho:
Phfftfng t r i n h (3) trd thdnh:
e('-q)6t(u -H d) - ud - e2c-<i'«'= cs'-8t
Chung ta thd'y rang, phUtfng trinh (4), (5), vd (6) vdn dung (khi ede sd h a n g ed bdc cao htfn 8t dUtfc bd qua) nhung vdi:
a = e'"-'"* (11) Vi the, theo each M l u nay eua q, thi thu tue
sd hpc bang eay nhi phdn cd t h i dUtfe dung gid'ng n h u trUde ngoai trff cae gid tri cu eua a dfftfc thay t h i bang cde gid tri mdi nhff trdn.
Khi dinh gid quyen ehpn, thi ehi so' chffng khodn, t i l n td, vd htfp dong giao sau cd t h i dfftfc xem nhff Id cdc tdi san tao ra hoa ltfi bid't trffde.
Trong trffdng htfp eua ehi sd chffng khodn, mffe hoa ltfi thich hpfp la mffe hoa ltfi eo tffe dffa trdn danh muc dau tff chffng khodn ctf sd eua ehi sd dd;
trong trffdng htfp eua t i l n td, thi hoa ltfi thich htfp la lai sud't phi rui ro eua dong ngoai td; trong trffdng htfp cua htfp dong giao sau, hoa ltfi thich htfp Id ldi sud't phi rui ro cua d i n g ndi td. Vi t h i , phfftfng phdp cdy nhi phdn cd t h i dfftfc sff dung d l dinh gid cdc quyin chpn trdn chi sd ehffng khodn, tien td, vd hop d i n g giao sau, vdi dieu kidn la q trong phfftfng trinh (11) dfftfc b i l u theo mpt cdch thich htfp.
Vl du 3: Xet mdt quyen ehpn mua 4 thdng kilu My trdn htfp dong giao sau ve ehi so, trong dd gid giao sau hidn hdnh la 300, gid thffc hidn Id 300, ldi sud't phi rui ro Id 8%/ndm, vd dp b§fp bdnh cua ehi sd la 30%/nam. Khi xdy dffng eay nhi phdn, chung ta chia thdi ky t i n tai eua quyin ehpn thdnh 4 khoang thdi gian ed dp dai 1 thdng. Trong trffdng htfp nay, FQ = 300, K = 300, r = 0.08, cr = 0.3, T = 0.3333 vd 8t=0.0833. Vi mpt htfp dong giao sau thi tfftfng tff nhff mdt ehffng khodn trd el tffe tai mffe ldi sud't r, cho ndn trong phfftfng trinh (11), q se dffpfe cho bang vdi r. Phfftfng trinh ndy eho kd't qua Id . cdc tham sd can t h i l t khdc d l xdy difng cdy la:
u = e'"^= 1.0905, d = ^= 0.9170 p = i L i i 4 = 0.4784, 1 - p = 0.5216
^ u — d
Cay tao r a tff p h i n m i m DerivaGem dfftfc trinh bdy trong Hinh 5 (Sd bdn trdn la gid giao
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NGHIEN Ctfu & TRACJ UOl
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Tai moi nut:
Gici tri tren = GiA tai san co'so' Gia trj dicdi = Gia quy§n chpn
Nhu'ng ch! s6 in dam la tai 66 quyen chpn du'P'c thyc hiSn Gia thyc hi?n =300
He so chiet khautren moi bypc = 0.9934 Bu'O'C thoi gian, dt = 0.0833 nam, 30.42 ngay He s6 phat trien tren m5l bydc, a = 1.0000 Xac suit b i l n dpng tang, p= 0.4784 So lan cua bien d$ng tang, u = 1.0905 Dp {d'n cua bien dpng gidm, d = 0.9170
300.0(
19.1€
327.1'
as.e-:
275.11 6.13
Nut thai gian
0.0000 0.0833 70.70
356.7' 56.7;
389.0(
300.0(
12.9C
252.2S O.OC
0.1667 70.70
89.0(
424.15
3 2 7 . 1 ' 27.1^
N j 275.11
O.OC
A
231.3f O.OC
0.2500 70.70
124.1!
356.7- 56.7;
300.0C O.OC
252.25 O.OC
212.17 O.OC
0.3333 70.70
Tai m6l n u t :
Gia trj tren = Gia tai san c a s a Gia trj du'a! = Gid quyen chpn
Nh&ng ch! so in d§m la tgi do q u y i n chpn d y p c thyc hi§n Gia thyc hi$n = 1.6
Hp s6 chiit khautrSnmoi bu-ac la = 0.9802 Bu'acthal gian, dt = 0.2500nam,91.25ngdy Hp so phdt trien tremoi bi/dc, a = 0.9975 Xac s u i t b i l n dpng lang, p = 0.4642 Dp lan cda bien dpng tang, u = 1.0618 Dp lan cOa b i l n dpng gidm, d = 0.9418
1.610(
0.071(
1.709f 0.0245
1.516:
0.113e
Niit thdi gian
0.0000 0.2500 70.70
1.815:
O.OOOf
1.927i
>d
0.047! 1.610(1.4275 0.175;
N .
0.5000 70.70
O.OOOC
1.709e
o.oooc
1.516:
0.090'
1.344J 0.255:
0.7500
2.046^
0.000(
1.815;
0.000(
1.610(
0.000(
1.4275 0.172-
1.2661 0.333!
1.0000
Hinh 5. Cay nhj phan dffdc xay dffng tff DerivaGem cho quyen chpn mua Icieu My tren htfp dong giao
sau ve chi sd
Hinh 6 . Cay nh| phan dff^c xay di/ng tff DerivaGem cho quyen chpn ban i<ieu lUly tren tien t^
sau; sd bdn dffdi la gia quyen chpn.) Gid tri ffde Ifftfng eua quyen chpn Id 19.16. Cdng thffc hidn nhilu bffdc thi kd't qua dat dffpfe cang chinh xde.
Vdi 50 bffdc thdi gian, DerivaGem tinh dffpfe gid tri quyin chpn Id 20.18; vdi 100 bffdc thdi gian, DerivaGem tinh dfftfc gid tri quyin ehpn Id 20.22.
Vi du 4: Xet mdt quyen ehpn ban 1 nam kilu My trdn dong bang Anh. Ty gia hdi doai Men hdnh la 1.6100, gid thffc hidn la 1.6000, lai sud:t phi rui ro eua dong USD la 8%/nam, lai suat phi rui ro eua dong GBP la 9%/nam, vd dp bd'p bdnh cua ty gid hdi dodi ding bang la 12%/nam. Trong trudng hpfp nay, SQ = 1.61, K = 1.60, r = 0.08, rf = 0.09, a = 0.12, va T = 1.0. Khi xdy difng cay nhi phan, chung ta chia thdi ky ton tai cua quyin ehpn thdnh 4 khoang thdi gian ed dp ddi 3 thdng, sao eho 8t = 0.25. Trong trUdng htfp nay, q = r^ va phUtfng trinh (11) eho k i t qua nhU sau:
a = e«'"»-""9>''''^5 = 0.9975
Cde tham so' can thie't khde d l xdy dung cdy Id:
u = e'"^ ^ 1.0618, d = -^ = 0.9418
1
u
i ' = f ^ ^ = 0-4642, 1 - p = 0.5358 Cdy tao ra tff phSn m i m DerivaGem dffpfe trinh bay trong Hinh 6. (Sd ben tren la ty gid hdi dodi; sd bdn dffdi Id gid quyen ehpn.) Gid tri ffdc Iffpfng eua quyen ehpn la 0.0710 USD. (Sff dung 50 bude thdi gian, DerivaGem tinh dUpfc gid tri quyen ehpn Id 0.0738; vdi 100 bude thdi gian, DerivaGem eung tinh dupe la 0.0738)
3. Ket luan
Bdi viet da eung cip mpt cdi nhin khdi qudt vl dinh gid quyen ehpn vd hpfp dong giao sau. Khi nhffng bien ddng gid chffng khodn hi ehi phdi bdi md hinh cdy nhi phdn nhieu bffdc, chung ta cd the xem xet ridng biet tffng bffdc nhi phan va tinh ngucfc tff cudi thdi ky ton tai trd ve diem ban dau de tim gid tri hidn tai eua quyen ehpn. Mpt ldn nffa chi cd ly lualn ve mdi trffdng khdng cd ctf hdi
Phat trien l(inh te - Thang Nam nam 2011
i^GHiEN Ctfu & TRAO D O I
ctf ltfi dfftfc sff dung, vd khdng clin cd gid dinh ve xdc sudt tdng hodc gidm gid ehffng khodn tai mdi ndt eua cdy nhi phdn.
Cdc eay nhi phdn dfftfc gid dinh rang trong khodng thdi gian nhd, 8t, thi gid chffng khodn hodc tdng ldn theo mdt ty Id phan tram u bode gidm xud'ng theo mdt ty Id phan tram d. Dp ldn cua u vd d vd ede xde sud't kdt htfp eua ehffng dUtfc chpn sao cho thay dli eua gid chffng khodn ed trung binh vd dp leeh ehuan phu htfp trong mdi tnfdng rui ro trung hda. Gia phdi sinh dUtfe bat diu tinh tai nut cud'i cung eua cdy vd tinh ngUtfe trd lai. Ddi vdi quyin chpn kilu My, gid tri tai mdt nut Id mdt gid tri ldn htfn vdi gid tri cd dUtfe nlu quyin chpn dUpfc thtfc hidn ngay ldp tffe vd bkng gid tri chiit khd'u ky vpng nd'u quyin chpn dupc giff trong khoang thdi gian 8ta
TAI LI$U THAIIA KHAO
1. Cox, J., S. Ross, vd M. Rubinstein (1979), "Option Pricing: A Simplified Approacli", Journal of Financial Economics, 7.
2. Rendlemam, R., vd B. Bartter (1979), "Two State Option Pricing", Journal of Finance, 34.
3. Chance (1998), D. M., An Introduction to Deriv- atives, phidn ban thU 4, Dryden Press, Orlando, FL.
4. Cox, J. C , vd M. Rubinstein (1985), Option Markets, Prentice Hall, Upper Saddle River, NJ.
5. Kolb, R. (1999), Futures, Options, and Swaps, phien bdn thff 3, Blackwell, Oxford.
6. l\/lcl\/lillan (1992), L. G., Options as a Strategic Investment, New York Institute of Finance, New York.
7. John C, Hull. (2007), Options, Futures and Others Derivatives.
8. Black, Fisher and Myron Sholes (2003), The Pricing of Options and Corporate Finance, MeGraw - Hill, 7rd Edition.
9. Saiih N.Neftci (1996), Mathematics of Financial Derivatives.
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