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i
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→ − I = (1, . . . , 1)T
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h
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]
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1
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L
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l
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²
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p
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= e−r(T −τ ) M S(T ) − ϕ[τ,T ] e−r(T −τ ).
F (τ ) = e−r(T −τ ) M(S(T ) − ϕ[τ,T ] ) =
c(t) = S(t)Φ(d1 ) − Ke−r(T −t) Φ(d2 ),
D
S
B
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X
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m
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L
c
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qm
C
@ V D
SH
B
0.0089 = 0.042 0.2009 ]
]
V B
B
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det V = 0.9999
N
D
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0.0089 0.2009 ]
MH D@ S
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L
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W
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NH
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P
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D
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LF
OT K ¤ ] B
H SX @ P CH
C
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] M
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M
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M
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L
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B
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W
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L
D
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A
IH
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Y
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ASH
Z
NH
AW
Bó
B
(RA −M RA )2 (RB −M RB )2 0.052 0.152
SH
A BF
LM
A BF
1 0.9999 (0.01, 0.21) ·
B
LM
→ − ν = (0.0127, 0.2861)
O
IB
B
]
Y
Y
D
E
H
I
N
U
V =
D
ZS
D
V
SH
H
F
B
] B®
O
C\
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Y
AY
OF
E
D
h
h
B
SH
H
B
L@
W
H
B
Z
MO
B
CP
k
¤
2 σA = M(RA − M RA )2 = 0.0025,
E
P
Á½
P
@
1 0.9999
D
@
OD
1 −0.01 −0.01 1
OD
Ë
À¿
→ − − V −1 (→ r − r0 I ) =
W
B MJ B
L
D
SB î oB
¥
i=1
2 X
B
→ − → − − − (→ r − r0 I )T V −1 (→ r − r0 I ) =
D@
O
C\
Z
X
V A
A
\M
AF
IH
R
A(0.1, 0.1) B(0.2, 0.3)
SP
→ − → − r − r0 I = (0.01, 0, 21)T
MH D@ S
WG @
A@ ? D S
K
@
] P
ID
NH
ν0 +
SP
]
[H
LF
L@
DH
NH
L
IH
¤
,
i,j=1
2 X
B
X
D
W@
A@ D S
C
1 det V
B
]
-
,
H MP
DZS
H
NH
L
H
] h e
∆Π = rΠ∆t.
MJ
GB D S
U
gh
V −1 =
A
]
m
B
OT
MP
H
SH
] B
O
C\
N
D@
C
σ2 =
r∗
MJ
AD HF
V H
B
P@
L
DL
@
DZS
L@
B
ρAB σA σB = 0.01
B
B
P
D@ ª S \F
L
OT
A@
MH
IB
@
[ \ OH
G
Oó
M
\B S
AY
SH
W
K
H
\U
L
B
U
G@
G
r∗ = 0.15
]
L
L
B
H
<
æ
@
j
]
B CB
L
DL
M
@
B
@
L@
M
I
L
W
H
AF
A
G@
0 0.03 0.5 M(RA RB ) = 0.015 Cov(RA , RB ) = M(RA RB ) − M RA M RB = 0.0075 0.1 RA RB
DL
@
OZ
¨ ì
I
NH
O
M
A
H
L V H A E
B
MO H
L
AT HF
MH
GF
H
\F
¤
∂f /∂S
M
S
U (σ, r) = 0.6r − r 2 − σ 2 U (σ, r) = C = −(r − 0.3)2 − σ 2 = C − 0.09 W
R©
§¨
¤
[H
S
LD
-
OF
wz
\m S
A
MO H
I
\@
L
AB
@
AF
C
Π ∂f ∂f 1 ∂2f 2 2 − ∆t = −r f − σ S − S ∆t. ∂t 2 ∂S 2 ∂S
G
±
,
\B
AY
SH
B
~x
∂f ∂2f ∂f 1 + rS + σ 2 S 2 2 = rf. ∂t ∂S 2 ∂S
-
f (S(T ), T ) = max{K − S(T ), 0}, M
t
~x
@
P
-
, C
AL F \S
G@ V \
Dj H
A
A
∂f 1 ∂2f 2 2 ∆t. σ S − ∂t 2 ∂S 2
h
f (S(T ), T ) = max{S(T ) − K, 0}, A
MO H
I
5
~
M
J
L@
gh
H
DN H
k
S
∂f ∆S. ∂S
]
B
L\ @
W
H
NB
e7 µ8
t
B
@
] B
L
DL
M
A
D
∆Π = −∆f +
,
H
L@
W
~t
~
t
s
qx
6 wz w |z M I|
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√ √ D(S(t + ∆t)) = S(t)2 er∆t − e−σ ∆t eσ ∆t − er∆t .
P
[B
]
V
AT
AD
C
M
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D(S(t + ∆t)) = S(t)2 (u2 pq + d2 pq − 2udpq) = pqS 2 (u − d)2 .
DL @
S CB
M
L
AB
CH
CU
qD
Y
F
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@
+ qd2 S 2 − (puS + qdS)2 = S(t)2 (pu2 + qd2 − p2 u2 − q 2 d2 − 2pqud).
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MA
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G H@
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AE
Nj \S
D(S(t + ∆t)) = M S(t + ∆t)2 − (M S(t + ∆t))2 = pu2 S 2 +
]
n
SH
H
B
NH
A
GL
M
S
Y
A
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IH
R
uS(t) dS(t) p q
D
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CG
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IH
í
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H
Im
O
GF
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D
p+q =1
L
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W
q\
\B
N
A HF
LT
σ = 20%
G
AB
A
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E
D
MJ
OD
OT
L@
B
< ï
I
D
I
NH
¤
er∆t − e = σ ∆t + (r − σ 2 )∆t + O (∆t)3/2 , √ √ eσ ∆t − er∆t = σ ∆t − (r − σ 2 )∆t + O (∆t)3/2 . √ −σ ∆t
IH
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H
H
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¡
D(S(t + ∆t)) = S(t)2 σ 2 ∆t + O(∆t)2 . S(t+∆t)−S(t) ln S(t+∆t) = ln 1 + S(t) S(t) O (∆t)2 S(t + ∆t)/S(t) − 1
D ln(S(t + ∆t)) − ln S(t) = σ 2 ∆t.
S(t)
L
OT @
NB
$40
DLC H
DH U V X @
L
S
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S(t + ∆t) P
B L@
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t
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u d ln(S(t + ∆t)) − ln(S(t))
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rA rB
r∈
r∈
1
A2
2 σΠ = ν12 σ12 + ν22 σ22 + 2ν1 ν2 Cov(R1 , R2 ).
h
k
@
r1 σ T − ∂L = ∂ν1 1 σT
ν1 σ12 + ν2 Cov(R1 , R2 ) (ν1 r1 + ν2 r2 − rf )
σT2
∂σT = σT−1 ν1 σ12 + ν2 Cov(R1 , R2 ) ∂ν1
+ λ. −5 95 100
S(T )
105
8%
è
90
H
H L@
W
DH Z
A
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H
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H
ó
CB
AY
L
M
S
ZS
D
D
W
D
LD
G
\
A
G K OL AD B ]
A
I
DL @
E®
D@
I
P@
DM H
D
\n OF
OK ] B B U
<
âä
S(T ), K,
ãø ùÚ
@
SH
SH D@
Z
G
V Y
[
F
AY IB
D
YMO
V
5
n
-
,
¤
-
,
] ® °
-
,
OD BF
OD BF
H
SB
L@
M
@
D
B
H
B Y
L@
E
L
H
] LB L¢ H
M
AF
A
M
AH
F
OK B YS
L
J
M
OK B B SU @
ASH
Z
-
,
ASH
Z
NH
M
Z
B L@
W
³
] ®
P
@ª
@
B L
JB
W
L
J
OTE @
] M
AD
H G@
NH
L
IH
D
AZ
\
H L@
W
Z
DH
H
NH
L
IH
D
AH
D
F A HF
V
]
]
G@
A k
B
C\
C
\F
L@
NB
L
I
Ll
JE
OLD @
OD H
O
H LG @
W
D
H
SH U
n
OT K B
ASH
Z
A
Ke−r(T −t)
Ý
V
D
\H
OZ
L
Z
DH
H
L V H A E
I
MB
DE
¯®¬ °±¬ ¡
B
D
G@
YC
FU
K
S(T ), K,
æ
Z
L
MGD V
B JB
W
o
V NB
B
@
A HF
S
OT X H
B
YI
L
DL
l
@
ª¬
AE
D
OR \H
S
U
OT K B
ASH
Z
L
J
H
@
] O@
AZ
t
Ý
K T
p + S(t)
éßáÝ
H AT HF
C
D
AY F
M
AD
G@
F
ó
B
OT K B
oD
OT
Y
A
IH
X
D@
Gm
L
D
OT
DB SP
< ï
â
max{K − S(T ), 0} + S(T ) =
è
]
S \S
D
[
YG
L
L
D
I
cH
lL @
OT X H
R
L@
]
ASH
D
H
H MP
@ lL @
N
C
\
Ll
JL
B
B
OD m
L
A@
F
@
< ð
â
max{S(T ) − K, 0} + K =
Þç
SH
K
ßèì
@ L@
M
S
M
L@
D@
MH
U
@
P@
m
U
D
DZS
D
@
W
L
G
H
G
A HF
LT
c + Ke−r(T −t)
îã
ÓÔ ÑÒ Þß ÒÕ çÚ Ø×Ö ç ÜÛ ÙÚ çÝ ÛÚ éáèÚ ÞßàÝ êÛÚ Ü ë áâßã éßáÝ æ Ýä Ý Þçì ããåÚ èÚ ß ë ÛÝ ìí æÚ
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@
M
A BF
] D@ E
P@ I
q
E
\
I
L
\L
S(T )
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îç îÛ éÜÝì æçÝ â Ú òÝñ ë ß ðó ããåÚ Ûè Û äè æÝÚ æå î ß ðàÝ çå è ð ïè Þì ß ô±Ý ìí
Ü Ü ãÚ ãÚ ç ç çßàè çßàè
[
max S(T ) − 100, 0 − 5
æäÚ
ν1 S(T )
ßÝ
U
ν1 r1 + ν 2 r2 − r f + λ(ν1 + ν2 − 1), σT T
ãè
Û Üè õ ãåÚ àöÝ Ý î óé ÜÚ Û ÞçÚ éçè à Ý éì æÝ ÜáÝ éàÚì çÚ í
MH
D
S MH
OD
C\
H
Y\ F
ó
@
YC
ZS
MH
M
C
DASH
C
F
F
@a
M
AD
G@
\E
L
JSB
A
ν1 + ν2 = 1.
AU H
H
OF
F
p
A
Y
IO
Lc
B
H
L@
AD
G@
O
GL @
@
lL
JL H
O
N
MH
GF
]
JS
@ª
lL
< ñ @
LJ
â
MH
G M
Π Π
L
\
DZ
D P@
Y
K
MH
D
OF
\B
@
MO
@
B A HF
Z
A
o
ZD
D@
\
H
H
DP S
D
G
AD
0.18
IH
U
SH
F
@
S
K
¶
OD BF
B
OT
A
−→ max,
éåè Û
H
H
MP
DZS
L
O
C
C\ B
CJ
IH
N
@£
B
X
D
B
lH
W
DL
G
\
m
@
SH
, L
YIG
Z
H
V
C
L
J
OT
F
A@
A2 FΠ
AF
ν12 σ12 + ν22 σ22 + 2ν1 ν2 Cov(R1 , R2 ) ]
ó
B AF S
YG
L
D
I
cH
lL @
E@
@
]
l
CJ
AD F
Z
\H
I
9%
ùâ
B
B
@
-
I
A
NH
o
ZD
M
@ Dn
J SB
Z
Y
] C
C\
A2 (σ2 , r2 )
E@
L
S
`
@£
Q@
Lc
N
Y
W
DL
G
\
@
-
MH
D
OF
\
A1
D
OF
E
D
MJ
Y
H AX F
D
I
,
]
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D³ <ì
] C@
C\ B
S
0.2
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H
@
L=
L@
Z
YIG
Q
6
- R¶
,
] ®
l@
OP 6ê
A
A1 (σ1 , r1 )
ν1 ,ν2
çÚ
B
H ] U
L
L
p ν2
+ 2ν1 ν2 Cov(R1 , R2 )
]
S
OK B B U
ASH
Z
l
CJ
H IU \H Z AD F
M
D
G
L
O
H
G@ ª
F
F
@a
@
[Z
H
H
NH
L
P IH
H DA
D
FU
SB
A HF
MJ
R
] OB K Y
ASH
Z
M
I
C
DH S
A
C
¯®¬ °±¬
ª¬
s = rΠ −
Þß
SH
B
O
C\
Π
ν22 σ22
þýû ÿ±Üû
Z
I
N
F = (0, rf ) + ν1 r1 + ν 2 r2 − r F
ü úû
H
SB NH
M
Z
σT = ν12 σ12
Π = ν 1 A1 + ν 2 A2
K S(T ) > 100
T
éß æÛ Þ é õÚ æÛÚ ðè êçÜè ðâÜÚ éøÝ çßè Ú é ßêÛ áé ç æìÝ ãø ù Ú âç ßá ðâ Ú éÚ ð éä Ü Ú ð Ú óÛÝ Þß çéè çÝ çÚ âæ çêè ÛÝ Û ô æçÝ ðçè çøåè ãà ßè è Ý ô öß æêó ã õÚ çÜ ßã óâ Ý ì ßé ÜÚ çßàè à áâ Ü ì áâ Û ÜÚ ßê Þß ã ù çÚ ßæ ßåÝ ð Ú ô ìçáÚ ðç éè ô í
SB
H CB
Z
Z
σT p
l
I
NH
¤
rF /σΠ
®
k¶
£
ª
ª§
£
`
¥
a¤
R¡
SR JSB
C
D
] ®
§
S(T ) > K . S6K
S(T ) > K . S(T ) 6 K
T
5 Π
100 0
110
ST
T
5e0.08·2/12 ≈ 5.07 max S(T ) − 100, 0 − 5.07
c
\
O
OP
SH
B
P
V
H
,
[
@
L V U
U
B
H
SH
D
L@
MA
@
®
] ® h
-
, ¬
®
] ®
Q
OP
@
B
@
®
-
, ®
-
B
P@
@ L
DL
M B
\B S
A
IB
D
M
MJ
B ] Ð Ìç
V
®°
V
<
\
H
SB
@
SH
E
I
M
C
LD
JL
E
B
@
P@
B
\H
OZ
YIG
Z
NH
M
Z
DYC
MH
D
OF
\
YF
T
P
OT R H
åò
e
®
] ®
-
,
V
AE
DL @
F
B
S [B
M
L
AB
CH
C
M
C
MJ H
GF
DB SP
Y
AY
OYM
X
OT K B
ASH
Z
®
®
] ®
-
,
F
@
ON H
F
@
V
]
-
,
f
®
] ®
¬
±
¬
De
L ] H
V
I
NH
SB ] \F ¤
K
S
H
AY
H
B
@
@
AB
G
M
AL @
I
S
DSH
YG
AT
L@
-
H
@
@
SB
S
@£
D
K
[
H
SB
@
SH
m
\
M
S B
@
F
B
AL F
M
M
Y
m
E@ A
G P@ O
YIG
Z
NH
M
Z
Y
MH
D
OF
]
AF
m E@
C
L
JL
B
CB
Z
Z
Lc
N
W
D
LD
G
\
lL
J
IH
m
[m
G P@ O
SH
OL
V
H
W
H
B
A BF
LM
ED
A
A
H
V
E@
L@
P
H
m DP
DZ
\
SH GH A
MP
DZS
Ll
J
@ GB
A
C
L
c\
¦
, DP@ ,
@
K
B
SH
ZS
L@
G@ A
A
GU
L
J
F
SB
m
B
OT
A
IH
-
,
-
E
T
A
E@
λ
rA > r B
Y
Y
L
C
D
L@
¦
®,
] ®
D
f
@ ] ®
M
S AL F
M
Y
cH
L
C
R <ì
6
6ê
V
C
CJ
AD F
Z
\H
F
B
¶
-
,
B
L V H
OT
E
m
TH
Z
ν1 ν2
-
,
B CQ B
C
H
B
H
I
Y
¨
j
+ λ = 0,
N\
S
B
@
V
D
L@
Y
IO
L
L
OZ
MJ
AY
]
V
L ] H
F
@
ON H
+ λ = 0,
M
C
B H SX @ P MH D@
V
S
L@
L
DL
M
G@
OYM
ν1 + ν2 = 1.
AT
IH
H
HF
ZS
B
O HF
Z
C
E
D
\H
OZ
\
V
H
σT2
D
DM
Z
W
D
D@
MH D@ S
Gm
U V
Ll
JL
B
ν1
Ml
OT K B
U
B TA HF
L
L
L@
E
\H
OZ
P
B
DH
M
Z
SB
r1 σT2 − ν1 σ12 + ν2 Cov(R1 , R2 ) (rT − rf ) + λσT3 = 0, r2 σT2 − ν2 σ22 + ν1 Cov(R1 , R2 ) (rT − rf ) + λσT3 = 0. C
OT K B
] B L@
-
,
ν2 σ2 + ν1 Cov(R1 , R2 ) (ν1 r1 + ν2 r2 − rf )
l
H
ASH
Z
L
JE
I
M
DB
AB
C
OD
B
A@
A
Sn \B OT P
C
DLc H
H ASH
Z
E
A
MO H
L
\± AF @
¬
A@
MH
IB
j
ν1 σ1 + ν2 Cov(R1 , R2 ) (ν1 r1 + ν2 r2 − rf )
JSH
A
B(σB , rB ) YG
IO
A(σA , rA ) L
Ì
Ü
P YF
T
\
OT R H
B
rT
] Ð Ìç
JS
M
Ê
D¬
-
®,
] ®
H
Y
Y
L
C
P@
]
C
¦
k
M
B
@
SB AL F
M
L
V
AT
MH D SB
]
B
σT2
CB
B
D
Þ
H
] EB L
E
A
IU ] CB î
L@
C
MH
Dc H
OF
H
m
A HF
LT
ZS
Y
F
@
]
L
O
σT2
L
DL
@
Y
l
MH
@
H
TL
n
r1 ν1 + r 2 ν2
S
@
I
NH
¤ V
@
E
A
[
ASH
Z
A
lL
J
Y
AH
SB
A
O
C\
SH
ϕ[τ,T ] = S(τ )e−q(T −τ ) er(T −τ ).
M
\
TU BF
A
\H
OZ
]
ν2 Cov(R1 , R2 )
D
OF
V H
L
V
L
AB
C
λ = −rf /σT
M
Ð
B
U
V H
L
B
= ϕ[τ,T ] e−r(T −t) > ϕ[τ,T ] e−r(T −τ ) .
H
Ìç
OT
A@
MH
j
CB
M
H
B
OT
A@
MH
IB
OF
]
H MP OT
DZS
rT
\
B
B
IB
O
[B
MO H
L
W
F
U
V
H
D
M
F
@
AF
¡
D
C
1 σT
AD
M\
L
DL
@
AZ
\H
D
I
AF
H
\F A
H
\L @
M
AD
G@
JM H
P
@X
U
A
F
-
,
H
H
AT BF
`
1 σT
G@
?
M
S
¦
C\
C
NH
L
IH
A
W
P YF
T
\ DH Z
AZ
\
OT R H
H JSH
A
D C
B
V
±
¬
-
,
ν1 r1 + ν 2 r2
@
D
OB
A
cH
L
\F
L@
B
] B
L
H
B
M
D P@
L
CJ
AD F
Z
]
De
¬
NH
SH
H
B
A
r 1 σT −
M
I
AD HF
< â
å
B JB
W
c
L
P
AD H
P (t) > Ke−r(T −t) − S(t).
L
AF
E@
[B
W
I\
cB
CB
B NB
L
I
L@
B
W
Z
A
GL
]
F
B
\H
YI
AY
] H
L
F
@
A
GL
r 1 σT −
OTZ
D
t S(t)
HF
n
B
DB
L
Y
D
A HF
FU
SB
W
B
H L@
L
W
Z
DH
< å \ AR
YMO
H
NO H
j
Πθ (t) = e−q(T −t) S(t) − F (t).
R <ì
]
] L
D
L
B
lL
MJ
R
H
L
D
S CB
M
L
AB
CH
F
B
¨ ì
R© Πθ (t) = e−q(T −t) S(t) − S(t)e(r−q)(T −t) − ϕ[τ,T ] e−r(T −t) =
Q
S
D
oH
OK B
JE
ODL @
P
@
H
V B A E
A
cD
V B I G
L
G
C
YM
AY
H
OF
D
OT K B
¯ θ(t) Πθ (t) > −C t ∈ [τ, T ]
CB
M
L
H
ASH
Z
H
NH
SB
@ L@ JS
c
L
K P (t)
L
AH
BF
cB
H
M
Z
@ ASH
Z
M
M\
B
@
AL F
M
SB
B
OZ
§¨
t ∈ [τ, T ]
AB
CH
CU
L
J
OTE @
L V AH G@
@
@
SH ] O B K DH U Y
MA
M
I
C
DH S
A
C
F
F
@a
L
AY
AX S @
T
L@
B
L@
W
Y
V
AD
G@
H
NH
L
IH
¯®¬ °±¬
SH
F
Πθ (T ) = S(T ) + ϕ[τ,T ] − S(T ) = ϕ[τ,T ] .
6
-
,
M
B
H
@
L@
E
L
G
I
NH
¤
H
DL @
ª¬
DS
¯ θ(t) θ Π (T ) = Πθ (τ )er(T −τ )
6ê
K
]
W
Z
M
D
ϕ[τ,T ] e−r(T −τ )
C\
A
OK B B ] OB K FU Y ZS O @ j M B I IOD B MH A@ MD OT B H
ASH
Z
H
NH
Ke−r(T −t)
Gp H D@
æ
ASH
Z
SH
DH
@
θ¯ C>0 1
¬ ] i
]
<
¨ ì
R©
C
LD @
A
Πθ (τ ) = e−q(T −τ ) S(τ ), θ¯
g
h
§¨
MA
T
B
H
¬
¡
A@
B
W@
A¡ D S
H
P@ E
D@
I
D
J HF
ZS
n
n
] ª¡
k¡
¡
¡
¶
B
í
k
¥ª
n
a
ª¡
í
¥
R
D ] DN B
[
]
H
F
F
@
B
H SU @
S
S
A BF S
L
L
L
A
C
A
M\
AF
A
M
AH
F
OT K B
ASH
R
θ¯
®
¬
§
ν2
r1 ν1 σT2 + r2 ν2 σT2 − σT2 (rT − rf ) + λσT3 = 0.
Cov(R1 , RT ) = Cov(R1 , ν1 R1 + ν2 R2 ) = ν1 σ12 +
r1 σT2 − Cov(R1 , RT )(rT − rf ) − rf σT2 = 0. (r1 − rf )σT2 = (rT − rf ) Cov(R1 , RT ).
σA < σ B
P
B
L
B
P
H
H
ó
H P@ J
O@
C
V H
L
B
A
C
F
F
U
V
OT K B
ASH
V
L
H
@
H@
H@
K
S0 (t), S(t), F (t) H
M
AD
[H
MH
GF
DB SP
X
B ] @ ]
O
@ IP
S
e h
]
O
C\
@
SH
`
ϕ[τ,T ] = S(τ )e−q(T −τ ) er(T −τ )
G@
A
P
lL @
DCSB
H
NB
qm
\S
L
AD H
GF
I
Dm
Tq H
]
I
H
S
B
@
@
SR
B
B
[B
S [H
M
L
F A
M B
H
AH
M
CH AB
L@
[
B IH
ZS
A
I
\
Y X
B
AL
@ B
AF
Y
DF
AE
D
OD
M
\
MGD D
W
@
A HF
Z
m G@ D
L
@
GU
W
D
G H@
M
A BF
A
E
IH
]
@
CB
B
]
L
W
H
ϕ[τ,T ] CU
L
J
OTE @
L@
M
NB
L
H
AZ
L
J
cB
L
YI
M
L
DM
D
B
Z
CH
A
G@
AH L S
G
C
L
J
F
SB
m
D
\
F
SB
@
] DW
D
G@
c\
IH
ZS
L@
A
G@
AH L S
GU
L
J
E
TH
M
ATZ
AF H
OD
M
¤ G@
I
NH
K
1
A
D
AH
D
A HF
F
B
CH
Z
L
ZS
L
B
E
K
C
L
@
@
B
W
D
D
G@
M
LIJ
I
D
M
D
I
D
AT @
M
IJ
O
MG
D
W
D
G@
ATZ
D
\
G
U
A B
OD F
E
Dn
J
X
V
V D
L
J
M
L S @
AB
W
U
]
M
M
DU S GF
L
JL
B
AF X
H
F
F
F
@a
Q m
NB
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∆w M Y (∆t) = a∆t + Y (0) M Y (2∆t) = a∆t + M Y (∆t) = a2∆t + Y (0) M Y (T ) = aT + Y (0). b∆w
I
M
DB
B
*
* 2
m
A HF
LT
Y
b( j0 , n, p) Cnj pj q n−j .
L
M
SB
F j=j0
n X
AH
M
U
ô
DL
C
L
AD HF
L
Y
AH
E
X
F D@
ZA
J
@
MH SB
L
M
I
L@
uj dn−j S(t) − K .
M
G¡ @ J P
P
H DL @
P
n c
H
D
D
S
Cnj pj q n−j
E L V H
NH
D
Cnj (pu)j (qd)n−j e−nr∆t − Ke−nr∆t
Cnj pj q n−j
OD
OL
\
SB
B
AF j=j0
n X
OL @
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CB
m
@
ET @
M\
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X
c = e−nr∆t
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AZ
J
F
lF
M
MH
j=j0
n X
CD H
I
B
@
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max uj dn−j S(t) − K, 0 = uj dn−j S(t) − K. j0 n ln(K/S(t)d j n−j )/ ln(u/d) max u d S(t) − K, 0 = 0
@
j=j0
Pn @
Y
M
P
j0 uj0 dn−j0 S(t) > K
S
c = S(t)
c
j=0
Cnj pj q n−j max uj dn−j S(t) − K, 0 .
\
lZ
H
D
j > j0 j 6 j0 n X
-
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OF
D
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c = e−nr∆t n
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cuu
c = e−2r∆t cuu p2 + 2cud pq + cdd q , = max u2 S(t) − K, 0 cud = max udS(t) − K, 0 cdd = max d2 S(t) − K, 0 V
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M
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B
H
H
H
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NH
SH
MA B
W
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L
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u − er∆t . u−d
L
D
S
AT S
MJ
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cd = e−r∆t (cud p + cdd q).
E
cu = e−r∆t (cuu p + cud q).
t c
MJ
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B
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NH
A
B
L@
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-
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Π(t) Π(t) = Π(t+∆t)e−r∆t Π(t) = S(t)+νc c = c(t)
JL
L
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L
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W
H
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E
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W
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E
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B
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L
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L
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n
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S(t), c
E
D
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L
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CU
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AF
R
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H IH
DZ
L
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C
H
c = e−r∆t (cu p + cd q),
L
IH
SH
AB
G
NH
NH
AD @
CB
A@
C
B
qm
C
V D
L
J
E
-
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ZS
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V H A
B
j
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t
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K
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S(t)u
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L
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F
L
B
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MH
U
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E
D
S(t) −
] B
O
W
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DH
NH
D
OL @
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YG
Y
B
JU
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B
L@
P
H
H
E
B
M
Dm
L
B
B
CB
C
∂f = 0. ∂t
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B
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L
M
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B
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t
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B
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B
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NH
D
OD
H
AD @
Y
CB M
L@
A@
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M
@
∆t + σ∆w.
P
[H
F
H
S CB
AD
ZS
JSH
H
B
A
A@
cB
σ2 2 ln S(t) + µ − (T − t), σ (T − τ ) . 2 ∆t
B
-
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C
M
L
AB
CH
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A
GL
I
P
O
MH
B
¤
M
L
Mn IB CB î
C
\
ln S(T ) − ln S(t) (µ−σ 2 /2)(T −t) ξ ∼ N (m, s2 ) ξ
N
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YJS
H
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B
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A
U
M
J
N
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U
U
BF
B
L
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C
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MH
LF
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1 ∂2f = − 2, ∂S 2 S
D
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Si (t)
M
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L
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L
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A
B
E
V
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F IB
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B
F
A
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∂f ∂f ∂f 1 2 2 ∂ 2f ∆t + o¯(∆t). ∆f = σS + ∆w + µS + σ S ∂S ∂S 2 ∂S 2 ∂t
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@
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L
J
E
A
MO H
I
B
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l
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NH
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B
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D
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c
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H
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B
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L
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s
H
DZS
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A HF S
Z
H
NH
G BF
E
AD @
* 2
S
∆(ln S) =
\
L
L
B
H
NH
D
AU H
@a
S
∆t
e
F
A HF S
[
B
L@
W
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L
L
S0 (t), S1 (t), . . . , Sn (t), D1 (t), . . . , Dm (t) S0 (t) . . . Sn (t) B
CB
A
∂f 1 = , ∂S S
OF
l
CJ
D
M
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B
G@
j
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L
L
W
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ln S(T ) ∼ N
B
P
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F
B
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NO
m
L
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f (S, t)
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S(t)uu,cuu
S(t)u,cu S(t)ud,cud
S(t)d,cd S(t)dd,cdd
S(t)(u − d) cu d − cd u −r∆t c = S(t) e cu − c d cu − c d
A HF
,
K > 0 1 B
H
B
@
θ¯ A BF
A
c
@
B AB H S
L
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B
B
B
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tx
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L
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θ¯ = θ0 (t), θ1 (t), . . . , θn (t), θn+1 (t), θn+m (t) .
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x
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S(t)u + cu ν = S(t)d + cd ∆.
w
t
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S(t)(u − d) . cd − c u
D
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t + ∆t
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S(t)cu (u − d) S(t)(cu d − cd u) = . cu − c d cu − c d
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t + ∆t S(t+∆t) S(t)u + νcu
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∆S = µ∆t + σ∆w. 3
r = r(t)
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