34
23
%'
1
* 0
*/
.
&
'
-(
%
"'
...
5 downloads
144 Views
591KB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
34
23
%'
1
* 0
*/
.
&
'
-(
%
"'
,
(
+*
!
)
% (
"
%
'
&
$
%
#
"!
hi
@
S
@
F
B
] fegf
V E
D SH F D MH
LI
G
O
O@
MH
Z
¨
¿Û
Á V+Á Æ À Ü » ½ ÃÇ Ñº Ä ÕÄ ¿ Ý ÀÄ ¼ Ë ¾ ÃÅ ½ Ú V+Á À Þ ÃÔ ÕÅ ½ ¿ Ã ß ¹Æ ¾ Ë àÆ À¿ ½ ¾ Ä À á ¿½ ɾ Ã Ä Ì Å½ Á Ú ½¾ Ç ÕÄ Ã Ê Å½ Ä Ë¾ Á V+Á ź ] eff À  ¬ À¾ Å Ë V Ä Å
ÀÁ
] © ¼» ] Ð À¿ ½ ¾ Ì
¬
®
À
Ä ½
Ö
¿
] g°
ÚÄ V °®
À
¾Ùº
Ç
·Í
ÿ Ø
×Ä Q
Ô V]
®
À¿ ½ ¾
ÄË
À¿ ½ ¾ ]V ·
Ð
ĺ
¿ ¾º
Ñ
À ¾ Ë Ò ÀÅ À¾ Ë Ä ½ V Äi V ®° É ¾È à g Ä Á¾ V e ËÓ ] ¼ Ž ¼ À¿ ¾ i i ¿ Ï µ
Q
ÄË
ĺ
¿ ½Ï ¿ ¾º
¿½
ÿ Ê Ã¿ Ê
Î ½¾
É
Ì ÃÌ Å ¿Í Î ½¾ É
] fef
Ð
Ï
e
Æ
¸ Å V ºº
Ä
ÃÈ Ä É¾ Å É ½¾ Ä V+Á à ÀÊ Ä Ë ½¾
½ ÃÇ Ä
Å
Ä V+Á À
¹Â ¹Ã ½
+VÁ
»
¼» À¿ ½ ¾
V ºº
¹¸
] ·©
] e© ] ¸
®¨
® ¨
®
ÃÌ Å ¿Í
Ì
ÕÁ ½ ] ¸
Ä
+VÁ
É
À
°
®
]
¶
³
k V ®° j °
`
] e Q?
V
]
CH
¤
¨
@
H
] GD
AD @
CB
CU A@
H
MH
LF
L@
H DU
G BF
E
AD F
l
AF
MJ H
L
F
MB
Y
S
D
ª
K
¡
]
] ©
f
®
a
`
ª
k
Q?
@
]
LI
G
O @
F
B
cp
]V Z ´
S
`
Q
W
D
D
AD BF
LM
OD B U
k
n
V²? SB ef V f® ] cp ]
µ
]µ
] LNO
@
] R SB ] F
¨
] ¦©
°
˜ X
SR BF F
«
V « ] eff f
[
CH
L
´
Q?
HF
B
H
B
x
¨
R
@
SB
P@
MD H
C
ZS
D
Q@
G
AD @
CB
A@
C
Y
MH
LF
L@
DL
f
f
¬
`
DC H `
] I@ E
]
¦ ] ³´ ©
D ] HX ] B
Z
L
E
z
}
|s
x
wx
tx
w
t
z
w
}~ |
]
D
®
¤
V
] ³
V ®°
S
h
?
] e Q?
]V
@
S @
P
MJ H
LF
L@
D
TAE @
\
D
l
Dq m
L@
E
DL @
MH
L
A
Z
DF
L MD H
I
\
YA F
{s
P
AB
]
O
yx wt z
wv
u
t
ts r
ODL @
@
K
i¨
L
B
V Y
L
D
B
cH
O
ZS
D
B
B
DB
A HF
L
Y
SB AH
M
m
DB SH
A
M
U
Mn IB D B
SB
O` OB
n
] ©
U
p @
H SU @
B
AX ^]J H
L
OT
Y
AH
A HF
C
YF
IO
F
D
B
@
@
@
P@
B
L@ Q
\YM
¦
X
`
X
² V ®° k ° ] °
k¤
ª
V ¯
JL F
L@
D
Q?
] P MH ]
H
HF OI B
D
Z
B
l@
LF
] E
Y
L
DL
c
ZS
\
L
J
I
E
D
I
Y
L
oB
S
JSB
C
ZS
D
L
JB
IB c
@
P
B
B
HF
B
¨
H
[
H
H¢
@
B
B
D
M
IJ H
A
C
L
JL
B
AE BF M
OD
\
C
S
S
@R
CB
o
A
¦
cp
V] ¢
]
K
] ±© ¤
B
GD BF
E
AD
SH
B
AB
L
J
MH
L
L
JB
cH
O
D
O
IP
CH S
I
B
H
F
m
C
LD
JL
B
OF
E
D
NH
L
YC
A
A mF D S \S
B V GU A H JSH D OB O
O
IH
C
Gn @
M
GD BF
E
SB @ ] DC l
B
AD @
CB
A C
ZS
D
ZS
D
V Y
L
D
IB
M
G
\
[
D
`
a
B
S @
F
C @
l@
CG
YM
A
IH
D
MH
ZS
@
SB O@
D
A@
CB
L
Ok P cH
B
]
]
F±
V °
¬ h
i
V
n SR H SH MH
l
g
SB
¤
[
@ª
B
A HF
L
AH Y
YM
DB SH
MH
L
P@
H
a
]V U Q?
]
@
H
P
SH
@
¨
ZS
D
D
DC B
I
G@ U @
H
MH
LF
L@
D
M
CH
MA
CH
L
L
EA
D
ZS
\
ODL @
@
C\ H
F
MH
¯©
V ¯ ]
@ V °® ]
]
B AF M
OT
A
M
SR D@
`
SB
S B
F
[L
L@
DH
Z
\
G
CH
MH
IH
N
\H
OZ
L@
L
@
HF MH
L
O@
D
A@
C
L
JL
]
K
L
D
B
B
cH
O
ZS
D
MJ H
F
L@
L@
D K
ZS
Ll
H
JU
E@
\
O mF
DB SH
A
F
D@
G
AD @
CB
A@ ?
@
IB S b
K
` VG @
W BF YV d F MH ef V] f® Q? V gi
U
H SX @
j
j
B MB
L
D
IB
M
M
S \F
\
GH
Z
W
D
G£ DB
MH
GL
Oj H IH DM H
j
]
]V
MH
G@
CB
] j £
¨ ] h©
] ih F
_
] fegf B
a
b
K
B
K
B
k
b
¤j V d ] eff ¬
H
AF M
OT
A@
IP
V Yd
L
D
B
cH
O
ZS
D
MJ H
LF
L@
D
D@
G
AD @
CB
¡
H
V
] D
@
S
?
Q?
] LNO
¨¬
V
@
H
L
I
DB
GD BF
E
AD F
l
A
Y
DL
M
@ B
\B S
L
J
OT @
W
K
K
j
G¡ F B LI OT @
¡ ] L ¡
K
] ©
DB
L
J
IH
MP
DZS
DB
B
L V Y
[
H
B
B
U
H SX @ P MH D@
L
W
OD
IH
V+P C
F
Y
Q
L
D
m
cH
O
ZS
D
D
DB SH
MA B
L
Mn IB D B
@ ODL @
L
J
SB
M
`
SR D@
ZS
D
Y
DC B
I
@
G@
Y
@
MH
LF
L@
D
Q?
AB
C
R
OT K B
ASH
] R SB
[
B AL F \S
D
MJ H
LF
L@
^]JL A@ ? ] _
@
IB S
V ` G @
B
AF M
OT
A
B
H@
Q P@
ODL @
NH
MH
LF
L@
D
IJ H
A
CB
GD BF
E
AD @
CB
A@ ?
< >
K
<=;
96:
678 5
HF
H
k
X
K
j
V « ] eff f
SB
B
AD BF
LM
D
D
L
D@
MH
LF S
D
L@
WD
M
oD
\S
F
B
GD BF
E
DF
@
OH
GB
D« D D AR ] W
[M
IP
QR
] LF MH
L@
D
DB DSH
A
MJ H
L
ª
`
¢£
¨ ] e©
j
a§
¤¦
a
k¤¥
£
¢ j¡
k
R
@ L B
@
° IH
W
DB
G U R
U
MQ H
rΠ = n X
νi ri .
H U
'(
]
M
M
AD F
G@
V
m
B
V
P
MH
M
AD
G@
Y
IO B
G
I
N
B _
νi H
A BF
C
CH
DC @
AL @
V
V
C
L
J
OTE @
L@
F
<
@
H LT
IH
l
YI
@
CB
I
Dc H
GF
S
] A HF
V
D
M
U
E
F
H U
Bî
@
GH
YA
A
I
P@
L H
AL @
I
H DSH
G
L@
U
\
@
B
H
L
DL
O
B
B
H
GH
AB
L
E
S AD @
I
M
GL
IB
]
L
J
E
D
OD
M
L
U
U
" !
B
H
E@
\
B OF
L
D@
I
cD H
F YF S
B
GF H
@ B
A
M
]
L
J
E
D
H
U MU OD B
L
E@
OF
AD
M
AD
@
G@
YC
A
M
J
A
@ F
[
P
[B
E
AD @
F
m
B
H [H
L
IH
l
I
G@
GY
\S B
B
SB ZS AD
L
A
DL
J
P
L@
CB J
OD
MB
L
U
JU
E@
\
B
OF
AD
E
A@
L@
]
V
]
F
DSB
Z
D
@
M
G@
]V
]
]
MH
M
AD
U
SX HF
Bî
D@ ] L
l
S
JB
AE
B
O HF
Z
G@
AD F
]
SB
AH
Y
TL
D BB
L
E
Y
@ª OX H B MB
n
D
< å
AH
SB
è < â
L@
B
í
P
B
H
L
DE
B
B
L@
L
Y
]V
]
]
L
J
E
D
OD
V GF
[
B
I
D F
CD B
L
D@
I
Dc H
C
GD BF
E
AD @
CB
A@
C
DC
L
JE
L
H
B
[B
MB
L
JU
E@
\
F YO F
Y
A
B
IH
M
M
OD
IH
CU
H
G BF
E
AD @
CB
A@
YC
L
H
F
B
B
] OB K Y
ASH
Z
AT HF
L
IH
l
YI
AY
O
H
H IB
ZS
DH
E L
p A HF S
Z
Y
O
å
Bî
ó
H
@
@
SB ] IH
DZ
B
OE @
L@
M
M
AD
G H@
NH
A
AT HF
C
D
H A mF
L
\E
L
L@ JS
Y
L@
L
I V @ I O D B NH OL @
ì
:
6êë é
]
<
[
S
Y
W
D@
D
]
B
U
@ B
X
]V
]
]
]
H
B
H
@
DZ
SB
B
OE @
L@
W
LD
MG
M
AD
m G@
AT HF
C
D
H
H AF U
L
E
L
JS
P@
@
SB
F
B
@ [P
S
G@
YG
AY
OF
E
D
IH MJ
DZ
L
IH
D
M
AD
G@
A HF
LT
lp H IH
V cB I\
<
åæ
H
B
Q
N
D@
C
\U
L
L
W
Y
L
Ìç
D£
D
I
OK B V CB N
ASH
Z
C
L
J
OT
A@
F
GF
N
D@
C
H \U
L
L
W
H
C
A HF
LT
[H
@p
åè
X
Ð
B
[
V
@
MH
_G
M
F
B
H
F
F
B
@
@
MJ
AD
G@
M
Y
A
M
IJ
O
MG
O@
AZ
D@
G
JSH DC
A
G
MD
D
AF
A
M
H
AH
]
å
< å
R
H
F
B
B
V
]
V ]V
]
]
]V
]
]
H DYC
O
I
D
CH
O@
ZA
D@
G
C
L
J
OTE @
L@
Y
AY
O
IB
ZS
OT K B
ASH
?
@
B
B
@ª
MJ
AD
G@
L
X
H SX @
F
DB
D
q
YM
Y
A
IH
lL @
N
C
\
lL
LJ
W
L
G
JS
B
cH
å
< â
] A
ν1
[H
I
L
\L
V
H
oB
MP
LD
ó V Gn L
MB
IB
S
I
DSH
H CD B
Yq
DYC
HF
AD
G@
L
J
DS R H MP OT H
Ai
H
@
¤
]
G@
M
L@
H
B
SH
AB GF
@
cH
YMO
G
YM
A HF
ä
A1
[E
A
I
NH
D RΠ
C
Y
*+ )
F
B
@
B
B
X
H SH
F
B
F
A 1 . . . An ν 1 . . . νn Ai Π = ν 1 Aa + . . . + ν n An
C
H \F U
L
B
A
M
P
√
L
Ë
J
L@
L
DE
L@
P
H
H
F
A
V
F
AD HF
G HF
Z
O
(σi , ri )
oB
MP
V
G
L
\
E
B
Y
V G
@
P
]
<
Ai Ri σi
M
V
V
MH
AD HF
r Π = M R Π σΠ = (σ, r) A1 An M
D
OD
M
m
B
P
A
M
J
L@
&
%
$ #
< ï
c
Y
F
B
]V
]
R
RΠ
AY
YMO
Y
]
G HF
TA F
A @
] SH
@
O@
AZ
CH
O@
AZ
D@
G
< ð
An ri
OK B
ASH
Z
H
OU
L
L
\U
H
F
H D@
GY
X
Ai
M
AD
G
P@
\
ν = −5 ν =5
Z
AT HF
AL @
\
OTZ AB
G
M
F
YD
OK B
O
IH
ó
c
AD
CD B
E@
\
DZ
U
ASH
A
[H
MH
GF
DB SP
Π
G@
rΠ ]
]
MQ H
OF
C
B
H D P@
L
E
X
Z
Y
OK B
ASH
Z
B
L@
W
DL H@
W
Z
DH
æ
(σ, r)
L
I
ν1 + . . . + ν n = 1 Π(σΠ , rΠ ) DSH
M
OD
IB
ZS
¡
JH
P
U
ì
:
6êë é
< ñ
Rn
B
IH
H
H
AD
G@
A HF
LT
IH
l
D
DYq H
F
A HF
OT K B
<
âò
A1
L
A HF
I
@
B
Y
B
Π
Y
G
B
I
N
M
H
F
B ASH
Z
A
c
l
H
n
V
F
F
@a DH S
A
C
ã
R1
IH
l
@ CB
I
% & D V $ AFHF " H Yq
I
N
B
V
L
DL
OZ
DL
AB
CH
C
C\
C
\F
@
L@
M
AD
G@
<
An
I
l
H
νn A1
CJ B
@
Dc
$
@
e(r1 −r2 )∆t −d u−d
G
]
I
N
]
]
N
= 50 + 2u + 2(4 − u)} = C4u (3/5)u (2/5)4−u 48.13
I
¡
'
V
V
AD
G@
H
< ð
T
c
-
cD H
B
M
AD
A
A
L @ó
< ñ
σT = 16
®,
@
I
N
H
G@
A
A
L @ó
OH
O
\
AZ
G
MH
L
H
MH
L
H
W
W
Z
DH
DH
Z
Y
Y
D
W
D
W
¡
<
âò
V
<
âä
]
<
âã
]
V
ý
ù
ý
þ
ÿ
û^
§
] ®
G
AD @
CB
A@ ?
MH
GL
V
õ
ô
p=
ciT
] °
S
OF
\
Oj H IH DM H
j
]
D
D
G@
I
L@
¡
< â
L@
G@
I
< å
â
T
DC B
I
GY @
Q¢ e
MH ]
]V
O
DS @
þ
n
]
ÿ
þ
W MB ]V
ý
ù
<
â
P (t) P (T )er(T −t) − max{K − S(T ), 0} 6 0 0.56 0.3739 ρ = e−r∆t (ρu p + ρd q) = 0 ν ν 2.5847 e−rT = b( n2 , p∗ ; n)e−r2 T + b( n2 , p; n)e−r1 T p∗ = pue−(r1 −r2 )∆t −1
@
MH
@
AT F
U
¤
ü úû
<
âæ
∆ tw(t) = w(t)∆t + t∆w C ∈ [1.90, 210.13] C0 = 19.29 S(6) ∼ N (10, 25) 6%
LF
L@
`
Q ? A HF ] D
SB AH Y
YM
DB SH
B
@X U
@
] n
ø÷ùö
âä
ri = 0.09 + 700ciT
V Y ®°
]µ
] MH
GL
Oj H IH DM H
H CH
LD
G
Mó
j ] G
]
]
] S IB
¨
] ©
®
T = (0.014, 0.23) σΠ = tσT rΠ = trT + (1 − t)rf t > 0
¬
±
h
P{ i
] ®
j
a§
¤¦
a
k¤¥
£
¢ j¡
k
R
k¶
£
ª
ª§
£
`
¥
a¤
R¡
§
An
Ri
RΠ = ν 1 R1 + . . . + ν n Rn ,
σΠ
i=1
O H U ]
¤
âè
]
<
âæ
â
]
< å
[
]
3 8 Π = 11 A + 11 B Π = 0.25A + 0.75B Π = 0.5A + 0.5B rf = (1 − 0.9)/0.9 = 0.09 r1 = 0.18 r2 = 0.32 σ1 = 0.067 σ2 = 0.14 Cov(A1 , A2 ) = −0.009
r = 20% νA = 1.75 νB = −0.05 νC = −0.7 ν1 = 0.85 ν2 = 0.24 ν0 = −0.09 ì
: â
DE
A¡
F
]
B
B
E
D@
V
V
B
@
P@
OT X
J
ZS
D
B
Dq m
I
L
J
E
D
@
L
G
JS
\
S
@
GF
D
GH
NH
L
AB U S OT @
L
Y
IO
C
MJ
IOD B
M
ZS
m
m
B
L
JL
E B
\H
OZ
AL
DM
U
X
U
Sj @
] NB
D
A@
AF S
lB
DH
Y
IO
OT X
J F
CH
D
L
V A
-
H [H
I
[
B
B
@
@
U
U
E F
Dn
@ J
NB
D
A@
A PF S
DH
I
c
G
M
A BF
q
C\
DZS
L
J
OT @
W
L
H
B DB
AZ
GD @
¢
[
OZ
MJ H
MO
S F
F
@
B
ó
@
L
] W
Z
H
DH
OH
O
G
M
OD
MD
AD
G@
CJ B
M
A@ D S
C
M
N
DL
T
I
AD
AT
Dc H
W
DY
D
I
B
U
[
H IH
l
\S
JL
AB
C
AL F \S
D
L
B
A
D
M
F
MG
YA @
@
F
L@
[H
L
U O K LD @ B ] S B U ] F® D P@
ASH
Z
JL
?
AD
G@
A HF
LT
OT
AD
]
_
LM
k V
S F
F
@
B
@
Q
DN B
D
A@
A BF S
M
I
A
M
A@ D S
C
L
] X B U
O
¡
F
B
@
B
MJ H
LF
L@
D
M
AT
S
H
Dc H
MO
Y
A
D
F
SH
A BF
K V
X
M
@
H A HF
LT
IH
l
YI
@
CB
I
cD H
@
_
[
H L
AB
W
ZS
0.05
P@
V
B
OD
MB
L
JU
E@
_ OF \
V
BF S IB
A
OD
ZS
H
B
H
B
A
M
A
F
A@
A
OT P
\B
S
S [P
A
L@
AD
G@
OD F
O
S
SH
c
S @
B
OZ H
B
W
Z
DH
OH
O
G
L
\B
W
A
OD F
On H @
B DL
Y
L
DL
OZ B
H
F
\B
W
AB
C
IH D@
] H L@
[
H
F
E
]
S CH
G
M
AD
H G@
A
A
L @ó
H L@
N D
D
MH
M
AD
P
G@
l
M\
I
H
H
MJ
P@
[ \ OH
@
Oó G
OD
IH
C
l@
CG
OK B ] B U
ASH
Z
l
CJ
AD F
Z
\H
IH
M
D A BF
OT
A
AH
D
AF
G
@
¤
cH
L
C
I
NH
105 n
@
B
[H
F
<
ï
OT K ] B
ASH
Z
E@
\
OF
CH
A HF
ZS
CH
C
Fn
B
@
L@
W
Y
q
\B
G
@
@
D
¤
SH
B
DH
L
JB
W
\
O
C\
A
IB
D
M
[
W
Z
DH
] \AZ
H O
OH
³
K
-
S
B
SH
cB
C
W
DY
D
O
GL
AB
W
]
DK
<
A
m
Tq H
CH
Z
, h
]
Z
H
ä
J
O
C\
SH
®¬ i
¬ Di
AZ
\
OD H
O
MG H
L
W
DH
@
B
A
[
ó
H DK
_G
MJ
L@
¢
B
OH
W
]
I
N
DZ @ S
O
G
H L@
W
Z
DH
L
JB
W
Z
]
S L@
M
M
AD
G@
H
SH
H
B
NH
A
GL
B LD @ C
E
D
OD
M
CD
L
JU
E@
\
OF
I\
100
I
J HF
H
V
A
OD F
E
Dn
J
Y
] l
D
TA F
R
-
B
,
¥
l
SX @ CJ
H
µ
ZS
M
L@
A
A
MT
V
NB
D
A@
AF S
l
CJ B
I\ B
\
V2
]
6êë é
V
V
]
V1
ùÿ
]
P{V1 −V2 > 0} P{V1 −V2 > 0.25Q} P{V1 − V2 > 0.5Q} P{V1 − V2 > Q} P{V1 − V2 < −0.25Q} P{V1 − V2 < −0.5Q} P{V1 − V2 < −Q} σ = 0.2 OF
,
[
S \S
OD
Y
F
m
]
DI P
V
M
AD F
Z
\H
H
0.08 0.04 Q = 10 000
]
V
C
ÿ
ZDS
D
OD K B
ASH
Z
SH
Y
A
IH
A
L
G@
D
AB MH
W
DK
I
σ r
V
÷
ü
þ
B þ
A
ZS
D
B
B
G
L
AD
@
m
lL @
OD K B
B
M
DK
DB
AB
C
B
S(τ )
K
j
-
,
-
,
] M@
AD
G@
@
- ,
L
AB
C
V
_
M
AD
@
G@
B L@
W
V
_
H
MG @
@
YA F
L@
L
AB
W
ZS
B
H
L
JB
W
Y
L
D@ X
O
_G
σ
V
V
]
< â
U
I
CJ
AD F
Z
A HF
LT
\S
OD
S
SH
AD
B G@ U
M
A BF
c
S
V
V
<
H
N
U
\H
IH
H
B ASH
Z
Gm
H
ó
GH
A
<
-
,
c
ã
A
GR H D@
H
V B A E
e
] ®
2
[
B
H
B
AL F \S
H
ND H
MH
LF
L@
DU
L
W
K
TA X
J
A
cH
C
S −2r/σ r
V
V
@p
< ï
O
GZ
\
MJ U
V
H
@
µ=r
V
]
< ð
σ Π1 < σ Π .
∗
Aª B
< ñ
ä
[
P
B
OK B B U
H
SH
ASH
Z H
σA r B + σ B r A . σA + σ B
<
B
YMJ
OTL
OD BF
A
B
L
CH
OZ
OD BF
CJ
A HF
L
Rj
< â
â
W
L
X
>?)
V
A HF
LT
A
[
G@
K
GK B
H
NU S D@
B
JH
E
A
GL
l
CJ
F
LO @
Y
Y
AD F
Z
\
ρAB = 1
L
G
AT @
ó
L
lSB
V
CH
V A -
,
B
ρAB
G
H
M
A BF
cH
M
U
=
GK
AD F
Z
H \H
I
A
m
D
Ri
\
M
A@ D S
C
L F
F
DH
G@
A
S
D@
YN
M
ACB
E
A
F
B
@
C @
Å
MB
" %
<%
F
B
IH
A
B
V Y
A
M
SH
e
Π1 = (σΠ1 , rΠ1 )
L
cH
F
F LO @
Y
Y
F
B
A BF
AY
lL @
A
MOD
] S
D
@ª
O K ] BP B B CU
Y
cH
L YMO
G
AB ρAB = −1
L@
@
AF
K
σ(r) \@
A
IH
r Π1 > r Π ,
I
SB
E
E
A
#
H
rΠ =
ρij
NH
ASH
OK ] B B U
H
@
P
C
AB
M
Y
@
SH
1
lL @
; %
V A
σ
Z
ASH
J
L@
H
V Gm A
A
B
OT K B
Π+
L
OD K B
.
B
E
Z
Π = (σΠ , rΠ )
Ll
J
M
ASH
Z
U
AB
W
B ASH
Z
~
0.4
D
DL
l
CJ
−1
GZ
K
K
GK AD B
M\
9 6 8 6 s {3u z / t y| t y7 | 0 ~ 8 s 3 w| 1 | w x 75 5x 4 z| t |s x } ~ 8 3 v s ~ | 2
x
w~ z
r
]
U
L
CJ
]
SX @ P
−1 < ρAB < 1
Gm D S
A
\
7
0.3
G
H
AD F
Z
J
M
W
D
DL
G
w|
0.2
\
\
H
\H
AD F
"
:
OD F
0
SH
OK B B U H X H P L@
I
OT K B
*
K 0.1
ó
ASH
Z
\H
I
ASH
:
¥
Π
YMJ
F
M
@
K
GK AD B
*
H
Π−
AY B
σ(r) A BF
M
A BF
cH
L
R
\ 0.1
H
B
Z
Ll
J
M
SX TA @
ρAB ACB
A
cH
L
?
C 0
GL
GK
AD
C
YMO
S
D@
N 0.2
]
a
I
N
O
C\
Y
AY
O
H
B
H
SH
F
B
B
IB
ZS
Y
U
K
OK B
ASH
Z
GF
D
S
L
J
E
D
OD
M
L B
H
E@
\
B
B
OF
Y
L
D
O
F
@
H
IB
ZS
Y
W
DL
G
\
_ó A
U
U
K
FS(T ) (x) S(T )
h
Y
W
D
i,j=1
v uX u n σΠ = t νi νj ρij σi σj ,
g
h
H
P@ E
D@
I
D
J HF
ZS
] ®
n
k¶
£
ª
ª§
£
`
¥
a¤
R¡
§
Π(σ, r)
σ(r)
Fζ (x) = P H
Y
L
DE B
F
U ∗ = U (σ ∗ , r∗ ) B
[H
I
U [H
I
lF
AD HF
L
V
GF W
D
DL
G
\
@
SB
B A HF S
D
D
r∗ M
P
B
L
D
@
K
P
B
H
P@
@
H
L@
L
GZ
AD HF
L
OZ
[
H B
D
G S F
AT HF
L
L
cH
OZ
F
@
DL
G
\
H
G@
G
W
L
SB
D
G DL
\
Y Y
L@
B
K
K
U
S IB
H
AT HF
L
MO
K
L
L
S MH
\
V
A
W
OD
D
H B
\
S GF
D
G
AT HF
L
L
cH
OZ
S
@
B
Y
L
DL
M
\
Y
A BF
N @
@
H
* =
- - & SR Y IB ( OZ Y
A
$
M F
m
@
P LM
J
L@
] SH@
[
B
YD\
&
Z$
Z [
, = L SB @ ) F
Ur0 (σ, r) > 0 r D
$
"
,
L
D
oB
a
& [ Z$ [ & ] ?> ?^ \ [ )' (& \ Z \ (% "X ' Z&
]
N
W
D
D
G@
A HF
LT
V
¥
B
&
%
$
'
=
$
$ %
Y
$
$
=
H
V H A E
L
L
M
A BF
AF
l
MJ H
IH
L
M
S B
@
H
SH
H NH
A
G
GF
S
OK B V DY
V
H ASH
Z
A
_
@
H A HF
LT
IH
l
I
Y
@
CB
I
Dc H
@
H
H
M
H
B
B
B
SH
B
@
SH
A BF
LM
D
L
D
M
A
MO H
I
\
OD
D
AF D
OT
MH
I
\
G@
G
TA @
C
L
DZ
L
cH
C
F
A
AZ
\
H
H
L@
W
Z
DH
L
JB
W
AD HF
C
D
M
D P@
H
F
B
H
B
H
B
W
D
DL
G
\
L
DE B
B
L@
H
] OF
E
DB
L
L
M
A BF
qB
M
M
D
AF
A
M
AH
M
A
M
D
K
¶
[
B
B
H
F
Z@
lL
J
OT @
A HF
Ll
JL
B
CP
L
D
Y
D
L
DL
OZ
DL
@
B
Z@ S
Y
W
DL
G
\
A K
V
[
_
_
@
H
@
AF
AT HF
L
IH
l
I
Y
@
CB
I
cD H
S GF
D
@
H H L@
W
Z
B
H
Z&
'
"X
B
(% Y
A
M\
AF
A
M
AH
F
\SH
A BF
LM
D
C\ H
U (σ, r) V
OT
AD @
DH
O
M
B
L@
DH
V
ASH
Z
Pó
B
H
H
F
m
c
A H B
[B
O
@
] SH A
B
A BF
LM
D
L
D L
IH
D
A
E
Z D
IB
ZS
A
A
D
M
B D P@
L
U
OK B B U
ASH
Z
M
A BF
cH
L
C
\S
] AD HF
L
OT
AH
K
H
H AD @
O
M
A
DCSH
F
JSH
A BF
LM
B
DF
M
I\ B ] A
H
NH
A
B
V H A E
@
D
OD
A
I\
O BF
L
Aª B
OT K B
B
B L@
W
c
L
D
CB
V A
D
D
G@ W
Z
B
@
L@
W
Y
F
B
H
W
Y
L
DL
OZ
F
q
\B
G
AB
o
MJ
CB
E
H
V B A E
A
DH
-
,
I
c
¢
[
B
DL B
W
B
c
æ Gp H D@
@ B
B
F
B
@
Ll
JL
L@
W
Y
A
o
L
T
C
\
OD
Y
D
A
M
E
D
OD
CP
B
M
D
ZS L B
B
D
\
AT
OD
IB
H
L@
W
Z
<
j
B
@
F
OH
O
G
H L@
W
Z
DH
L
JB
W
AD HF
C
D
H
HF
H
D P@ M
ZS
M
L
JE
NH
D
OL @
] lS@
A
C
S
U
@
H
S
B
K
L
M B
H
GK AD B
V óH
L
OT
A@
MH
IB
O
MH
AD
G@
GF
D
D
AT HF
L
IH
l
I
m
C\ B
I
] M
j
[
B
B
B
H
F
Z@
Ll
J
OT @
B
A HF
Ll
JL
CP
L
D
Y
LD
DL
OZ
DL
JB
W
L
DB
DE
OD
M
\
ZS
D
OH
O
G
[
H
F
B
@
@
H
H
B
B
B
H
H
B
@
B
B L@
W
o
MJ
CB
V A
W
D
D
G@
L@
W
Y
q
\B
G
AB
o
MJ
m
@
B
Dc H
A
M
L
DW
H
S CB
P@
Hy
t
\
Y
W
LD
G
\
H
V B A E
A
SR MH T SB
K
L [B
M
@
SH
B
\B S
AY
M
A
MO H
I
I
N B
V
_
@
H
TL
Z
D
MJ
F
SB
[H
C
M
E
m
@
F
JSH
A
L@ G
M
U
~
W
w|
w|
v
u
s
wt x
us
w
z
w
}~ |
2y
t
s
w |z
x
4
t|
~
~
t
23
t
~
|
t zs
ys
5 7 | ~x V 8 yIx ~
t|
~
~
5yxs 5x 4 4 5x ~x M v y s| ~5t t {s 9
w
u
{
K
S L
B
CB D
M
L
AB
CH
G@ AD
CU
L@
H
B
F
B
JSH
A
GL
YM
AY
O
B
W
4 1wx ~ . z z 5t| ~~ s S 7 x9 T8 U w| 44 w~ ~z yw x ~ s t 35 Mt ~
H
]
@
M
L
AB
C
D [ B ] L \ OH
G
t
OT
AD @
O
M
L@
DH
W
Z
DH
Y
L
DL
OZ
F
DL B
W
Ll
JL
CB
E
H
V B A E
A
c
m
Oó
L
D
s
[
B CP
L
D
ZS
D
H
L@
W
Z
DH
Gp H D@
< å
H
L@
W
Z
DH
¡
OH
O
G
MH
GL @
IH
D
JSH
A BF
LM
B
DF
M
E
A
V Y
A
N
OZ
SR IB
B
F
B
@
B
F
B
@
B
B
L@
W
Y
A
o
L
T
C
\
OD
Y
D
A
M
E
D
OD
M
\
AT
OD
IB
ZS
] AD HF
L
V
O
C\
Z
B
SH
H K
B
H
F
H
B
F
@
F
SB
m
U
[
OZ
H
F
CB î
LD
I
YD
L
DL
OZ
LD
W
E
T
L@
B
< â í K
E
D
MJ
Y
L
DL
F
B
(
N
"
'
M
w
9s
y |s
}
M|z
5
L s|
| x
w
Iw~
K
J
t
~
H
I
6 3w
|
x
3x
s|
|
55|
y
5 }
6~
35
F
/G
8
G7
4 |
3x t
~
r
D
OZ
Y
W
IH
W
D
LD
G
\
Y
L
M
A BF
q
L
W
D
K
U (σ, r) = C
H
L@ ZS
B
@
P
@
H
B
,
D
D
OD
-
H
LD
G
\
U
H IH
l
I
Y
@
MU H ZS HF
JB
W
L
DB
DE
B
¡
DH
AZ
\
Π∗
DL
l
I
m
A
M
J
GZ
\F
J
O
ZG
\
D
E
A
V Y
[H
I
Y
IO
B
A HF
L
_
@
CB
I
ZS
] AD HF
L
E
G
K
@ C\ B
I
cD H
U
\U S MH
@ AT HF
q CB
H U MJ
L
L
JF
L@
SB
B
B
B
AH Y
M
L
D
O
IB
F
@
@ª U
σ
\
CB î
L
L@
OD BF
L
G
\F L
D
OD
I
M
DU
L
EJ
Z
LD @ S
j
B
CB
) [ "
S ZS
AD
L
JE
NH
D
OL @
OD
M
\
ZS
D
AZ
\
(σ, r)
L@
V
@
ß
D
Y
cD
¡
B
] lS@
A
C
@
a2 r − b2 (r2 + σ 2 )
P
)
S GB
O
A
e
F
B
r(σ) U Uσ0 (σ, r) < 0
NB
]
B
A
@
U
_
H
] AD HF
L
OT
AD @
S
E
qD m
B
Π∗ C
V D
P
B ] AD HF
L
SB
, L
m
B
@
N ] IH
M
I
P@
H
B
]
]
W
D
í
O
M
(a)
o
&
'
R
OT K B
ASH
Z
σ
OT X H
L@
DH
Y
F
B
@
D¤
G ] \
L
DL
M
@
SH
B
\B S
AY
M
A
MO H
I
L
L
E
\H
OZ
A HF
LT
D
G@
L@
B
è
l2
NB
H
A
M
Y
\
V
H
<
r
F
B
@
QL @
W
D
D
G@
í
W
D
D
G@
OT
AD @
O
M
c = e−r(T −t) St er(T −t) Φ(d1 ) − KΦ(d2 ) = St Φ(d1 ) − KΦ(d2 )e−r(T −t) ,
M
A
P
S(T ) −1<x = S0 = P S(T ) < (x + 1)S0 = FS(T ) (x + 1)S0 , N
F
@
L@
M
S
L@
(S(T ) − S0 )/S0 =
H
V
E
∆S(T ) = µ∆t + σ∆X. S(T )
J
D ] S
M
@
Fζ (x) 15%
OZ
S
B
B
B
U
ζ
f
T f = S −ϕe−r(T −t)
L@
Y
L
D
O
D
OD
YM
L@
E@
\
O HF
B
L@
S(T )
IB
ZS
F
@
K
_A
ó
R <ì
AD HF
L
IH
l
$40 30%
YS
W
LD
G
\
S(T )/S0 − 1 Bî
¥
AT HF
L
Q 6
OP 6ê
t ϕ
AD F
]
lp H IH
S = S(t) t
r
g
h
h
] h
?
@?
B
W@
MH
D
GS
OT
IH
®
] ®
£¶
[ ¦¡
¢£
£
p
¥
?¡
§
_ a b
(b)
Π*
l1
σ σ
U (σ, r)
σ∗
_
C
I
N B
V = Ri
e7
xρ(x)dx = e
M max S(T ) − K, 0
σ2 (T −t)+ln S + r− σ2 0 2 2
(T −t)
Φ(d1 ) = S0 er(T −t) Φ(d1 ).
v
1 Φ(d1 ) x
2xp
5
5
√ s=σ T −t
z
/S
2 | 4
8
e7 c | d}|s o 5M|9 3xz
f
/G
8
e7
|
3t5s
6
s
m
t
~
3w3
z
x
9 ~
~x
5
tx
2 |
t
w~ z
I I| w
~
t
|
s z }x
I
6w
~
t
wx
}x
5z Hy 4
~x
tx
5 Ms
}x
at z
|s
d|
6
~
yw
sv
|
u
{
t
t s|
9 y z
~
f
l
k
j
i
5z
~
I
5 W5
tx
}x
s
k
d}
~x
5tx
}x
|
x
z
sv
5~x
t
x
z
x v
~
z
}~
~
3x
~
}t s
1~|
3
y~ z
Ix
w~
xs
F M
| | x 3 s 2z
U
w|
|
35ts
tx
3wx
~
3 ~tI s
3ws
x
~x
l
f
h
g
ln S(T ) − m K−m √ > √ σ T −t σ T −t
8
0 1 − Φ(−d1 )
tx
}x
ζ2 2
z
e−
~
v }x x
u
1t
~
~z
Iw
s
v
ρ(x)dx = KΦ(d2 )
/e
e7
z
−d1
∞
1
x
s2 +m 2
z
dζ = e
5~
+m
|s
2
t
~
|
s
5
dxs
w
tv
u~
x
3
w~ |z
M
35ts d}x
1 − Φ(−d2 ) = Φ(d2 )
}~
eζs+s
~
f
d}|s o
f
m = ln S0 + (r − σ 2 /2)(T − t)
s 3~x 3t5M
3
ζ = (ln x − m)/s − s u
//
e n
F
z
w
}~ |
u
y z
ln S(T ) − m √ > −d2 σ T −t
|
u
f
z
| v
Y
W
5yxs
w
{s
~|
y
5M|
wx
m = ln S0 + (r − σ 2 /2)T √ (ln S(T ) − m)/ σ T − t 0 d2
~
x
35
M
t
`~
|
z w |z
ρ(x)dx = K P S(T ) > K = K P
−d1
|
(ζ+s)2 2
t} s
t
z s
{
1 tt3x wx x
y s|
5s
W
[
6
4 ~5 x
w|
x
u
K
s
}
K
M 3 6 d|| s 1~ | wI t t
e−
∞
w |z
|
d~| 5sx t| x w 3 9s dxs t3 x
t
K
w
K
y~ z
[B
xρ(x)dx =
l
L [H
K
∞
w~ |z
@
A
IH
e
t
K
wt |
∞
x
xρ(x)dx =
_ zs w w3 s xs w3 x ~ wt | I x
f
6|
Y
∞
x
5M
tx
F
q
\F
D
lL @
Å
t
Φ(x)
t
s
f
w
s|
[
C
K GK AD B
ó
F
s
x
z
ρ(x)dx = K P
|
k
Is
dx ~5 x s ~t 5
w
~t v
W5
} x
K
G@
@
X
D
LD @ ] S
{
∞
6 e 87 hl ~ /G 8 e7 /e 8 5| Is x 6 3w I
/S
∞
sv
∞
It
s
5z 5t}zx
[B
S
N
C
\
P
B
H AD HF
L
OZ
t
K
9
~
F |
[ K
r∗
IH
D
L
_A
M
lL
B
B
U
OT K B
ASH
Z
t|
9
7 yw 4H Hy 4 T8 ~ y ~~ w~ | ~ ~ 3s a 5 5 | t5M 5t| tx9 I x9 u w~ |z v b w| ~ z ~ w| 5 4 a L 5x Is M ~x9 s
x w |z
~
~
G
σ
s
[
ASH
K
JL
W
L
K
L
J
OT @
C
Z
AD
6 ~
yw x
z
F
g
t
[
ó
@ó
GK AD B
C
J
B
¬
I
NH
G
W
D
LD
G
\
Y
L
S MH
\
L
~
~
~
.
a y~ _78 s 44
|
~
Π*
6 ~|
U
H
@
F
V Å N I
] S
D
OK ] B B U
¡
~x
5 t
5~t t sz
ys
Iys
l1
j
i
∞
5M
H
l
M
¤
L@ JS
D
D
OK ] B B U
s
s
~
t sz
|
s
K
tx
9s
y s|
f 6y 5 s Mz 5Mq| L 6x Ix 3
w s
L
ASH
Z
Z
,
]
H
$
ODY
L
ASH
Z
ASH
Z
wt x
ys
~
yw
T
k
|
-
@
Y
A BF
cH
L
CH
NH
L
M
H
GP
B
D@
F
@
|
|
x
t
s
5
dxs
w
| t~ v
w
/S
8
e7
c
(b)
Iws
yt
T
¬
L
IU
K
GK AD B
Er
V
,
→ − r ∗ − r0 − −1 → → − T −1 → → − V ( r − r0 I ), → − − ( r − r0 I ) V ( r − r0 I ) ] ®
H
M
MH P@
I
)
&
"
F
B
@
P
S
T
L
OK B V B
OK B
ASH
Z
Y
A
M
ó
H
L V H A E
B
@
G
l
w
9 Iyx a w
us
y |s
Π∗ [F T )
E
B
@
CB
I
M
AD
G@
F Dc H
OK B B FU
Ll
JL
AD HF
L
IH
H
U
Y
F
OT
AL
DH
M
A BF
cH
L
B
CJ
AD F
Z
\
Er
CB
B
ASH
OT K B
l
I
H
J
L@
YA
B
M
U
H
"
E
GH
MA
l2
D
ASH
L
D
oH
L
@
MH
ASH
Z
@
CB
F
B
@
F
@
AY
AD
G@
OZ
I
C
cD H
H
CH
NH
L
FT F
OD
Z
M
MH
B
GF
S
G@
YG
A
G
YMO
MJ H
GF
X
OZ
:
* 2
*
r
M
Ll
JU
E@
Z
F
D
OD p H
I
E
OK B B U
ASH
Z
B DB SP
A
F = (0, rF ) FT
H
AH
l
M
S IB D
F
F
cD H
OT K B
SH
H
CH
M
A BF
SR IB
T
M
AD
F
m YF
YA
B
A
IH
U
ASH
@
OH U \
A
MG
E
GH
M\
AF
Er
\
G@
Ll
@
O
IB
GF DU S ] CH GF
H
R
V
c
σ
OF U
W
JL
MH
ZS
H
L
MH
] M
AD
G@
@
T
D@
MH D@ S
r ∗
GF
V
D
lL @
cH
X
lL
LJ
MH
CB
A
FT
G
S
S
OT K B
MU H CP
H
m
SR
G
O
F
D
D
C
B
ASH
Z
U
O
D
]
(a)
W@
H
→ − ν = (ν1 , . . . , νn )T n Cov(Ri , Rj ) i,j=1 OD
→ − ν = DLo
T
C
L@
r∗ D
L
J
M
GK AD B
Π∗
_I
D
m
A HF
LT
T
Er FT
A@ D S
OK B I V DB @ C CB B H D I U qm l H IH DU L@ L AT HF C T B DLo m
cD H
r
¬h
±
]
@
SH
h
`
k¶
£
ª
ª§
£
`
¥
a¤
R¡
P@
@ [ \ OH
Oó G
O
C\
e
] ®
§
,
= K 1 − Φ(−d2 ) ,
K
(ln x−m)2 1 − 2s2 √ e dx. 2πs
√ ln K − ln S0 − (r − σ 2 /2)(T − t) √ − σ T − t = −d1 . σ T −t
−d1
dζ.
K
= S0 er(T −t) Φ(d1 ) − KΦ(d2 ).
e
M max S(T ) − K, 0 = K ∞
x(ρ(x)dx − K ∞
S(T )
ρ(x)dx.
αi
-
Cov(Ri ,RT ) 2 σT
]
[
B
B
U
X
L
AB
H
H U
Bî
B
OT K DB
ASH
Z
L
JE
L
JS
M
DU S GF
L
J
N
C @
@
\H
E X
S
HF
B
L
JE
L
JS
M
H
GF
D
A
DL
M
L
U
H
@
X
[B
L
YF
X H
B
H
G BF
E
DLC H
G
] M
AD
G
H@
NH
MH
GF
DB SP
AD HF
L
V A
-
H
V
H
B
@
H
@
A HF
LT
IH
l
I
Y
CB
I
cD H
E
A
A
I\ B
OF
] P®
D
h
-
,
S
H
F
B
B
H AT BF
A
OK B CB
ASH
Z
C
L
JE
L
JS
A
B SH \S
OD
LG
M
~
|
,
]
V
YC
MG
SR
JS
H
AD F
_G
I
N B
=
=
"
&
_
H
@
H
Y
A
M
J
F
B
@
P
L@
A HF
LT
IH
l
YI
@
CB
I
cD H
OK B CB
ASH
Z
C
L
JE
L
[
Y
HF
@
MH D@ S
G
QY
l
lL
J
AL @
I
H DSH
@
H
OK B
] Y
ASH
Z
NH
L
E
L
H
M
AD
D
@
G@
AD HF
L
IH
l
YI
W
D@
MH D@ S
H [H
L
IH
l
I
Ai
W
ZS
H
MH
GF
DB SP
o
L
T
F
B
@
B
OT
AW @ P
@
H
A
CH
H
IH
l
AD
G@
\
YF JS
W
D
r8
e7
] ® h
-
,
/G
G7 y8
35
3
4 |
3~x t
rT − r F ri = r F + Cov(Ri , RT ). σT2
g,
Ri = αi + βiT RT + εi , ] ®
βiT =
L@
E
C
Y
A
M
J
V
A
M
@
OD F
)
$
e7 2 x v s/ 8
Ri
CB
L
B
@
D
B
F
B
@
G
V H OD BF
NH
A
G@ K
Ai
S
OD
YM
A
M
P
HF DH S
A
A
DL
O H A BB
I
L@
@
¥
3
t
~
l
hl
4
~
u
35
|
s
w z
V
B
Ê
?
B
@
SB
M
AD
@
G@
H
m NH
OY
O
IH
E
A
A
I
c
M
\Ì A
OT
IH
X
Þ Ü
]
V
³
h
-
,
@
@
O@
AZ
D@
G
)
\
)
$
F
B
@
L D@ ª S H EJ ] F L C ¬ A Z H S ASH B O K G PH B ] CB
_ JS
]
Y
A
M
H
F
B
L
D
B U
H DO U
L
E
L
YJS
AY
YMO
FT
DL
M
\Y
F
B
B
J
L@
AT HF
L
IH
H
M
H
NH
M
S
2 wx
~t
4 wIzx s
| w z v s
z
Cov(Ri , RT ) RT T y = rF + (rT − rF )/σT2 x T
AY
O
IB
@
Ai
H
l
I
Y
ó AB
L
D
s
}|
Ai Cov(Ri , RT ) = 0 ri
ZS
&
M
AD
G@
@
] OT B K
ASH
Z
3x
~
t} s
3 ds ~1 |
z
|
~
~
w z
wx
g
9s
5Mx
w
T
Y
M
@
Ai
CB
I
AD
G H@
A
8
s
t
s
s
z
w ||
|
y |s Y
L
DL
OZ B
H
Q
L
W
Z B
B
H
H
@
SH
F
L
JB
W
O
C\
Y
A
IH
D
lL @
L
E
NH
D
OL @
K
H
[
B
U
P
B
L@ J
OD
IH
C
M
Ì
Ê
OT K B
ASH
Z
L
J
OT
A@
F
@
Þ Ü
[L
OK ] B B U¢
ASH
Z
M
A BF
c
F
B
CH
C
AZ
\
H L@
W
Z
DH
&
'
P
"
&
\
óm
@
~
) !
K
H [H
L
CB
L
M
GK AD B
A
M
B
SH
}|
DH
L
JB
W
O
C\
B
Z W ] H L@
L@ J
H
cU
A
D
K
M
AD D
H
@
] D BF
D
S
F
MH
L
M
YM
A
IH
D
lL @
GL
JS
l
l
a
B
L H
IH U
V
[
H
S
S P
@
@
GL @
IH
D
JSH
A BF
LM
B
DF
MH
E
A
CH
A
YM
F
m
B
G
@U
MJ H
GF
DB SP
A
m
@
B
@ B
CJ B
I
cD H
A
M
L
DW
H
H
V
A
E@
A BF M\
H
S
B
B
\
L B
B
oB D
AY
O
IB
ZS
L
DL
M
S B
@
DM H
D
OF
\
ZS
D
] Y
MH
X
[H
C
M
lL
JL
@
MH
IB
O FF
D@
IH
Il H
Z
Ll
JE
OD
M
q
\ BF
L
G
L@ JS
MH
M
AD
G@
GD F
D
D
AD HF
L
IH
l
I
@ [ \ OH V P@ D ZS M IH A Y G
G
Oó
U LD @
LoA
,
¤
H SX @ P CH
C
GD @
] h ±
]
Da V X I
@
F
HF
@
H
B Y¢
O
I
[\
[
F
\
] MGD
AF U @
cD H
//
e7
s
w
m
) !
(x − K)ρ(x)dx, H
|
0
L
I
t|
(ln x−m)2 − 2s2
w
s
w |z
c(t) max S(T ) − u
u
ξ
s|
v
k
}
F
DC H
A
K
¦ ¡ ¤ ¢ £ ¥ ¦
g
}
t
z
S7
k
,
8
I ~t
5~x ~
H
|
3s Iws ts z | v xs M w3~ w U H ~5 x ~t I d | | I y~xz |s }x
Is t
z
6 u8 x ~5 z
wx
}s
8 c(t) = e−r(T −t) M max S(T ) − K, 0
/S
K
|
u
s
|
z
t
~
H 5 r |
Φ(−d2 ) − S(τ )Φ(−d1 ).
e7
= ∞
v
55|
1 e ρ(x) = √ 2πsx
|
v
y
u
~t
|
t
yz
r
P (τ )
M max S(T ) − K, 0
Is
x
ρ
t
t
z
|
x s
ln S(T ) ∼ N ln S(0) + (r − σ 2 /2)(T − t), σ 2 (T − t) .
µ
f d}|s o
f
x
5z
H
õ
}x
~v
` 5wt t5xsx t Iws v yss9 xs ~ z | t t z e 7 3~ x w |z w |z M | qs r wM 8 3s 5z 5tM s a ts w
d}|s o
z
x
55|
y
−r(T −τ )
hl
~
~x
v
s
d}
gh
-
,
f (S(t), t) = c(t)
u
f 5~x ~t I
t}
x
t} s
x>0
v
v
x
z
w
5 c t5M | t F | 3 5 wz 6 u M yw ~I5z d|| s | t y e7 | ~ x ~ s / s |
ds
P (τ ) = Ke
|
t
g
k
x
s2
Is
5z
M
~
z
}~
~
T
hl
|
K, 0 ~u h I~5z d| W w w 5t s s ~ y M z 6 w3s | ts e7 xs ts w3~ 6 r 8 } Ix 5t 5v yz {s ~ | x 1 | 1~|
~
ln ξ
w |z
5wt
a
ρ(x)
g
k
x
B
H
óB
H
[m
O
G P@
Y
L
B
cH D
OZ
IB
ZS
B
GD BF
E
CH
LD
G
L
J
MH
L
HF
Bî
] C@ N¥
\
Ll
JL
W
X
-
,
H
B
L
G B
L@ JS
DC @
M
AD
G@
I\
P
HF
P@
B
@
cB
C
Y
T
M
C
D
M
A
M
JF
Z
D
OT
IH
Ì
Ê
?
Þ Ü
*
*
"
w
:
z{
xy
l
H
V
]
M
AD
@
G H@
NH
MH
GF
DB SP
TLA HF
IH
_I
ln(S(t)/K) + (r + σ 2 /2)(T − t) √ d1 = , σ T −t √ ln(S(t)/K) + (r − σ 2 /2)(T − t) √ d2 = = d1 − σ T − t. σ T −t X
I
N B
V
_
@
H
B
MH
M
AD
G@
Ll
JL
@
MH
S GF
D
AD HF
L
IH
l
I
CJ B
I
cD H
→ − r = (r1 , . . . , rn )T r0
i
h
d}
] h
OT
B
IH
¬
] ®
Ì
Ê
?
Þ
£¶
Ü
[ ¦¡
¢£
£
p
¥
?¡
§
→ − I = (1, . . . , 1)T
Ai
Cov(Ri , RT ) < 0,
βiT
K
h
σΠ = q 2 + 0.752 σ 2 + 2 · 0.25 · 0.75 · Cov(R , R ). 0.252 σA A B B
]
¨ i
)
1
0 ]
]
A
I
F
B
@
Z@
¬
¬
-
, V
H
@
E
A
A
H
B
I\ B
OF
M
B
@ AL F
M
S
ϕ[τ,T ] = S(τ )er(T −τ ) Y
L@
L
F
U
L
J Z [H
D
H MU OD B
L
E@
\
OB
L
D@
I
cD H
B
H
G BF
E
AD @
CB
E
F (τ ) = S(τ ) − ϕ[τ,T ] e−r(T −τ ) . P
V Dm
OL
\
C\ H
AH
Zó
A@
C
] B ¬
®¬
-
,
L@
M
SH
H
M
E
T
B L@
W
] B
O
C\
Z
L V H
¬
®¬
K
-
,
@
F
SB
m
K
@ MGD
A HF
Z
L
AB
CH
C
]
S @
@
F
SB
m
L@
M
E
T
B L@
W
K
L
AB
CH
CU
L
J
OTE @
L@
[
@
MH
IB
O j
S(T ) − ϕ[τ,T ]
B
@
K
V
H
K
B
MH
M
AD
G@
F
@
F
SB
F
@
m
H
B U
S
] GL
JS B
GF \
D
G
CH
L
AB U S OT @
L
L
I
c
\F
S
Tq H
CH
Z
E
m
n
SB @
B
BF
L
JB
W
\
O
C\
SH
CB
IB
M
MJ
C
ZS
D
M
EA
Gn @
K
[
T
MGD
@
A HF
Z
K
ϕ[τ,T ]
E
S
@
F
SB
m
L@
M
E
T
U
L
O
C\
SH
@
OT
A
F (τ )
Y
OK B
ASH
Z
L
DL
M B
@
SH
B
\m S
AY
M
A
MO H
I
M\ H
L
AB
C
AL F \S
lD
MJ H
LF
L@
] m ]
®¬
K
,
V
-
H
S GF \
D
G
L
J
AB U S OT @
L
JSH
A BF
LM
D
E
A
¬ V D
V
AT @
AE
H DF
L
cH
C
AD HF
L
D
L
AF
E@ JB
W
[
@
c
L
DZS H
]
D
L
J
M
@
H
B
B
G@
B
A HF
L
IH
l
I
Y
L
O
OT
B
H
F
B
j
B
SH S ] G\ F
D
MH G
Y
n
@
L
B
L
OT
B
MH
B
F
H AT @
O
I
L
cH
C
S
H
S B
@
B
@
CB
Am
A
I
@
OH
Y
L
DL
M
Y\
L
oB D
D
V A
GF \
S
D
SH
A BF
LM
Y
LD
D
oH
A@
B
CJ B
I
MG H
U
B Dc H
A
OD F
E
IB
O
CH
M
H AF U
M
V H D H ZS AX H D Z L H oH GF D DB L D SB oB D LM B L A BF
L
cH D
OZ
IB
ZS
L
J
H
MP
,
J Dn
] S
]
H IH
l
@
D
MH
I
AD HF
D
MH
M
AD
G@
AD HF
L
M
AD
SK H V AU D lH
D
U
H IH
Å
F
@
X
W
D
IB
ZS
S
H
SB NH
L
C
M\
I
W
\
OD
AB
A
M
T@
A HF
j
l
I
D
L
J
M
S
X
B
H
@
B
B
H
A
I
Bî
Z@
\@
AL
C
OD BF
V H
L
L
M
AF
A
M
AH
F
SB
S @
@
[
A
A
Y
A
M
B
H
H
F
D P@
L
P@
[ \ OH
G
Oó
L
DB
DL
M
\
B MB
L
JL
CB
Z
G@
G
G@
¤
H
N
A
I
H
G@
AD HF
L
IH
l
I
H
CH
SB
¥
X
L
DL B
] gh
] B
Bî
Z@
V A
[
B
H \@
AL
C
OD F
AL ]^J B
C
CB
L
SX HF D@
]
D
MH
M
AD
P
H
H
G@
l
M\
I
D
G
OT
AU
YqD
-
,
H
Y
AY
O
F
B
B
IB
ZS
H
A HF
LT
IH
l
I
l
BF
L
M
B
A BF
q
\F
IT
@
H A HF
LT
IH
l
YI
@
CB
I
Dc H
H
L V H A E
k
¶
[
M
M\
A
S @
IH
D
Ml B
L
µ
OR \H
B L@
W
L
AB
CH
C
L
J
OTE @
L@
M
@
S(τ )er(T −τ )
-
³ , ´ ;
)
!
F MH
M
S
S
T
P@
[ \ OH
@
NH
L
G@
S(T )
G
@
B
B
L
H
rA = M RA = 0.1 · 0.5 + 0.2 · 0.5 = 0.15, rB = M RB = 0 · 0.5 + 0.3 · 0.5 = 0.15. R
τ
G
¤
B
E
G@
U
S GF
D
D
TLA HF
Π = 0.25A+
O
C\
\
0.2 0.5 0 O
A Q
B
0.75B
* 2
SH
K
D
MH
0 0.3 0.5 0.5 RA RB Q
D
A HF
L
H IH
l
Im
]
B
@
rΠ = 0.25rA + 0.75rB = 0.15,
²
L@
D
L
H IH
l
YI
B
RB
*
Π = 0.25A + 0.75B M
M
AD
G@ U
B
A HF
L
RB P RA
@
0.1 0 0.5 E
B
0.2 0.5
D
OD
M
lL
L
D
B
@
C\ B
I
D
MH
M
,
0
Oó
O
C\
SH
`
S
H
B
JU
E@
O
X
cD H
B
ß
0.2
IH
l
YI
L
B
F
@
IB
ZS
S
@
A
OD F
E
Dn
J
AD
B
-
Y
RB \RA 0 0.3 D
O
\
O BF
W
J
OD
¬ « ª ] ®¯¬ °±¬ , Ť
,
0.3
OK B
F IB
ZS
Q
Ë
B
ASH
L
D@
I
B
Dc H
GD BF
E
AD @
CB
0.1 0.5
@
SH
X
-
D
A
Z
A HF
LT
NH
L
SB
¤
A@ ?
0.1 A
,
H
C
\M
I
W@
OD
@
,
RA P
IH
l
YI
@
ß
¨
Sm
F
@
B
H
[H
A HF
V N
C
\
lL
JL
W
GL
X
B
H
H
B
G D
S
MH
SH
A BF
LM
Y
D
L
D
oH
L
A
A
MD H
SH
A BF
LM
D
L
D
AH
Z
U
H
H
Y
A
M
B
H
F
D P@
L
L
!
Fn
@
@
DB
] MG
AY F
L@
L
AB
W
ZS
D
AT HF
L
OT
AD @
O
M
V Y
M
@
E
V
M
AD
S CB
H IB
ZS
A
] G\ F
JS U
L
J
A@
L
D
DC
C
DH S
A
C
F
F
@a
< â
ì
R©
§¨
B
@
R
[
B
@
G@
L@
W
Y
q
\B
G
ó AH
_A
P@
@ [ \ OH
G
Oó
L
DB
DL
M
S
M\ B
L
JL
CB
SB
V
[
S
P
H
H
B
AB U S OT @
L
JSH
A BF
LM
B
DF
MH
E
A
A
E@
L@
GL
JS
GF \
D
GU
L
J
D
] G\ F
L S
-
,
AB U S OT @ G
L
J
±
] ®
Cov(Ri , Rj ) = βiT βjT σT2 .
i
@
¡
, A
CB
I
Dc
Ë
B
A H
OD
B
IH
H CU
L
E
L
JS
Y
l
MH
D
S
JS
\
H
lL @
DCSB
A
l
GD BF
E
DLC H
G
ó
S GF
D
G
C\ H
L
AB U S OT @
L
@
B IH G
l
] G\L
\n OF
n
F
H
F
B
Þ ] ÊÜ Ì
YF
A
\F
@
F
B NO H
OT
IH
YC
L@
E
L
JSB
OF
CJ
MJ
H
@
N
D@
C
\
A HF
LT
IH
l
I
Y
CB
X
F
H
B
B
Z [B
CH
IH
A
C
Y
l
L
D
cH
O
ZS
D
l
DG BF
E
DLC H
G
Mó
Y
C
Dq m
E@
B
H AF S
M
AF
E@
]
SH
H
F
B
h
CH
An
ó
] B®
O
C\
Z
Y
AY
OF
E
D
K
-
,
Ai
] ®
_
GF
DH S
NB
@
A B
B
B
p
P
H
HF
P@
Y
F
CB
\
OTZ
M
@ [ \ OH
G
Oó
Y
L
DL
M
\U S @
L
D
oB
YS
L
D
I
cH
lL @
Y
O
@
B
I
I @
cD H
@
NH
B
D@
I
Dc H
H
G BF
E
AD @
CB
A@
CB
H
MB
OL
\Y
] L
[
B qm
C
D
¤
OK B DC B
I
N B
V
*
*
[
F
S
B
@
H
F
ASH
Z
C
L
JE
L
JS
Y
qm @
B
SH \S
OD
GL
G
Do H
Y
L@
E@
\
_O
X
:
²
;)
;
(
(
U
εi
®
³
]
\
S GF
D
G
A HF
LT
AB U S OT @
®h
] ®
ª
k¶
£
ª
ª§
£
`
¥
a¤
R¡
§
= 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
¢
X
¯®¬ ±°¬ ¦
ª¬
]
\
m
K
AD HF
L
c
\S
H
B
H
NB
qm
C
@ V D
SH
B
0.0089 = 0.042 0.2009 ]
]
V B
B
O@ p
det V = 0.9999
N
D
_ I\
@
G
AY
0.0089 0.2009 ]
MH D@ S
¬ ¢
K
-
,
]
Y
L@
SX A@
¡
Cov(A, B) = 1 0.01 0.01 1 Y
W
D@
-
,
K
F
B
B
B
H
m
[P
O
B L@
W
G@
YG
AY
L
CP
Y
D
U
U
@
U
@
P@
D
I
I
M
D
YM
A
M
B
F
B
@
JF
F
CH
S GF
D
C
L
J
OT @
C
L
D
DC
Y
OK B
ASH
Z
B
cH
lL @
Y Z
@ E@
I
@
¯®¬ °±¬ ¶
ª¬
D P@
AD HF
L
H IU l H IH
L
L
I
ZS
D
L
B
G X L
L@ JS
c
AX S @
DS
D
MJ H
GF
DB SP
X
OT K B
ASH
Z
B
X
OK B
ASH
Z
L
JL
H
A HF S
] h
h
]
-
,
AD HF
L
IH
l
]
W
YD
D
O
LG
S
B
SH
AB
W
DK
B
B
U
@
D
M
AD
@
G H@
NH
MH
GF
DB SP
D
X
F (0, 0.09) IU [H V
H
CB
I
cD H
ZS
D
DU S GF
o
LD
T
C
L@
D
qD m
C
¢ V D H DK
ó
¨ ì
R©
§¨
H
P
[L
@
MH
S GF
D
l
M\
I
DY
AH
A HF
F
B
JSH
A
OK B B V GU
ASH
Z
OT K B
M
AD
G@
Ll
J
]
Sj D IB
-
,
< å
H
¬h
-
MH
B
E
A
C
E
D
\H
OZ
M
ASH
Z
A
IU
D
L@
V
V
]
®h ,
-
]
De
, e
Y
] Y
@
¬h
-
,
]
D
B
S
P@
Y
S
@
CB
M
Y
OK B
ASH
Z
L
JB
W
Y
L
DL
a
L
M
H
O
j
SH
H
] L
M
S
B
@
] B®
O
C\
Z
Y
i
K
-
,
OK B V
ASH
Z
GF
B
V H
L
OT
A@
MH
IB
SB
F
B
] I
c
G
B
H
@
m
A HF
LT
AH
Y
M
YD
L
DE
P
B
@
L@
A
C
L
D
ZS
D
L@
E
D
] e
h
-
,
F
B
B
[
OD
YM B
L@
E@
\
O HF
L V H A E
I
M
DY
L
D
O
IB
ZS
@
SH
SB NH
L
C
\M
I
W
J
OD
U
AP k @ X
B
@
L@
W
I
NH
@ L@
M
YS
OK B
ASH
Z
]
H
@
C\
C
\F
L@
W
D
D
B G@
A
Z@
\H
GZ
N\D
C
\m
L
\
IH
MP
DZS
S
@
OH
O
I
L@
G@
D
X V
@
O
I
MO @
ASH
Z
MJ H U
D
P
AD H
D
A HF
LF
OT K ¤ ] B
H SX @ P CH
C
GD @
] M
AD
S GF
D
¡
H
@
H @
F
B
H
M
C\ H
AH
óR
H
AF
A
M
AH
] LB L
D
DC @
L
A
AL F
GY
@
B
E@
\
OF
CH
L
L
I
M
D
L
J
E
D
OD
B
MB
L
JU
E@ F
X
\j
O
N
D@
C
H \U
L
IH
YA m
B
H MP
DZS
W
J
L
D
IB
YMO
A
IH
ZS
Y
OK B
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\
Z
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|
tx
T
L x {v L } e 7 w w xz ~u 3 u ~x ~ µ 8 t tx z c 3wMx 3wxx | 9 | |s9 z s| L 3 } s 1t| 3wxz ytxz L ~ s 3 | w 6 | tx }| 6 t} | ~M x I ~| ~5 z s t 3tx y wM L t|x | | 3 z | s 1| x } wU }~t x w xz 5 x ~ z | w sz w z 3 5z w~ tt M | ~| x Ix 2 qx s 2 2 3 x x |
~
t=T
−
Z
s|
3x
t
~
~x
t
wx
us
¶ 9 ys
M
DH
H |
y
t
s
t
w z
}~t
v
S
¦
Π
OH
O
GY
L
B
B
~
~x
~
Hy 4 t
s
f
D
O
1t|
~x
t
5
~
3 qx
3|
~
t
∆Π =
∂f S. ∂S ∆t
hi
H
IB
ZS
\
O
Dn F
t
s
Π = −f +
]
_
H
m
x
w~ z
Π
L@
W
Z
DH
AZ
\
B
L@
W
M
D
∂f /∂S 1
−1
°
fh
@
] h
SR JSB
C
D
£¶
[ ¦¡
¢£
£
p
¥
?¡
§
2 σB = 0.0225.
ρAB = 0.5 σ
νj νj ρij σi σj −→ min νi ri = r ∗ .
F
H
X
]
B
]
[
IH
D
®h
De
-
,
, e
U
] ®h
-
,
H
I
C
H
@
B
CB
ZO
NH
F
] W BF
ZS
¤
V
[L
@
HF
@
MH
L
L
AB
V
S L
B
CB D
M
L
AB
CH
@ C
M
M
AD
G H@
NH
MH
P
@X
H
C
K
AL F \S
H
ND H
MH
LF
L@
DH
NH
-
[
P
C\
Y
A
\
OTZ
D
L
SH
F
m
P
H
F
DB
cH D
O
D
MH
] M
AD
YG @
L
n
K
B
B
B
H
F
B
@
@
H
H
[H
YC
L
B
cH D
OZ
IB
ZS
DC
L
JL
L
OZ
YMJ
A
N
OZ
IB
ZS
O
@ IP
CH S
An
ó
ý
üý
ý
¼
½
º
^û
»
ÿ
¹^û¸
·
B
HF
H
@
@
@
] L@
M
S
S
OH
O
IH
L
OT
A
DL
AY
O
\S
L
H
X
H
AT HF
L
OT
AD @
O
M
@
B V m
OT
IH
H CU
L
OT @
DLC H
DH U
L
IH
SB
X V
@
F
F
B
B
[H
M
Y
AY
L
C
l@
S
OH
O
I
M
Y
O
B L@ \S
W
CH
M
AD
G@
C
F
GF
DB SP
F
B
MJ H
DM
D
Y
A
DZ
[
B
H AF
A
M
A
[H
I
M\
A BF
X
K
S @
H
m
GF
D
A
W
LD
]
X
[
I
cD @
OK ] B B U
ASH
Z
V
[
H
MG
@
@
AY F
L@
L
AB
W
ZS
S
MH
\
L
@
F
B
M\
m
IOD B
M
ZS
A
OD @
AZ
S
H
SH
A BF
LM
ID
NH
¤
S
O
\S
QF
\
G
B
C
M
SH
A BF
LM
ADY
O
ZG
D
N
DL
T
YI
Y
AL
B
F
-
, V
l
MJ H
IH
N
IH
Z
V B A E
X H DL @ CH B A V E U H B DL @ A \ B ZS OF L X @ H S H E A M H B @X F O Z C W V Y H G AB \H Y U CJ B A L L CB CH E l @ L \ D @ MH L DE JH AB GF E B X Y L OF L L m L AF B MG H B OZ YI ZS O B AY ZS \H @ B M H G JSB @ F ] B \ X JSB I L Y DE ] NB Y ] M A P@ B D F M ZS L \ TO L AE H B V L R H YI H AT F O @ W S AF R B A O @n @ A HF B F L M M L V NB A Y L@ \S H H L@ \m L@ Y D YA F V Z O _ L Z S H X L@ M I V @ G C O @ MG MJ B î Y OH A I ] \S SB H @ O C V A @ OH O \S V SH X G \S J @ \B H MB î S lH U Dn O ZS @ MOY X Dq MH ] S E IB Y @ @ B F U [Y _ U [B [ Y [
A
H Dc H
OZ
SR IB
¡
D
D
ODY
L
F D@
Y
l
O
\S
OD
D
@ S l@
X
@X
B
B
@
@
OH
O
I
M
L
G
M
N
DL
T
I
A
M
IJ
O
[
B
@
@
H
U
AL F \S
D
MJ H
LF
L@
U
DU
L
J
IH
MP
DZS
Dq m
I\ B
OF
A
N
SH
L
G IO B MG
âä @ ZS
k
<
A BF
LM
] AB L
C
K
V
H
MG @
@
YA F
L@
L
AB
W
ZS
B
V H S @ M L@
L
M
JSB
ZS
L
Y
@
CB
Y
OF
E
DL @
]
V
@
B
H
H
] l
MJ H
IH
N
H Y
L@
L
Y
A HF
Z
L@
W
Z
DH
Y
L
DL
OZ
F
B LD @
M
AD
G@ U
CJ B
M
A@ D S
C
F
A HF
LT
IH
H
í
K = 41 9%
[P
AT
M
D@
A HF
L
l@
O cH
C\
SH
K
GD F
SB
MH
L
NH
L
SH
H
B
±
F
m
B
cH
L
C
G
\
@
GB
E
F
GH
A
YM
A
IH
D
lL @
l
Im
C\ B ®
V
j
,
-
,
B
OD K B
ASH
Z
CJ
AD F
Z
\H
I e
] ®
D
L ] H®
I @
F
NO H
cD H @
¯®¬ °±¬
ª¬
[
m
H
BF
H
@ F
L@ S
AZ
\
L@
W
DH Z
NH
L
E
Y
C
AD BF
o
L
\B
W
q\
\B
G
AB
A
OD F
E
D
MJ
D
]
P
B
H
@ L@
M
SH
SH
H NH
A
G
AD HF
L
OZ
Y
SB
C
OT
m
F
B
IH
C
L
IH \
DZ
AD F
,
H
B
BF
-
V
@
L@
M
@
H
IH
N
A HF
LT
OT
AD @
O
M
H
NB
H MH
L
W
Z
DH
Y
L
B [B
o
A
M
T@
A HF
H S X JM H @ ] l P MH D@
L
W
W
Y
S
V
]
j
σ = 20%
C
W
M
∆w D
D
F
BF
H
_ DM
IH
A
GL
L@
B
Y
AY
O
O
GL @
GY
l
CJ
OT K B
ASH
Z
m
AB
C
W
D
[
U
L
G
\
@
SH
A BF
LM
Y
D
IO
K
OT K B
ASH
Z
L
J
OT @
C
Z
B AD H
A
@ª D IU
< L@
B
]
D
âã í
@
@
L
D
B
S CB
M
L
AB
CH
CU
Dq
\B
G
MA
S L@
M
M
AD
G@
H
SH
H
B
NH
A
GL
¢
S
H
B
SH
B
A HF
L
IH
l
I
l
D
W
YD
D
O
G
L
AB
W
U
DK
H DK
ó
F
m
SH
B
H
L
D
m
B
oH
L
AH
AY
M
A
MO H
I
\
<
¨
@
B
B
M
AD
@
@
G@
M
H IH
G
OT
Am
A
M\
A BF
q
è @ª
\F
N
C
\
Ll
JL
W
L
G
JS
ì
R©
§¨
X
V
]
H
P
B
H
B
@ K
S
D
@
ZS
H
H
SH
B
®
®¬
Bî
B
] U
O
C\
L
JL
O
IB
K
-
,
MJ H
@
B
W
[H
Z
L
G
\F
LD @ S
GB
O
A
L ] B
G
\F
MG @
@
AF
Y
L
L@
N
L ] AD HF
OZ
Y
W
DL
G
\
A
SP @ AF
M
CH
B
e
Bî
LD @ S
L
JL
c
ß
P H SX @
V U
DB
SR ] l MH T SB A
H SH
H
A HF
L
c
\S
G
AB
âè
B L V B M G A
D
MO
@
B
@
P
@
L@ ZS
A
M
J
G
IH
V B A E DH E OF
<
q0 p + q0 = 1 0
G@
W
ZS
U
H MP
DZS
W
H
H
B
B
SX @ P MH D@
L
W
OD
I
H
IB
ZS
B
AB
C
L@
G@
G
AT @
Z AD F
@
M
I
M
A BF
cH
\H
F
F
@
H
A@ D S
C
S
U
GF
D
∆t = 1
L
DH
W
Z
DH
Y
\S
@
M V B
B L@
]
H
U
X
D
B
¤
L
L
C
D
L
J
OT @
C
Z
S
r
L
GD F
H
AD
W
, e
]
-
, e De
J
O
L
O
GL @
AD
L@
JH
OT
F
@
@
A@
G
I
NH
] GD BF
n
DX
SB MH
Im
_
R
S(t)
L
M
G@
w(t) o¯(∆t) ∂f ∂f ∂f 1 2 2 ∂2f ∆f = σS ∆t. + ∆w + µS + σ S ∂S ∂S 2 ∂s2 ∂t CH
TA F
\
H
B
]
L
L
E
OH
Z
CB
\S
W
A
B
E
cH
C
A HF
LT
W
D
LD
G
K
$1 = 30
CD H
G
DH
H
A
L @ó
H
NH
f (S(t), t) AT HF
L
E
HF
k
L
L
DB
SB
IH
D
ZS
L
V Y
E
GH
A
DB S
A
@
H IH
l
I
\
Y
D
r%
C\ H
E
A
A
A
S
MH
\
§
AH
C
H
U
K
K
]
K
C
Z@ S
m
@
m
r2
óR V
B
LG
O
B
GL @
L
JL
OF
E
@
dr + 2σ = 0. dσ
MH
l
CJ
AD F
Z
\H
I
M\
A BF
cH
D
MJ
Y
M
L
DM
@
B
W
D
DL
G
drΠ drΠ dσΠ 2σ = : = . dσΠ dt dt 0.3 − 0.3t ∗
S(t) 40
p0
¬
σ 2σ = . 0.3 − r 0.3 − 0.3t L
C
GH U
L
B
S
SR
Q
L
B
L
D
D
OD
OK B B MU
ASH
L
\L
I
P@
AD
D
rΠ = trA + (1 − t)rB = 0.3 − 0.2t
AD
D
2(r − 0.3)
Π = tA+(1−
C
F
@
A@
L
D
oH
L 2
U (σ, r) = 0.6r − r 2 − σ 2 −(r − 0.3) − σ 2 = C − 0.09 \
Y
L
MH
Z
l
Π = tA + (1 − t)B
t
]
]
H
H GU
L
CB
S
k=
\H
O
BF
AH
B
L
\
V
U (σ, r) rΠ
OT
B
B
GL @
L
H
OT
B
CJ
L
2 + (1 − t)2 σ 2 + 2t(1 − t)0.5σ σ = t 2 σA A B B p = 0.03t2 − 0.06t + 0.04.
Π∗
U (σ, r) = 0.6r − r 2 − σ 2
@
CB
dr/dσ = σ/(0.3 − r) k @
L
L
B
D
L
D
D£
D
C\ B
q B(0.2, 0.3)
GL @
¶
G
S
C
E
F
@
A@
H
oH
L
AD F
Z
¡
σΠ =
B
C\ H
\H
OZ
G
dr/dσ σ
L
V
H
AH
\
σΠ
L
E
OK B DB V U
AE
DL @
Z
F
t)B
B
\H
OZ
ASH
Z
¶
A(0.1, 0.1) ρAB = 0.5 j
V
H
B
F
B
S @
H
L@
M
DL
M
JSB
ZS
L
Y
AY
OF
E
LD @
L@
L
Y
A HF
Z
r ]
] ®
h
k¶
£
ª
ª§
£
`
¥
a¤
R¡
£¶
[ ¦¡
¢£
£
p
¥
?¡
§
°
f
®
§
7%
$1 = 29
r
r1
σ
t
H
[
F
M\ l
B l
M\
[
SR H CH
S
SH
V
YC
L
DE
S
B
V
C
L
J
M
@
0 0.1 0.2 V
V
B @
H AT HF
L
IH
l
Im
C\ B
I
Dc H
A
OD F
E
H
GD F
D
AD HF
L
IH
l
IB
a
QW
OD
MA
L
J
G
\
B
@
P@
@
X
B
V
D
MJ
AD
G@
M
N
DL
T
I
AT @
MH
S
AD BF
LM
B
@
@
H
Y
OK B
ASH
Z
S GF
D
D
AT HF
L
IH
l
Im
C\ B
I
D
MH ]
[
@
F
@
H
B
B
G@
B
A HF
L
IH
l
I
Y
L
D
O
IB
ZS
S
H
SB NH
L
C
M\
I
W
\
OD
AB
A
M
T@
A HF
U
X
X j
-
,
@
H
F
H
B
AD
]
H L
IH
l
I
ó
m
A HF
LT
Y
SB AH
M
DL
L
M
]
D
L
J
M
@
S
MH
M
AD
G@
A HF
LT
IH
l
Im
C\ B
I
B AF
A
M
AD HF
D
B cD H
A
OD F
E
Dn
J
[H
A HF
Z
S
H
BF
Bî
NH
L
E
Y
C
l
,
SB
H
H
@
-
U
I
m
AT HF
L
Y
Mj SB AH
< å ]
L
IH
DZ
L
IH
l@
CG
Sn @
H
L S
B @ L@
M
B @
H
@
D
F
L@
M
S
] OB K Y
D
ASH
Z
AT HF
L
IH
l
Im
C\ B
I
Dc H
A
@ª IU D
MH
M
AD
G@
A HF
LT
IH
l
S CB D
M
L
AB
CH
CU
qD
] l
MJ H
@ \B
G
MA
M
AD
G@
H
SH
H
B
NH
A
GL
IH
[
H
AH
0.2 0 0 0.3
[B
I
cD H
B cD H
A
OD F
E
Dn
M
AD
@
@
A
L
N
A HF
LT
í L@
M ] SR H W AB
L@
M
V
@
H
B
L
J
M
S @
AD HF
L
IH
l
I
CJ B
I
cD
] lH Y
O
I
¡
] l
MJ H
IH
N
V
R
Bî
V
lL
J
M
M
l
JY
S
@
AP
M
A B C rA = 20% rB = 10% rC = V
D
MH
M
AD
G@
l
AP S
D
AH
D
F A HF
OT K B
ASH
< â
V
H V
H
B
F
B
L
M H
JSB
ZS
L
Y
AY
H
SH
H
B
H
NH
A
YG
L
DL
OZ
F
E
OF
B DL @
W
M
AD
LD @
L@
L
Y
Z G@
A
L @ó
A HF
@
@
OH
O
G
H
L@
W
Z
DH
A
MG
YA F
L@
L
H
NH
L
E
Y
BF
(
N
"
'
H
B
B
H
H AT HF
C
D
m AF
q\
\B
G
AB
A
OD F
E
D
MJ
MH
L
W
Z
DH
Y
L
H SX @ P MH D@
L
W
OD
IH
]
B
S @
@
¯®¬ °±¬ ¡
ª¬
SB
D@
] DW
Z V
]
OD
D
L@
S
@
JS
GF
D
I
NH
H
¤
@
SB
H
L H
IH
DZ
L
IH
l@
CG
S
U
n
]
G
C
M
@
H AF U
M
HF
R
F
B
@
@
H
Y
AF
Z
\
WF
Y
BF
CH
E
A
V Y
A
N
OZ
SR IB
JS H
MB
DE
B
M
U
L
A BF
MP
Z
H
H
óH
@
ASH
A
A
SH
A BF
LM
D
OD X
J
ZS
D
AD HF
F
D P@
GK
U
V H A E
H
S
D@
N
A
\F
NH
L
E
L
B DL @ S
L
D
C
]
M
M
AD
G@
¤
H
D
@
B
@
L@
W
Y
q
\B
G
< ð
L@
M
S
AT X
J
A
@ cH
C
M
AD
G@
B
L@
W
C
B
P
@
H
B S
B
P
@
S
k
[B
M
JL
AB
CH
C
L
JE
OD
M
NB
L
I
AD HF
C
D
AF U
W
L
D
YS
SR B L B SX NB @
H
L
M
YS
B
@
OK B
ASH
Z
NH
L
E
L
W
D
D@
MH D@ S
]
L V H
OK B
] Y
V
]
V
GF
DB S
A
@ª IU
D
]
YMJ
B
YF
IO
L
J
IH
N
W
AD
@ G@
M
H IH
G
OT
A
H
@
@
U
OT K B
ASH
Z
L
JE
L
JS
D
M
V
<
¨ ã
H
H
@
B
B
A
M\
B
A BF
q
\F
N
C
\
Ll
JL
W
L
G
L@ JS
E
A
C
cD H
OZ
SR IB
ì
R©
§¨
X
H
W
Z
B
L@ JS
H
H
DH
Y
Y
L
OZ
U V ODL
X
]
P@
B
H
W
Z
H
DH
Y
L
DL
OZ
F
L@
DB
í
@
D
W
D
Bî
P@
m
G@
A
IH
ZS
SH
A BF
LM
< ï k
H
W
D
m
G@
\
A
L @ó
B
L
]
V
V
O
C\
q\
I\
E
A
C
E
D
H
\H
OZ
\
SH
m
IJ B
ZS
M
Y
IO
L
K
G@
@
L
DL
OZ
DL
B
H
F
m
IH
G
M
AD
] Y
SB
]
V A
\E
GH
A BP
E
A
IH
D
L@
W
Z
B OH
O
G
A
Z@
\H
GZ
H JB
W
A
l
]
B
H
F
DH
Y
L
DL
OZ
L
AD B
CH
Cn
F AD HF
C
D
M
D P@
D
X
D X L@ H ] F
V OD m
] X ] GL SH
A1 A2 T = 0.2A1 +0.8A2 Cov(R1 , RT ) = 200 Cov(R2 , RT ) = 220 T σT2 = Cov(RT , RT ) = Cov(0.2R1 + 0.8R2 , RT ) σT2 = 0.2 Cov(R1√ , RT ) + 0.8 Cov(R2 , RT ) σT 216 ≈ 14.7 I
B
@
B
OX H B MJ B
A
O
I
SH
A BF
D
B G@
A LM
\H
GZ
SH
Z@ H
B
A BF
LM
D
L V H A ED
L
M
A BF
AF
¥
¤
I
@
NH
t = 3/7
[B
G
JL
AB
W
DK
DK
H
P L@
Dn
H JSH l
A
G
AD
cH
B
G
ó
GB
A
DH S
l
@ CJ B
¡
S GF
D
D
A
IP
OK ] B B U
ASH
Z
]V
,
J
@
I
Dc
MJ
AD
G
k
, J
M
B
MG
m \B
W
q\
\B
G
AB
A
OD F
E
D
MJ
B
OT
B
L@
@
@
C
H
S @
H
[B
c
MJ
YA
Yó
MO
@
AF
IH
R
B
V
S
B
¬
¡
@
B
H
H
B
L
H
LG @ O
AB
C
D
G
OT
LA
Y
L
S MH
\Y
L
D
ODY
CH
GF
GD @
G
G@
M
H L@
W
Z
DH
OH
O
G
A H
¤
H [H
ZS
DL
H
P@ E
D@
I
D
J HF
ZS
n
n OD X
J
] ª¡
k¡
í
R
k
¡
¶
ZS
D
AD HF
C
D F
¥ª
a
D P@ M
¡
ª¡
í
¥
n
U
]
GK
D@
N
A
\F
DL @ S
V NB
L
I
k
IP
B
50
t
H
A
C
F
F
@a
Dq m
0.12 0.20
lD
L
JL
IU B
L@
B
V]
V
V
OK B B V U F AH H M A B ] AF M\
C
SH
A BF
LM
æ ]
OD
IH
H CU
L
JM H YA F
L@
L
] l
IH
B
YqD
V
V GK
B
V
AT HF
L
<
BF [H
L
E
Y
C
Y
B
SR H W AB
N
]
L
DL
H
F
A
H
@
H
@
IH
l
YI
C
@
L@
@
SH
IH
AT HF
SB
@
L@
S
H
OZ
U
OD
IH
CU
B
lD
] AD HF
L
S
D@
N
V
L
J
M
W
]
M
SH NH
C
DZ H
H
L
IH
l@
CG
Sn @
A mF D
q\
\B
G
AB
M
DL
M
JSB
B
DH
L
W
F
H
AD @
O
MH
@
NB
A
YqD
H CH SB
L
S
ASH
í
MG
@
YA F
L@
L
H
B
H
]
L@
M
@
@
B
F
B
H
V
S
C
r = 0.3 − 0.2t
@
OK B B V U F A HF
OK B B V U F A HF
D
ASH MH
A
ASH
Z
IB
D
S
G P@
CH
L
IH
U
V
A B
M
AD
G@
l
DF
DSB
Z
B
V
Z
OT K B
ASH
R
L@
B
D
D
G@
B
L@
W
MJ H
] l
A
YG
L
DL
OZ
F
A
OD F
E
D
S
ZS
L
Y
AY
L
OT @
1/2
Z
IB
D
F
F
L
B
B
L@
A
V
S GF
D
Π = tA + (1 − t)B
S
Dq m
\S
B
JL
E I
D
C
B
P@
U
V
0.3 0.4 0.5 0.6 0, 7 0.8 0.9 1 ρAB −1 −0.5 0 0.5 1 W
DY
D
@
]
MH
L
SR H W AB
] l
MJ H
IH
N
B
DL @
W
M
MJ
M
AD
G@
B
OH
O
G
H
L@
DLC H
< ñ
S
Π∗
DF
DSB
S
SH
C C\
L
B
D@ ª S \F
-
,
P
IH
SB
B
OD
H
IH
N
V
H MH
L
W
Z
DH
W
OF
E
DL @
W
Z
X
1/2
Z
F
OD
G
L
D
H DC U
A
\S
L
AD B
X
D
H
DN B
DZ
W
D
W
Z
DH
Y
L
@
L@
M
S
H
L@
M
@
AD
G@
S
L@
Y
L
@ Y
q
V
L@
L
DH
D
r Π ≈ (0.141, 0.214)
I
D
C
C\
L
AD
m@
-
,
[
W
H
_
@
SB IH
DZ
DL
M
JSB
ZS
B
@ H
A
M
A
L @ó
AZ
\
\B
G
H H SX @ P MH D@
¤
A
D
DC U
G@
M¢
O
L@
F
D
D
Yq
H
B
H
B
B H H SX @ P MH D@
L
W
L
Y
F
B
AD
G@
A HF
H
L@
W
Z
DH
<
Y
30%
H
A HF
LT
H
C
U
@
DL B@
G@
B
L@
W
L
G
M
Y
L
DL
OZ
F
W
L
H
@
B
AY
OF
E
DL @
LT
OT
39 r
G@
OZ
n
W
M
S
í
F
A HF
V Y
H
G
M
35
,
Bî
D
G P@
@
IH
MH
L
H
OK B
ASH
Z
H
NH
MH
S CD H
G
F
H
V
OD
IH
V
H L@
L
Y
B
âò
r
N¢
l
41 SR [H
] l
MJ H
H
]
OD
W
Z
DH
m
GF
] W
X
F
H
CU
L
n
L
W
t
S
20% IH
SB
D
35
LA
N
DZ
_
Y
L
H H SX @ P MH D@
DB SP
D
G@
SB
H
L@
30
\B
W
B
ND B
Yq
B
H
@
30
L@
A
H
W
L
G
YM
SB IH
DZ
B
H
B
\
A
C
L @ó
@
L@
M
SB
H IH
DZ
L
IH
7%
@
F
A HF
V Y
B
H
W
L
L
W
B
W
Z
S
@
SB IH
DZ
B
OE @
H
c(t)
M
_
A HF
LT
L
DL
OZ
F
MG
@
OD
IH
B
DH
OH
O
B
H
< â
On H AD @
9%
SH
OD F
E
D
MJ
OK B
V
D ZS
H MH
L
A
IB
D
M
YG
L
DL
L@
A HF
Z
D
42
SH NH
A
B
OT
OD
IH
H
HF
DC H
G
S
L@
CU
H
W
Z
DH
SR
D
L@
OH
U ] G O
D
OZ
â
N
$42
GY
ASH
Z
H
NH
MH
GF
X
F
M
L
n
M
F
20%
L
H
AD @
H
40 O
M
m
DB SP
H
L@
SB
H IH
DZ
L
IH
W
@
M
40
B
V
D ] W
SB
@
S
W
@
IH
Z
SB
DH
DZ
B
OE @
< å
D
D
σ = 20%
CU
L
L@
M
@
@
C
ZS
B
AZ
\Y
B
L@
â
S
r
DL
H
OT @
DLC H
S
G@
\
A
L @ó
D
A
IB
D
M
L
DL
SR
D
H
32
OZ
X
M
AD
H MH
L
W
Z
DH
AZ
OZ
G@
32
F
B
DL @
W
DH U
L
G@
L@
IH
¤
\
M
σ 40 38
T
®
H
L@
W
Z
DH
B
@ W
Y
q
\B
G
SB
DZ
l
<
âæ
]
34
OH
O
GH
M\
I
l@
CG
@
D
31 41 7% S(t) $30 $33
]
V
BF
S
$40
NH
L
E
Y
C
$3 S(T )
®
³
¬
§
B
B)
Π = 0.5(A+
0.15 0.45
i
HF
Y I
B
H
100 Y
L
F
@ MH
S
[
L
AY @
SB
B c
@
AX S @
U
Y
AY
L
OZ
F
B
H
F
P
B
SB
L
\B
W
@ª AD U
IH
] N
E
V
N
L
J
M
S
B
@
B
V
l
MJ H
IH
1%
B
@
OH
DS
H
B
LIJ
]
MG @
@
AF
Y
L@
L
SR H W AB
±
¬
-
,
-
,
]
_
¢
V
L [H
cH
CP
H
MB
L
J
c
AX S @
BF E L [H
Y
C
l
M\
IH
Z
MGD
@
A HF
Z
Y@ DS
H
B L@
W
@
NB F
H
B
@
m
AY
YMO
Z
GD @
S
@
F
B
H
B
@
B [ U
M
o
L@
W
Y
q
\B
G
H
] LB L
M
AF
A
M
AH
l
¤
H
@
@
B [ U
M
D
M
MG
YA F
L@
L
AB
W
ZS
Y
L@
M
MJ H
B JSB
ZS
L
Y
,
j
-
AX S ] @ c
L@
B@ DS
BF
A Y E
C
IH
N
D
L@
SK @
L@
M
L
G
M
H
@
S NH
N
@
B
B
H
@
H
H
AH
D
A BF
L
NB
E
L
D
A
OP
L
DL
V B A E lH S
A
H cD H
OZ
SR IB
] l
MJ H
IH
]
S H
F
@
@ L@
M
IH S
N
L@
CH
G
A
G@
AH L S
GH
NH
L
S @
@
L@
M
MG
@
AY F
L@
L
SR H W AB
_
F
@
H
Y
L
\ P@
IH
NB
A
H
OP
L@
B
< â
n
í
[
m
E
TY
L
DL
OZ
F
SB
m
B
H
F
DL B@
] W
L
D
í
K
] L
M
@ B
V
D
L
_
S
@
D
GD F
D
D
MH
M
AD
P
G@
l
M\
I
Y
]
l
CJ
AD F
Z
\H
I
Sj D IB
M
(
N
"
[L
'
B
H
B
AF
A
M
AH
F
[
@
H IH
l
I
B
CJ B
I
cD
¡
]
V
@
P
B
B
B
CQ
E
D
\H
OZ
V Y
L
DE
L@
L
JL
OF
E
DB
L
JL
IU B
L@
Y
Y
MO
AF
IH
V
D
L
J
M
S
@
@
R
S
B
D
S GF
D
_ D
[
A
M B
H
AH
F
V
D
MH
M
AD
G@
lL
JL
@
B
SH S MH
OD
GL
[
H
@
H IH
l
IB
CJ B
I
Dc H
JSH l
A
YG
IO
N
I B
H k
l
M\
I
lD
YqD
P
H V
@ Y
@
CB
I
Dc H
GF
S
,
OK B B V U F A HF
U
AD HF
G HF
Z
O
H
ASH
Z
l
CJ
AD F L
AL @
D
H
DSH
G
L@
B
\H Z
è
H SX D@ AP
P
I
-
<
IH
M
A BF
cH
L
C
A HF
LT
IH
l
I
@
NO H
¬
j
-
,
]
D
H
OK B B
ASH
Z
M
MH
M
AD
G@
B
lF
M
OD p H
MH
] M
AD
G@
l
DA P
ó
ND B
S
H
m
B
SH
@ ] L
M B
MH S
M
AD
G@
Z@ JS
O
W
DY
D H
Yq
O
A HF
^]MJ H
F
GL
Y
LG @
AB
OK B
IH
D
U
X
V A
AD
G@
V GB GD @
cB
GD @
D
B
L@
í
Y
YS
W
L
[
DK
DK
H
ó
D
]
¢
L
@
S GF A
D
D
AT HF
L
D
[
ASH
Z
GF
DB S
H IH
l
Im
@ª D IU ] M A
@
C\ B
I
cD H
V
-
, AD
G@
H
H
G@
A
M\
m
A BF
q
\ BF
L
G
L@ JS
E
A
C
cD H
OZ
B B
Dq m
C
MH V D
M
AD
]
I
@
NH
¤
S R M@ IB ]
B
-
¯®¬ °±¬
ª¬
F
V
L ] H
³
OD
L
DE
B
B
P
B
@
L@
A
C
L
D
ZS
D
L@
H
AT BF
H
L@
M
H
F
L [B
OZ
DL
AB
CH
C
M
OH
O
G
H
L@
W
Z
DH
L
J
M
L
@
L
D
S
@
B
S CB
M
L
AB
CH
C
M
M
AD
G@
JB
W
]
V
H
P
E [B
L@
lL
J
cH
CP
M
Mp @
V
]
V
V
V
C
O
C\
@
SH
H
¬
R
K
-
,
³
B
H
B
@
c
@ @
IH
ZS
Y
@
SH GH
A
G
L@
B
SP oB
S
AT BF
H
AU
L
OT
AW @
DH S
A
AT X
J
A
cH
C
,
I
N
B
@
HF
H
n
,
-
,
AD
H
M
M @
Y@ G
O
I
V B OD BF
E@
\
OF
M
N
LA
G
Z
\
ZS
I
OK B B
ASH
Z
D
MH
M
AD
G@
l
]
E
C
l
A
V
B
F
B
S @
H
MJ H ] l
IH
N
L@
M
DL
M
JSB
ZS
L
Y
AY
OF
E
DL @
L@
L
Y
H
S
H
BF
H
B
B
H
Y
L@
L
SR H W AB
Y
L
DL
OZ
F
DH
L
W
F
OH
O
G
L@
W
Z
DH
U
Bî
NH
L
Z
B Dq m
A HF
@
S \S
OD
SH
Gm
Y
MG
@
AF
H A HF
LT
OZ
@
$41 7%
DL
5 O
@
CB SH
B M
D
L
S(t)
I
A
M
D
AF
A MO H
I
W
D@
@ L@
M
S
]
]
g¬
K
\F
SB
m
Dc @ l
M\ p
G
\ < å
[
J
DU S ] GF
L
J
OT @
C
L
D
CD U
MH S
M
AD
B
G@
W
DY
D
O
S
SH
AD H
D
AD HF
L
q=
lm S
P@
H
\F
F
I
D
M
Y
L
IH
$100
L
W
L\ @
H
M
H
Dp
D P
IH
N
OZ
] l B
SB
D
SH
B
E
K
D
[
DLC
B
Dq m
C
D
@
A HF
F
H
] LB L
M
AF
u = eσ ∆t d = 1/u σ = 0.2 ∆t = 0.4 u = 1.1348 d = 0.8812 S(t + ∆t) t + ∆t S(t)u = 45.3936 S(t)d= 35.2473 max S(t + ∆t) − K, 0 cu = max S(t)u − K, 0 = 4.3936 cd = max S(t)d − K, 0 = 0
√
AT HF
H
Z
A
AD
IH
E
NB
F
]
J
O
C\
A
@
GF
V
OT K B
V
OT K B
ASH
Z
J
R
OD K B
ASH
ã
er∆t − d , u−d
C
D
AY F
H
P YF
T
\
G@ U
MJ H
IH
N L
JB
U
c
Gp H D@
T
DW @ DS
W@ S
@
D
X
] Y
OK B
ASH
Z
A HF
LT
$700
DH
W
Z
DH
] N
Y
L
B
1
OH
P MJ H
@X
DL
H
W
D
D
H
SH
H
B
G@
@
A
GL
í
<
C\
V
M
ASH
_ DC R
H
H IH
l
Im
C\ B
@
LG M
AB
<
p=
O
AZ
OT R H
K P (t) \
U
OZ
E
@
S -
L@
B
L@
M
Y
L@
<
AD F
[H
0.6600
GU
BF
B ] B
L
W
X
,
æ
B
H
D
Y¤
OK B
ASH
Z
B
L@
]
AD HF
L
U
Y
L
D
I
cH
lL @
@
I
B
$500
L
JE
Y
L@
B
H
JSH
F
Z
O
MJ
A
M I\
JSB \m
B W
ZS
L
è
V
S
AD
G@
l
H JSH
H IH
l
I
H
U
E
\
P@
cD H
MH S
W
M
AD
G@
DK
DK
AD HF
L
c = e−r∆t (cu p + cd q),
SR B L B SX NB @
] S MH
W
H
H
D
G@
P@
8%
C
l
Z
B
CD H A
GL
H
G
S
HF L@
L
F H
@ W
SB
m F
E DH Z
AR
\
K
M
$0.650
@
B
<
[
OD
K
3%
OH
M\
L
D
S CB
T
I
B
V B A E
A
c
D@
H
M
L
AB
CH
C
T
B
B
SH
8%
O
A
Z@
\H
GZ
SH
G L V B I G
L@ JS
ã
[H
L
JL
E
C\
NH
MH
@
CB
I
cD H
H
@
I
U
A
OK B B U L@ D IU
ASH
L¢ H ó H
r
L
DL
OZ
M\
B
M
@
0.2
AL F
M
[
@
0.1
A BF
k
CH
A
D
\H
OZ
F
B
rA = 0.2 F
10%
F
LM
ä
[
I
MH S
M
m
A
YG
X V IO
MH D
M
AD
G@
L
L
P@
B
GD F
DB S
Π
U
B
T@ M
@
Y
j
A
GF
DB SP
V D
V
@
I
ZS
A
K
A
OD F
]
B
H
A
I
(σ, r) P@
A HF
@
CB
I
@
A
Do B a
A HF
LT
Ll
JL
MH
S GF
D
\S
OD
SH
r∗
M
H GU
U
0.09
L
B
L
J
M
E@
H
m
IH
l
I
V
D
GF
S
C\
j
Z
0.3
A
@¡
L
AD
G@
]
@
C\ B
I
l
M\
IP
D
AT @
S MH
E
D
0.3
F
AD HF
S
L@
P
H
H
V B A E
AL
DY F
X
H
Dc H
NB
AD H
ä
0.2
Y
]
ó
A2
A
Π
LD
<
B
l
ADY
¡
V
D
<
(σ, r)
AT @
@
A1
L@
M
D
D@ W
MH
MH D@ S
M
AD
G@
AD HF
L
]
@
P@
m
qm
P
A2 0.1 0.5
OD
S
@
] LB L¢ H
M
H IH
l
B
ZS
D
E
\
I
L
D
σB
G
U
M
F
G HF
Z
H
H V U A E H NB H cD H ]
AD
G
B
H
AF
A
M
I
] DO H
IB
L
J
B
F
A HF
A B rB = 0.3 σA = 0.1 σB = 0, 4 ρAB = 0.5 rF = 0.05 r∗ = 0.30 \L
C
0.5
BF
AT X
O
L
G@
V A
F
AH
@
CJ B
I
cD H
B
@
A1
SB
A = (0.1, 0.1) B = (0.14, 0.20) C = (0.3, 0.3) V
B
H@
NH
MH
GF
X
D
OT
AW @
DH S
σA
Z
C @
A OK B B JU
L
D
DB SP
AT HF
L
S GF
D
< ï \ R AT F
B
SH
A BF
]
V
F
AL @
I
O@
C
L
D
DC
MH
M
A 1 A2
ASH
Z
H
DSH
@
m lp H IH
OL
\
l
Π
@
H
G
L@
S
GF
D
AD
P
G@
D
TO K ] B
ASH
L@
D
0.2
LM
M
A BF
cH
L
V
D
V
@
L@
M
S
A HF
LT
IH
Z
M
A 0.4
Y
D
D
MH
M
H
CB
V
MJ
C
E
P
AD
G@
L
Q
AD
G@
< ð \ R AT F
A
0.1
D
OD
@
X
l
M
GK AD B
K
DC @
E
F GH
AY
m
0.2 A B
B
B
M\
IH
H
ó
OD B
A
cH
A
0.3
SP
S
G
OT
AB U
W
DL @
?
Π
MJ
D
G
OD
A
N\ H
?
X
D@
0.1 0.1
n
B
B
X
< ñ
l
rB
B
W
] AU B
L
\
H
D
rA
A
¬
e
®
H
P@ E
D@
I
D
J HF
ZS
] ®
n
k¶
£
ª
ª§
£
`
¥
a¤
R¡
§
u − er∆t . u−d
c = 2.4795
p = 0.5804
P (t) 6 Ke−r(T −t) .
L@
M
@
B
m @
c(t) F GD H
U
X
A HF
LT
IH L
AB
CH
CU
W
B
ZS
D
C
M
AD
@
G@
C
DA H
[
D
M
I S F
@
Q@
M
AD
G@
AH D S
H S
B
SH
H
B
A HF
L
IH
l
I
W
YD
D
O
U
l
V
N
@
U
B
H L
IH
l
B
L
I
S
SH
B
I
0.5
A HF J
OT @
W
YD
D
O
U
D
AD HF
L
IH
H
S
@
V
D
V
S
@
D
GD F
D
_ D
_
l
A
L
J
OT @
C
U
JSH
GY
D
Z
AD H
σB
IO
B
A
σA
V
D GD F
@
B
S
@ª D IU V
M
B
B
H
H
M ] LB L
AF
A
M
AH
F
I
N
@
U
L
@
H
] GF
_ DS
SH
LM
B
A BF
A HF
LT
D
L
D
E
AH
Z
IB
@
U V
H IH
l
YI
@
ZS
Y
W
`
S
D
AD
G@
l
M\
IP
D
Yq
DH
F A HF
@ CB
I
cD H
DL
G
J
M
MH
D
L
J
M
V
@
SH
A BF
LM
_
A
_
H JSH
A
C
Z
AD H GY
D
IO
σB
V
B
A
σA
S
B
GD F
@
] GF
_ DS
SH
A
D
MH D
M
AD
G@
l
A BF @
@ª D IU V
D
L
M\
IP
U
] LB L¢ H ó H DK DK W AB GL
M
l
IB
â < å \
0.3
MH
M
AD
H
B
H
OD
Zó
G
OT
A
@
LM
B
] L
B
@
B AF
A
M
H
AH
OT K DB
ASH
Z
0.3 U (r, σ) = r − r2 − σ 2
B
A HF
L
l
L
JE
L
G
M
B AU @
SH
lp H IJ H
S
U V
D
L
D
E
AH
M
S
F
@ CJ B
I
cD H
L
J
E
D
H
@
H
B
V
OD
GL
l
M\
I
D
S
@
B
SH
B
P
DYq H
A HF
F
@
SH
A BF
LM
[L
D
AD HF
L
IH
l
I
CJ B
I
Dc H
JSH l
A
YG
IO
D
U
U
OT K B
ASH
Z
S MH
V
MH D
M
L
J AD
G@
OT @
C Ll
J
V
@
H
S
@
AT HF
L
IH
l
I
Y
@
CB
I
Dc H
_D
_
V
B
@
SH
A BF
LM
D
L
D
E
AH
Z
IB
ZS
Y
W
DL
G
U
â < â \ `
[
B
Z
AD H
A
@ª IU
D
] GF
I
N B
U (r, σ) = r − r2 − σ 2 σ
GU
H
P
G@
r2
L
H
IB
A
@ª IU
D
qm
A BF
LM
I\ B
ρAB
IH
l
I
D
AD HF
L
] m
A HF
LT
J
M
D
A HF
LT
IH
H
Z
IB
U (r, σ) = r − r2 − σ 2
E
AB
B
U
] OT B K
ASH
U
H
IH
l
B
L
DB
D
q
SX @
Yq
DH
l
YI
@
ZS
Y
D
MH
M
D
ρAB
OT
SB
E
F
@ A@
G
B
IH Yq
X
Z
m
I
] U
MJ H
0.4
B
A
H
L
J
W
OF
M
L
JB
IB
H
D
0.2
I
l
ZS
@
B
V D
M
AD
G@
YM
F A HF
@ CB
I
W
`
AD
G@
B
H
SH
0.2
\
CH
I
OT
D@
SB
AH
Y
M
F
GF
M
V
@
SH
A BF
cD H
σ
H V U
Π = ν 1 R1 + ν 2 R2
0.2
L
¤
A@
@
CJ B
I
Dc H
DB SP
ZS
F
AD HF
L
IH
H
rB
F
MH D@ S
G
D
Y
A
] LB L¢ H ó H DK DK W AB GL
M
l
LM
_
DL
G
\
@
AM B
rB
NH
@
X
@ ] B L
M
B AF
A
IB
rA
σ
O
]
D
F 0.9 q = 0.6 q = 0.3 q = 0.1 1.00 1.00 1.00 1.05 1.20 1.10 1.30 1.00 1.05 0.9 A1 1.20 0.3 F r1 σ1 σ2 Cov(R1 , R2 ) IH
IH
S
M
H
AH
OT K DB
I
NH
U H MU OD B
L
E@
\
OF
DF
DSB
Z
σB
H
YC
GL @
H
M
V
D
D
F
@ CJ B
I
âæ
¤
B V B A@ G@ c
AT
S
K
rA
GB
A
I
V
M
CH
B lL @
L
G
MH
M
<
H
A
U
F
σA
ZS
L C
oD
O
s(ν1 , ν2 ) LG @
rΠ = ν 1 r1 + ν 2 r2 , OT X H
h
@
F A1 A2
DL @
M
cD H
@
D
JS
@ª
ASH
Z
E
A
B
SB DH
M
Z
B
SH
B
I
B
B Dm
L
D
O
IB
H
R
B
C
D
H
L
M
S
lL
JL
A2
E®
l
I
A HF
V
L
DB
D
A1
CJ
AD
G
MO
LT
n
cD H
]
V
CB
YA
F
O
C\
L
IB
O HF
Z
M
CB
ZS
H
A 0.4
D@
I
V
Y
AH
SB
B
P
EA
SB
@ª
D
H
-
,
P
H CB
\
OTZ
HF
B
B
@
F
H@
NH
C
ZS
D
AD
G@
A
GD F
D
\H
OZ
B
âè
E
A
g
®¬
K
M
D
A
F
]
A1 P@
S
MH
GF
L
<
B CB
C
D
J
SH
@
CJ B
N
@
O BF
H
\S
Z
Z
D
0.1
B
SB
D
l
A2
E
D
D
@
MH
G
0.1 0.3
@
DM H
X
cH
MO
H
V
V
]
O
C\
P
[0, 0.4]
D
DB SP
CB
S
E
A
C
E
D
\H
OZ
@
SH
U
¤
B
DB
L
DE
B ASB î
M
r ∈ [0, 0.4]
Y
R
<ì \ R AT F
V D
M
S
OB
YI
M
P
L@
Y
@
0.1 0.2
YC
Π C
MH
M \n OF
D
H
[H
L
E
A
F
V
V
\L
H
F AB
L
Z
ó
Y
MO
[0, 0.4]
SR
<
AD
G@ âã
H
Im
A HF
LT
E
H
AF
H
rf A1
E
Q
¶
E
√
6
OP 6ê
V
AE
DL @
L
B
A
B
YG
@
@
√ √ D(S(t + ∆t)) = S(t)2 er∆t − e−σ ∆t eσ ∆t − er∆t .
P
[B
]
V
AT
AD
C
M
IJ
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
DSB
Z
@
+ qd2 S 2 − (puS + qdS)2 = S(t)2 (pu2 + qd2 − p2 u2 − q 2 d2 − 2pqud).
H
X
¡
@ \B
G
MA
M
AD
G H@
@
P@
F
Bî
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
AF
IH
R
uS(t) dS(t) p q
D
S l@
CG
] l
MJ H
IH
í
F
H
Im
O
GF
@a
D
p+q =1
L
\B
W
q\
\B
N
A HF
LT
σ = 20%
G
AB
A
OD F
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
H CU
H
H
AD @
O
M
¨ ì
R© @
AF
¡
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
§¨
AT F
S(t + ∆t) P
B L@
E
D
OD
YM
L@
E@ U
\j
¯®¬ °±O¬
ª¬
]
W
OD
YA
A
I
B
@
F
U
Bî
@
P@
\
L
E
D
OD
M
lL
S
B
JU
E@
OY F
F
F DSB
Z
I
D
X
H
YC
L
B
cH D
OZ
IB
ZS
]
L@
M
@
H
U
ä SR MH T SB A
¨ ì
R©
§¨
ZS
ZS
D
H
H
V B A E
<
[
F
F
@
H
B
H
B
K
F
@
SB
P@ E
D@
I
D
J HF
H AD HF
j
-
,
¬
¡
¡
¡
n
n
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
] ª¡
ZS
@
âò
±
®¬
]
CB
q
JF
L@
L
DYC
L
cH D
OZ
IB
ZS
MJ
OD
L
J
NO H
L
<
ó AD U C
SH
H ] U
O
C\
D
L
D
B
SH
H
@
B
DB
D
O
IB
ED @ q
M
AD
ZS
E@ V U
\
@
SR
O BF
A
C
I\
D
\
S GF
D
G
AD HF
L
L
B
cH
OZ
σ 2 ∆t
¬
H
t
SB
L
O(∆t)2
IH
DZ
L
IH
D
p
S(t + ∆t)
u d ln(S(t + ∆t)) − ln(S(t))
®
±
¬
§
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
OT X
H
óR
[H
L
N
C\ H
AH V D
]
K S(T ) > 100
ËÊ
5 J
ZS
D
E
A
C
E
S
B
H
H
H
B L@
W
OD BF
G
OT
V LA
@
MH
P
OD B@
A
I
L\
W X
H
@
] Y
L
DL
OZ
L
B
H
F
AD B
CH
C
M
M
AD
G@
L
JB
W
A
W
S
B
B
@
F
H
EA
D
\H
OZ
m
\SH
A
V G
OD X
J
ZS
D
AT HF
C
D
M
B D P@
A
IU
D
L@
V NB
L
I
Y
L
DL
l
B
V
[
OZ H
V
Y
L B
BF
@
SH
H AT HF
C
D
AY F
\S
DL
Nk
W@
] Y
C
DL
OZ
LD
W
SB IP M
E
C
qD m
Z@
\
AF
L@
Y
L
S
B
H
F
H
B
F
U
M
AD
JSH
A
Yq
DH
G@ U
@
F
B
F
A HF
B
DL
YG
M
lF
V
I
L
M H
@ U
V
[
F
CB î
LD
I
DM H
S
@
OH
O
100
U
¨
SH
I
ZO
D
\
GY
W
@
H
B [H
GL
L@
L
@
H
@
] @P L@
G
I
O
B
@
óB
H
OK B B U
ASH
Z
l
LAD
W
M
S AL F
M
A
B
OF U
A HF
L
I\ B C\
W
M
S B
@
P
S B
H
DM
D
AF
A
\F
A
ZS
D
L
CB D
M
L
AB
CH
C
M
k
OK B B U
ASH
B
@
AL F
M
Z
[ U
D@ ª S \F
cH
`
] L
]
_
GF
DH S
NB
@
L
H
V
CP
M
Ll
J
c
SH
DH W
@ AX S @
DS
P@
OH Z
S CB D
M
L
AB
CH
CU
L
J
OTE @
L@
M
@
S MG H O
[
F
D@ S
M
H IH
G
OT
B
OH
O
U
X
A
L
B
D@
Z
SH
ì
R©
§¨ è < ¢
[
G@
MJ H
GF
DB SP
B
DB
D c
H SX @ q
@
F
G@
A
YM
@
L@
M
YS
OK B
ASH
Z
H
NH
L
B
T
S(T )
]
B H
AT HF
L
IH
l
I
AB
C
MD
AD
G@
MJ H
GF
U
[
B
H
B
U
K
Þ
@
H IH
l
I
Y
@
CB
I
¡ ] Ü Ê cD Ì
OD
B
IH
YC
L
cH D
OZ
IB
ZS
A
\S
OD
C\
SH
j
B
ó
h
K
H
H ASH
Z
M
A BF
cH
L
CB
L
M
OD BF
OD BF
Z
L SB
H
IH
W
D
LD
G
\
U
K
P
D
] P® °
CD
S D@
J
AF
IH
Z
\ m
B
H
B
@ L@
M
YS
OK B
ASH
Z
H
SH NH
MA
L@
W
Y
L@
E
L
¢
M
[
@
H
B
L
\
m
IH
MP
DZS
A
IU
D
F
B
@
G@
C
AD
L@
Y
AL
A HF
L@
NH I H
NH
YA
U
M U ] ® O ] p ³ ó H
B C
K
C
C\
F
L G BB î ] L
U
U
D
D@ B
@
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
ÜÚ
ããáÚ
@
M
A BF
] D@ E
P@ I
q
E
\
I
L
\L
S(T )
çÝ
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
,
]
] ® °
D³ <ì
] C@
C\ B
S
0.2
çê ÜÛ ç ßê ßÝ ãù Û ß æçÝ î éßáÝ éáÚ Ý æ ÛÚ æÝ âÜñ öÚ ðÚ ã æêåÚ ðç óÚ éè âß î ÷ç ßÜ é ãìÚ ÜáÛÚ ßÝ ë éå ç çßàè ÛÚ èä ÞßàÝ æßÝ ô î éÚ ßðì Þ éøÚì èí ÈÌÈÊ ¾¿ ÂÁÀ ÍÎÊ Á à ¿ÏÊ ÄÅ Á Ð ¿ÈÉ ÆÇ
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
D
A@
A mF S
q\ m
I\ B
OF
C
DH S
A
C
]
]
S
B
S @
B
SH
L
J
MH
OD
GL
N
D@
C
§¨
\
O
C\
A
SH
B
±
¬
Gp H D@ c
B
B
K
-
,
Fn
BF
P
\B
L
JL
W
< â
V
¯®¬ ©R °±¬ ì¨
ª¬
I ] DAY
_
U
X
F
P
H
Ll
JL
W
l
S
U
B
Bî
JB
AE
D
YqD
A HF
V L
IH
D
Y
Q
]
O
!
±
] ®
\
SH
B
OK B C\
AL
D
SR
B
@
C
D
R <ì
ASH
Z
M
I
C
H
Q 6
K
N
D@
C
X
d
t
|
ô
5 s 2 ysU 5d~| t | I 59s | w 5z tszs }x 5 w x ~ ~ 9 wsx G7 z / 6 8 wI
-
,
]
H
B \U
L
L
W
V
]
35s
4 |
d}|s o
b( j0 , n, p0 ) → Φ(d1 )
]
@
F
m
B
B
D
CB
S
D
H
B
E
@
A P@ S
F
S
t=T
SB
A
M
L
D
I
c
M
L
AB
L
E
\H
OZ
V
T
J
M
F [H
Y
OK B
ASH
Z
^]MJ H
LG @
IH
D
N
@
B
B
F
H
DK
DK
D
F
F
@a DH S
A
C
SH
@
D
N
@
@
L@
M
S
b
B
N
S
D@
C
H \U
L
L
W
X
d
AX B @ H
H
åã p
@ L@
M
YS
OK B
ASH
Z
NH
L
E
L
YJS
F
F DSB
Z
D
V G
<
@ L@
M
A
B
@ª D IU
ô
b( j0 , n, p) → Φ(d2 ).
L @ó
E
F
F
@
B
C
E
CH
B
@
k
S(τ )e−q(T −τ ) ϕ[τ,T ] t=T
T
DM @
AD
S
H \F
L
L
D
C
c
H
V
G@
A
DM
D
AF
e−q(T −τ ) eq(T −τ ) = 1
G
NH
H
@
\H
OZ
B
F
A
A
L
D
L
D
H
H NH
L
E
L
JS
C
\
lL
LJ
W
l
M
OD
I
Y
l
OK B
MA
V N
C
X
Ll
JL
W
Y
W B
D@
MH D@ S
s|
A
YC
ZS H
L@
M
AD
U
G@ U
MJ H
GF
DB SP
OD
X
@
Ð
|
v
F
s
~x
5tx
}x
z
~
~
Hy 4 t
M
3t5
s
|
| w z v s
z
|
Ìç
Dc B £
6 | t3
tx
5M
W5 w
} x
<
åè
S
B
GK
D@
N
CH
L
IH
L@
Ìç
@
Ð B
?
D Ìç
L
D
D
OD
AT @
MH
F
DL @ S
OD H
L
cH
] AL @
I
H
HF
DSH
G
D
F
AT @
Z
DI H
Z
A
Þ
B
D
E
GD H
AB
] C@ N¦ G Dc @ A
\
Ll
JL
W
X
H U
[
ª
ln K/s(τ ) − (r − σ 2 /2)(T − τ ) j0 − np √ → −d2 . ∼ √ npq σ n∆t
V
H
B
H
m
m
5
l
U
Ìç
Ìç
Ð
Þ
F
sr
6 xx
L
D
OD m
L
\E
L
JS
L
D
OD m
L
\E
L
JS
A
D@ ª S \F
U w|
k 3|
I
TX
\
P@
B
] Ð Ìç
YC
ZS
L@
] H
B
A
G H
AT HF
L
I
@
V
I
N
H
A BF
MP
H
NH
SH
L V H U A E cD H
qD m
C
V D
U
TO K ] B
ASH
Z
L
JE
L
JS
k
H
S GF
D
D
A HF
LT
IH
l
Im
C\ B
B _
j0 − np ∼
L
@
V
B
IH
D
M
GD
D
ù
B
± û
û
S GF
D
OK B B V U OD BF
X
Gd ó H
AX B @
b
<
\
D
M
AD @
G@
V
[
k
H
ln K/S(τ ) − (r − σ 2 /2)(T − τ ) √ . 2σ ∆t
X
@
B
B
S
@
H
H
m
y
ys
y
9
|
w
|
6 wx I
}x
5z
35
~{
}x
{
}
6~
Ll
JL
@
MH
GF
D
M
I
G
OT
A
A
M\
A BF
q
s
@ª
\F
N
C
\
Ll
JL
W
L
G
JS
l 5|z t
IH
l
I
M
AD
G
@U
MJ H
GF
DB SP
D
<
åæ
X
o
4
n∆t = T − τ
L
O
I
Y
O
A
] @P L@
cH
P CB
\
`
@
LD
C
er(T −τ )
GL
H
SH OD @
C
G
I
[L
JS
H
H
B
ÿ
û
ý
ÿ
20%
JS
E
A
C
B
L
L
U
¹û
DE
ASH
Z
lH
e−q(T −τ )
MH
V
H Dc H
OZ
SR IB
L
B
M
L W
B
A BF
LI
A
DB
¥
H
F
H qD X l
A
A
D
AF
CH
M
D P@
@
N
B
ÿ
m
@
b
AX \B
npq ∼ n/4
GF
DH S
NH
L
IH
M
AD
G
N
[H
W
L@
]
U
M
AD
G@
t=τ
Z
C
IH
N
H
OTL @
W
M
AD
G@
C
L
LJ
@
F
H
P@
F
"
û
DH
ASH
Z
lH
30%
DH S
@
H
@
DSH
Z
ZS
M
I
L
L
Y
M
L@
B
;
B
B
]
DU S GF
C
OP 6ê
T
@a
M
M
G@
G
OT X
J
ZS
10% $100
[H
W
DSH
H
@
AD
OT
G B@
L
A@
B
D
B
@
V
W
H
%
÷ÿ
B
AB
åä
1.20
Z
ZS
M
AD
G@
W
MH
IB
j
A
AD
H
G@
A
OT X
J
ZS
D
Y
@
AD HF
L
L
E
A
C
þ
AY
X
X
F Cov(RB , RT ) = 0.0007 0.23
L ] AD HF
OTL @
W
DSH
H
L
JB
W
O
H
B
H
@
CB
I
V
(
ù
ü
DH
\
490 A
Z
ZS
L
B
ZS
D
M
L
DW
SH
A BF
LM
D
OD BF
Bî
B
MO
C
1
AB
B
@
$20
L
] l
V A
M
M
AD
G
B@
B
H
O
H
cD H
OZ
n
B
W
DL
MJ H
IH
U
V
H
IB
ZS
B
Dq H X
A
DK
N
M
L
W
L
E
H
cD H
E
A
L
:
@
B(0.14, 0.32)
B
CP
B
DH
CB
@
10%
DK
G
C
E
I
OT X
J
B
H
V
*
* 2
D
470 B
LD @
X
Dc H
∆t µ H
ZS
D
A
L
H
AT BF
A
@
SR IB
AF
A(0.067, 0.18) 0.09 H
H
] Y
OK B
ASH
Z
NH
L
E
L
JS
GF
DB S
A
V
]
V
[
@ª D IU
j0 ∼
¬
@
JS
ZS
AT @
AE
D@
G
B
SB
S∆t D
B
H
B
M
AD D
l
DF
A
CB
M
L
DW
c
G@
G@
MJ
0.014
S
M
]
O@
SB
H
M
L
A
A
§
L
OTL @
D
]
A BF
MP
k
V
< å J
å a
k
V
V
V
H
L V H A E
√ ln K/S(τ ) + nσ ∆t √ . 2σ ∆t
l
Bî
Q
D
MH
M
G@
AD
l
AP S
D
AH
D
A HF
F
OT K B
U
6v
~x
5tx
}x
|
|
z
d5|s W t
H
ASH
Z
L
JE
L
A1 A2 A3 T = ν 1 A1 + ν 2 A2 + ν 3 A3 ν1 = 0.3 ν2 = 0.4 ν3 = 0.3 Cov(R1 , RT ) = 230 Cov(R2 , RT ) = 280 Cov(R3 , RT ) = 250 T
¬
±
®
] e
k
ª
n
a
SR JSB
C
D
¡
¡
¶
ª¡
í
¥
£
p
¥
?¡
ª
n
a§
a
¥R
ª
¥
§
Φ(x) = 1 − Φ(−x)
¯ = 0, e−q(T −t) , −1 . θ(t)
i
x
@
F
BF
[0, 1]
[
B
CB
B
[
@
B V H N S
L
OT
B
SB IH
l
Z
U
B
C
L B [ U
M
A
N
OZ
J
PF
h CB
Y
DF
A
E@
] HU
H
S @
m
@
@
H L
J
E
D
S
B
H
U U
L
M H
@
∆Wh
L
D
X
B
\H
OZ
OD
M
L
H AL ^]J B
YC
L
SX HF D@
m
Tq H
CH
Z
U
L
J
E
D
H MU OD B
L
E@
\
O BF
L
I
X
P AT HF
L
@a
V
C
M
L
J
E
D
OD B
A
GL
B
CB
@
F
QB
W
OD
YA
U
Bî
@
V
S
L
B
CB
D
M
JL
AB
CH
E
H
C
OD
B
IB
ZS
D
H
OF
D HB
SH
H
h>0 Wh (t) t=0 h ∆Wh = Wh (k+1)h −Wh (kh) A
V
V
@
F
BF
H
H
SB [B
OF
C
L@ D ZS
ATE
D
\H
OZ
L
cH
C
W
ZS
H
NH
GF
SB MH
]
D
J
M
SB O@
L
A
Y
D BF
Dq m
@
B
BF
B V U S @ M L H DU
F
DSB
V
lX
F
Jm
O
ZS
D
CJ
D
M
B D P@
L
V
V
SX @ P qD m l
\
H
F
@
BF
B
B
F
CD B
L
D@
I
cD H
C
GD BF
E
AD @
CB
A@
C
C
JM B
OL
\
L
D
O
IB
ZS
S
D
L
J
E
D
OD
B
MB
L
JU
E@ B
H
L
\
J
U
[
m L
SH
AB
C
D
="
$
=
F
B
@
V
OD BF
B
AY
O
]
H
H
(
'X$
)
B
B
IB
ZS
M
HF H AF U
M
A
L@ ] DC
E
D
OD
M
C
LD
JU
& V \
Y
A
M
P
F
BF
H
L@
W
ZS
L
JU
E@
\j
O
L H
B
@
\
YO
AY
YMO
F
F
m
K
@
C
S F
F
@
AD HF
L
W
D
DL
G
\
Y
A F
m
@
H
B
B
Bî
B
L
JB
W
TLA HF
L
O
IB
ZS
L
A
M\
A BF
q
\F Bî
M
L
F
H NB
M
GL @
B L
O
@
@
B IH
G
AD
G@
I
H
B
p
ZS
@ª
IB
ZS
L
Y
L
AB
S
F
H
@
D@
Z
DY
O
] M
AD
G@
A@ D S
C
N\
L
DB
DE
P L@
CH
I
c
G
ZS
D
B
JSH
A
V G
[
S CB D
M
F
S B
CB D
M
] L
YG
A
DC B
AF S
]
X
ZS
D
O
B
H
H
B
[B
ZS
OD
IH
SH L B
B
oB D
N
¥
B
B
@
L
AB
CH
C
M
M
AD
G@
L
\B
W
AY
O
IB
ZS
H
V
H
H
H
B
B
X
H MH
L
W
Z
DH
Y
L
SX @ P MH D@
L
W
OD
IH
CU
L
OT @
H DU
L
IH
IB
ZS
]
L
J
SB
C
L
J
OT @
W
DB L
I
D
[
B
B
U
H
C
YM
A
S @
F
IH
D
MH
ZS
L@
W
DH
L@
K
SB
H
O@
M
L
A
LD
L
B
Z
K
S CB
M
B
OH
O
G
L
JB
W
L
D
OF
E
Dn
C
OD
IP
I
N B
SB
H
MH
GL @
IH
D
MH
IH
DZ
H
\
D
SR
H
] O@
CH
L
E
A@
A HF
_I
B
@
Y
A
F
MH
DL
@
AF
] B
L
DL
M
S
®e
-
,
U
Y
L
DL
@a
OZ
D ] L
S
B
H
]
L@
M
@
H
S [B
CB
M
L
AB
CH
C
M
H L@
W
Z
DH
NH
A
Ló
\B
W
C
OD F
E
Dn
J
]
[
]
-
®e ,
∆S = µS∆t,
M
DY
L
c
L
J
OT
IB
ZS
OF
D
F
P@
U
]
GK D F \S
D@
¶
L
M
D
G
F
w(t2 +s)−w(t2 )
h = 0.01 ]
D
c
A@
MH
j
L
@
E@
B
DH
L
E
D
M
Ll
JU
E@
H SX @ ] P CH
GK
D@
Z
t 1 t2 s (t2 , t2 + s)
OD
ZS
D
IB
O
C
M
S
]
\
F
DE
L@
L OB
P
U
C
\
O HF
C
SB
OF
E@
?
K
U
CU
L
OT @
DLC H
DH
L
IH
SB
S(T )
P H SX @
C
Do H
Y
@j
F
F
T@
O P@
@
H
δ
Y
AT P
H
ND H
L
SH
@
] BF
A
w(t)
MH
L
E
F
BF
H
G
E
A H
V Y
L
B D@
I
DE
H
MU OD B
L
E@
OR H Dc H
M
AF U
Z
(t1 , t1 + s)
YI
lY
h→0 M
W
ZS
U
D(∆Wh ) = δ 2 D(∆Wh ) Y
Dc H
\
YO F
F
B
S
D OT @
C
[
\
B
U
B
B
V m
L
D
O
M
L@
j [ DZ
-
H
SB
DLC H
B
DZ
Bî
Y
[
J
IOD
∆t → 0
H
NO
L@
]
ZS
D
GD F
MH
SB
Y
AT
@
?
qD m B
B
I
D
Y
B
B
w(t) w(t1 +s)−w(t1 )
A@
B
A@ D S
L
P L@
P
MO
AF
Wh (kh)
A HF
AY
Y
h YMO
C
F
F
@
SB
Wh (t)
IO
SB
L
L
B
H
Y
H IB
ZS
L
cH
∆Wh P
I
V
l
J
O@
BF
IT P
G BF
E
B DF
A
E@
\H
OZ
]
Wh (0) = 0
C
ZS
D
C
ZS
L
AD @
CB
A@
C
AY
´
-
] L
D
O
SH
H
H
H
F
IB
ZS
L
@
NO H
e
,
W
Z
DH
A F
D IH
B
lL @
CB
L@
W
k
S(T ) = S(0)eµT
B
P
S GB
@ª
D
H DB
I
\S
LA
@
CB
L
i
,
s
H
@
H ] FBF
W
ZS
@
Wh (t) A
L@
W
H
H
M
AF U
I
NH
V
2h . . .
NH
E
H
V G¡ A E \ @ J JH P X E@ M D @ I JU A Ll B F V @ F D Z I M V Y DSB A DO F B E F D DH Z M Y D S B E F @
OF
YMO
I\
F
®¬
O
C\
L
OT @
NB S
L
A
DR H
]
®¬
K
V
G@
A P@
l@
G
∆t
ZS
H
B
CB
A@
K
]
]
ZS
D
H
] U
O
C\
SH
F
B
A
@
B
H
@
F
CD H
L
L
D
S
B AB
C CH
M
L
AB
CH
C
qD
\B
G
MA
M
AD
G
H@
SH
H
B
NH
A
GL
B L@
W
S CB
í
F
M
AD
H G@
A
A
L @ó
OH
O
G
H L@
W
Z
DH
Y
L
DL
OZ
F
DL @
B
AT F
\
U
R
@ L@
M
\
M
AD @
SH
B
L
DL
M
S
m
B
@
\B
AY
M
A
MO H
I
S
NH
GF
F
M(∆W h) = 0 √ h
MH
SB
1)h
GF
L
M
2
MH
A HF
SH
m
n→∞
GK
¤
[
A
,
SB
AF
8 u
wx
t|
F IB
ZS
I
/
G7
@
DH ZS
@
K
j0 − np0 √ 0 np q
D
5sz
I
x +.... 2
2 5sz
x
t
P
Φ(d2 ),
~x
M
U x
z
y
E
A
YS
W
LD
G
\
Y
L@
ô
L
`v
5x J
L@
V
C
E
SH OT @
C
L
@
F
@
Y
AY
B
j0
dsx
x
x
}x
~x
H
L@
W
D
\H
OZ
_
b( j0 , n, p0 ) → 1 − Φ
t
1t|
Z
V
d
6 | 8
DH
59
OH
c = S(τ )Φ(d1 ) − Ke −r(T −τ )
8
G 7
wx s
D
B
OF
E
-
u
|
ex = 1 + x +
9
G O
L
JB
W
Y
5M|
8
D±
,
x
z
/ ~
G7
,
Z
O
@£
n→∞
w z
t
s
8
®¬
[
@
O
I
n→∞
n
L
|
/
d1 =
ln(S(τ )/K) + (r + σ 2 /2)(T − τ ) √ , σ T −τ √ ln(S(τ )/K) + (r − σ 2 /2)(T − τ ) √ d2 = = d1 − σ T − τ . σ T −τ
IO
]
5| | 4 ~ 3 I 5 6 3w x G 7 I / | ~
}
u
i
CB
B
B
dt
G7
35
,
L
D
O
IB
j0 − np √ npq
[τ, T ]
M
√
er∆t − eσ ∆t √ p= √ . eσ ∆t − e−σ ∆t
u
u
®¬
H
ô
S
@
n
6
∆t √ √ 2 σ ∆t − (r − σ 2 /2)∆t σ ∆t + (r − σ /2)∆t √ √ , q∼ . p∼ 2σ ∆t 2σ ∆t
vz
]
K
ZS
1 −t2 /2 e −∞ 2π
~x
P@
SH
m
P YF
T
\
B
M
ln K/(S(t)dn ) / ln(u/d)
t
35
4 \
O
C\
L
\L
I
N
b( j0 , n, p) → 1 − Φ
wx
|
@ [ \ OH
U
B
OT R H
K
t
t|
d
4 | ]
G
H Oó
I
N
Rx
|
w z
r
@ª IU B
c
sz
u
9
9s
5Mx
w
y s|
O
C\
Φ(x) =
~
(T − τ )/n
|
3
I
~
5|
}
n
z
~
q ~
p
6 s
wz
F }|
~
τ T
D
t
s
3u . ~ yz 6 3s 5}s 35 us ~u 3 yU
ds S(τ ) τ
®
e
¬
[
H
B
H
H
H
A
M\
G@
B L@
W
H
B
AF
A
\F
A
L
JB
W
L
E
L
JS
"
'
"
U
Bî
B AT HF
L
L
O
IB
ZS
L
OD F
V A
*
¥
w * 4
] ¬
¡
¡
¡
]
P@ E
D@
I
@
@ MG H
L
A HF
®e R
n
] ª¡
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
§
S 0 = µS.
T
w(0) = 0
−δ δ 1/2 1/2 Wh (k+
L B
D@
I
M ∆Y = a∆t + b M ∆w = a∆t, u = eσ √ ∆t
, d= √ 1 = e−σ ∆t . u
-
H
[B
IH
C
L
U
B
F
F DSB
Z
I
DF
L@
B
B
U
B L@
E
D
OD
M
Y
L@
E@
\
OF
U
X
] g
®¬
-
,
-
B
]
L V H
OT
A
M
D
AF U
B
p
SB
D
U
AD X
JH
F
AD HF
L
AH Y
M
V
]
B
H
N
H
MH
GF
DB SP
Y
IO
AT @
MH
S
B
SB ZS AD
L
A
DL
cH
C
M
L
JL
³
¬
X
-
,
JS
L
J
H
B
@
OF
E
DH
E
A
C
AD
C
P@
@
OE @
L@
Y
O
] GL
V
F
HF
B
S OT B@
M
Y
Y
A
L
SB
U
p
H
B
H
H
H
NB
L
G @
JSH
NH
MH
GF
B
H
GL @
IH
V L
cH
CP
H
MB
L
Y DB SP
ª
qX
M
GL
JS
Y X
S GF \
D
GU
L
J
MH SH
@
B
F
B
X
-
,
AB U S OT @
IO B
L
JL
O
I
V Y
L
D
Im
O
B
@
L@
L
J
c
MB
JSH
A
GL
[E
H
A@
A HF
I
e
e
!
] ®¬
h
-
,
M S(t + ∆t) = S(t)er∆t .
±
B
M
JSB
B
O
IB
ZS
V
] LB h LH
Bî
IB
M
M
Z
H
F
V
] m
B
(
"
V
S
V
SB
E
A H
AT @
I
c
M
A
\L H
cH
C
L
B
CB D
M
L
AB
CH
C
M
AD HF
L
AF
E@
n
^
V
[
] ¬
®¬
-
,
M P (t2 ) = er(t2 −t1 ) M P (t1 ).
D
AT
OD
IB
ZS
ZS
L
F
JH
E
A
,
V
D
[
A
C
F
@
Z@ S
@
AH
L
DL
M
@
\B S
!
|
V
M @
P
L@ J
U
Bî
B AD HF
L
L
O
H
B
B
IB
ZS
L
W
L
D
IB
YD
FU
qD m
Y
YMO
D
O
C\
L
J
Y
M
F
H
SH
P@
U
K
] e
e
-
,
H
@
L
D
B
S CB
M
JL
AB
CH
C
M
L
AB
C
AL F \S
H
DN H
MH
LF
L@
DH
NH
L
H
MP OT
DZS
L
J
K
B
@
CB
L
DL
S
V
B
L
J
E
D
OD
B
D
MB
L
JU
E@
\
O HF
E
A
A
E@
L@
P
H
V
W [B
B
qD m
E@
L@
P
H
H X
P ( t2 )
H
Y
V
m
¨
G@
G@
L
G
JS
H
SH
H
B
B
NH
A
YG
L
DL
ó
B NH
L
OT
A
MH
D
H
qD m
SH
%
)
(
L
AB
W
DK
DK
M
\
B
B
Y
AY
O
F
B
B
IB
ZS
H
E@
\
OF
CB
H
H
H U
@
]
B
B
M
M
AD
@
G@
L
W
ZS
D
H
B
B
P
F
A
O
H
óB
B
IH
C
l
@ CJ B
M
A@ D S
C
S
@
Y
A
\S
L
F
B
AD H
GF
I
D
L
AB
C
AL F \S
H
G@
SH
H
H
H
m
L B
B
S IB
F
CH
A
G
M
LG
JS
A
@ DN H
V
J
L@
"(
$
)"
&
$
U
MH
K
H
OD X
J
ZS
D
Y
O
I
Y
AE F
A D
B
@
B F ] H
A
D \H
OZ
lL @ F
S
D P@
L M
@
A
@
Y
L
D
c
MJ
Y
LF
DH
NH
L
H
MP OT
DZS
L
DE
P L@
B
SB
G@
V LG
M
H \L B
MH
GL @
IH
D
SH
M
U
H A BF
LM
ED
A
L@
DCSB
An
B
@
AF
L
L
AB
W
ZS
Z
] MG
H qX
M
MH
M
AD
G@
L
JB
W
Y
L D
D
IB M
AD
B
B G@
L
W
M
B
?
MH
Z
OT
IH
ZS
D
V
P
H
H
A
B
E@
L@
A
B
B
@
@
D L
B
cH
H
*
H
B
²
w
;
;)
;
(
Bî
B A HF
LT
L
O
IB
ZS
L
H
B
@
H
H
@
H L
J
OTL @
W
DSH
Z
ZS
M
AD
G@
L
JB
W
Y
D
q
S
ZS
D
E
A
L
V B A E
*
A
H
¤
G@ K
OZ
B
H
B
V
A
H
] OF
E
D
IB
ZS
L
L
M
A BF
A BF
A
c
S
L V B MJ
W
I
N B
V
W
ZS
@
F
BF
H
GD F
SB MH
L
_ DM
U
_
L
OT
A
H
B
H
SH
H
B
B
H
B
cD H
OZ
A
GL
ô
]
ô
e
®¬
-
,
c = S(t)b( j0 , n, p0 ) − Ke−nr∆tb( j0 , n, p).
,
V
SB
@ F
B F
A
L@ S
-
,
er∆t − d . u−d
B
³
¬
p= G@
A
YA
O
IB
ZS
H
F
B
@
M
B AF L ] H
M A F
@
-
,
B CP
D
J
O@
A
N
Y
B
(
(
&
&
H
¢
P (t1 )
C
L@ D ZS
OT @
C
SH
\F
@
NO H
F
B
L
OX @
J
]
]
A
@
H
AH
F
e¬
C
B IH
Z
M
M
A
MO H
I
V \
J BFF
H
t
L V H
?
MH
D
OF
\H
A
D
JL
A
I
Z@
I
NO H
j
]
Q
@ A BF
E
G
M
C\ H
W
ZS
B
#
=
m
ó
V
AE
DL @
] DL
D@
A
¶
SB
FU
F
\m
AT HF
L
AH Y
M
L
D@
AZ
J
F
DP
D
OD
OL
\
SB
@
SB
B
H
@
SB
S
H
F
B
@
@
H
AD HF
L
AH Y
M
L
M
E
A
A
N
OZ
IB
ZS
A
M @
ô
B
L\ J
Bî
AD
AF
L
CB
L
V Y
V
F
IB
S
V A
I
@
NH
E@
\
OF
CB
oU
B
A HF
ZS
X V GB
H
@ P@
U
@ E@ ®
I
I
H
NH
M
LG
JS
MJ H
LI
H
AT BF
V D
L
B
@
GH S
Z
U
j=j0
n
h
¤
K
-
,
H
OF
E
D
[
lZ @
B
\
C
SB
U
Pn
®¬
@ cH
] M
AD
G
Y@
L
H SX @ m X P MH ] CH D@
@
u d ln(S(t + ∆t)) − ln(S(t)) σ 2 ∆t t O
HF MH
[
SB MH
L
M
_D
V
H
B
J BFF
W
ZS
L
JU
E@
AH
óR
L
* 2
* 2
A
n
p0 + q 0 = 1 b( j0 , n, p0 )
]
SX @ P CQ H
H
H
Y (t) L
AL F
H
B
@
@
H
F
q
C
ZS
D
Y (∆t) = Y (0) + a∆t + bw(∆t) − bw(0). \
G
A¡
M
SB O@
L
A
V
OF
E
DB
L
JL
A
Y
A
M
B
H D P@
L
L
L
B
L
JU
E@
\j
O
t2
B
m
E
[L
DL @
∆t
W
¢
A
[
F
@
SH
H
B
CD H
A
GL
GK D F S MH @
DB
L
J
H
V
D
CB
\
OF
@
OE @
cH
^]J
O HF
L
$100
Dq m
OT
V
E
D
MJ
H
GP
B
F L@
Y
B
P E@ A
\
D¬
, e
OT
MP
H
S
H
F L@
C
S
∆S = σS∆w,
I\ B
OF
B
H
@
O@
M
SB
SB O@
M
L
S
H
X
[0, T ]
SH
L
B
H
A
B
A
CB î
H
B
@
] B
L
DL
M
S
DZS
∆Y = a∆t + b∆w,
[
B
H
A
W
LD
G
TP
@a
@-
,
] H
L
A@
S
¦
M
A
Y
A
F
D
∆S(t) = µS(t)∆t + σS(t)∆w,
cD H
MA
G
H
HF
CH
DLW
H
G
MB
O
M CH
B
L@
@
e¬
B
_
F
H
] W BF
B
w
H
G BF
OT
A
A
DL
]
F
BF MH
W
ZS
B
@
G
I
N
ZS
D
M
D P@
S
E
AD @
F
BF
Y
CB
M
W
B
S
] B
L
DL
M
\
V
U
I
N
Y (t) Y
A@ ?
-
,
H
L
L@
H AT @
C
L
DZ
G@
¢
GD F
σ>0
] e ¬
] Y
H
ZS
DE
L@
P
B B
P
L
DE
L@
Y
B
SB
-
¶
∆t
E
¬
, e
]
L
J
IOD
J
O@
M
SB
T
B
b∆w ] m
Tq H
CH
Z
F
F
B
A
t ∈ [0, T ]
L
SB
Y
AY
O
b σ
DL
V
C
L@ D ZS
a S(t + ∆t) − S(t) /S(t) $20
ZS
@ª ¶ DS N @ @ ] C F CB AD B H h B C GZ V H A @ E P@ ZS J H @ L O F b M \ D H ZS U E@ C E@ DA F U JSB L AT HF \ A S @ d lS M M Bî
@
OF
w(t)
µ=0
¨
³
®
@
Z
J
F
CD H
I
c
MG
B
F
B
CB
lF
MH
M
lZ
Cnj (pu)j (qd)n−j e−nr∆t j0 p0 B
] e
]
LG
JS
\
H
S GF
D
GU
L
J
AB U S OT @
¬ h
¡
ª
k
ª
n
a
¡
¶
ª¡
í
¥
£
p
¥
?¡
ª
n
a§
a
¥R
ª
¥
§
t1
p
S(t+∆t) = S(t)u S(t+∆t) = S(t)d M S(t + ∆t) = S(t)up + S(t)d(1 − p)
S(t)up + S(t)d(1 − p) = S(t)er∆t .
u d
p0 = pue−r∆t, q 0 = qde−r∆t B
H
>
'X$
]
lX
Jm
O L
J
E
D
OD
OF
B
[
I\ B
∆w
[m
I\ B
OX H B B ZS A HF
ZS
D
H
MU
L
U
]
-
ge ,
]
F
Q
DF
DSB
Z
I
SH DB
O
C\
G
A
IH
D
M
ZS
D
I\ H
h
K
-
, e
B
B
B
B
H
B
H
B
@
@
L
E H
NH
D
OL @
M
B
@ AL F
M
S
NH
L
L
E
\H
OZ
L
DL
C
ZS
D
L
OT
A@
MH
IB
O HF
R
V
@
j
V
_
E
A
H
C\ H
AH
L
Q
V
F
\
U
L
J
E
D
OD
H
B
MU
L
E@
OY F
F
DSB
Z
D C
CJ
lF B
p
B YM
V H
L
OT
A@
IO
MH
IB
O
]
H
B
B
B
L
OT
A
B
H
B
cD H
OZ
o
L
T
C
L@
D
h
-
, e
]
H
H
V B A E
G@
AH
OF
E
D HB
,
B
] B L@
qH
ZS
\
R
AT X
J
A
cH
C
f
®¬
`
H
H [B
ZS
A
]
D
¡
_
o
L
T
C
B
B
B
L@
D
AT F
\
]
W
S
@
F
BF
V
F
Bî
@
P@
H
@
@
[B
N
YO F
A
I
M
AD
L@ ZS
V
H
B
H
B
E
A
H
CB
C
DL
OT
AE @
L
G
B
B
O@
I
G@
A
V
V
@
m
AE
DL @
OL
\H
L
M
¶
S B
@
B
SB
H [H
ZS
NH
GF
MH
L
M
DU
L
D
q
ZS
D
L
D@
I
cD H
B
H
G BF
E
AD @
CB
A@
C
G@
G
G@
A
] f
®¬
-
,
z
~|
v
H
u
x
|
s 5tM ~ F 3 | x
w|
d ~. ~| z t9 e
6
~
|
zz
~x
t
s ~9
d
s
L
K
Q
@
-
,
[ °¬
Y
A
c
O
F
B
@
H
IH
ZS
qm
X
H
B
SX @ P CH
YIO
H
H
@
SH
F
qX @
] Y
V
O
C\
Y
A
E@ B
C
\H
L
JL
B
A BF M
OZ
B A BF
A@
A
Sn \B OT P
SB
[B
IH
H CU
L
IH
DZ
V
8
OF
k
)
D
E@
L
OF
\
AD HF
L
¶
B
F
B
@
]
CB
E
V B
o
L
B
@
SH
a = 0.1 b = 0.15
MH
A
\
&
(H
B
SH
P
H
OT @
C
L
ED
A
C
E
B
Y (t)
@
ATE
D
\H
E
A
A
I\ B
D
OD
M
Ll
JU
E@
V
B
@
M
MJ
I
M
D
YM
A
M
JF
Z
D P@
F
L
AB
OT X H
]
O
C\
OD
IH
C
6
C
CB
OZ
H
V
B OF U
L
D
I
c
\
OF
A HF
LT
AD
C
∆Y
YO
L
-
@
SH
B
OT @
C
L
A
H
L@
W
Z
DH
L
JB
W
Y
C
C\
D
G@
∆Y
F
B
e
, e
S
B
\ FF
Ll
JL
IB
M
ZS
D
P
k
D Y (T ) = b2 T.
P
cH
C
B
( \ $ ' = =& $ = = & Y$ \ ^ & [ > '$ ] & Y | ^ ! $ > \ Y= " & Y& $ Y' '$ $" & " ( " % *& *
f
C
OT
A
H
B
H cD H
OZ
B
] OD
4
A
\
H
] Y
L
DL
M
U
U
,
B
ó
B
O@
M
M
D
G
L
H B
SH
H
V GH A
B
OW
Y (t) ∆t = 0.01
OTZ
p
!
!
B
@
H
¨
IO
@
SH
K
B
lF
D
O
I
2
F
H
ND H
AY
ó
@
H \P S
ED
A
F
B
S
H
E
A
C
E
D
\H
OZ
l
@
@ CJ B
N
O
C\
AE
OF
E
D
0
O
-
, e ] ¬
]
@
V Y
A
@
w
L
ZG
V
V
@
OF
M
]
V
YM
IO
∆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 @
] DL
CB
m
@
ET @
M\
ZS
X
c = e−nr∆t
D@
AZ
J
F
lF
M
MH
j=j0
n X
CD H
I
B
@
DL @
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
-
G
j0 p M
OF
D
¶
c = e−nr∆t n
®
F
B
SB
@a
H
SH
H
´
&
!
w
w 0
H
SB L H
IH
DZ
l
\M
I
M
L@
W
Z
DH
L
\B
W
TA
E
U
"!
&
"
'
*
*
OD F
D
JM U
Dq m
Y
O
MP
Z
C
A
D
N
O
n
,
\
C
]
I
N B
V
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
] °¬
-
,
2
®¬
@
B
F
¬
C
E
D
\H
OZ
¡
¡
]
A
C
H
@
CB
e¬
k
£
n
] ª¡
¶
¡
k¡
í
R
k
¥ª
n
a
í ª¡ V
D
Y ¥
] IO
cd
]
lZ
\
cu
®°
f
¬
§
y
1.2
0.8
0.4 0
t
D(∆Y ) = b2 D(∆w) = b2 ∆t. Y (T )
t>0 Y (t)
t
b2 t
Y (0) + at
D
S
cu SH
Y
B
@
F
B
@
@
[B
E
Y
A
c
S @ l@
CG
M
H
B L@
W
@
SB IH
DZ
NH
M
SB
B
H
H
H
Z H
@
SB IH
DZ
NH
SH
MA B
W
L
G
M
CB
L
B
H
P@
@
H
H
±
L
D B
B
OF
E
Dn
J
] F
L@ D ] S
L@
GZ
L
E
AD @
CB
lF
"!
&
!
& $
I
N
B
] ³
¬
-
,
u − er∆t . u−d
L
D
S
AT S
MJ
OD F
cd = e−r∆t (cud p + cdd q).
E
cu = e−r∆t (cuu p + cud q).
t c
MJ
B
B
Z
W
L
MG
OD F
DL
OZ
F
DF
"
'
)
%
q=
L
D
JM H
L
cH
C
H L@
W
Z
DH
] LH W@ ¥
GF
M
B
AD
OT
IH
YC
L@ L@ G@
SB IH
DZ
w * 2
*
B
H
L H
OT @
SH
B
B
OD
D
[
I
D
D
F
CB
L
D@
I
Dc H
D
B SH
L B
H
OT @
C
L
AB
C
@ B
V
[
L@
P
H
AD HF
L
AF
E@
L V B M
DL
M
\H S
A
\
L
J
E
D
H
U MU OD B
L
E@
\
B
@
YO F
S
[B
MO H
I
V
G@
A
G ] OO H k
V
Y
B
OK B
ASH
Z
B L@
W
MJ
]
Z
Y
AY
H
F
B
OF
E
D
K
O
C\
L SH
H
F
@
NO H
ó
®¬
K
j
B
@
H L@
W
Z
DH
L@
W
Y
q
\B
G
_A
I
N
V
O
C\
L
JH
SH
H AF U
N
Ij \S
B
SH
H
S L
B
CB D
M
L
AB
CH
C
YM
OK B
ASH
Z
NH
A
B
L@
W
] B
-
,
Π(t) Π(t) = Π(t+∆t)e−r∆t Π(t) = S(t)+νc c = c(t)
JL
L
-
,
L
\B
W
H
ATL @
E
H L@
W
Z
l
M\ p
-
G BF
E
er∆t − d , u−d
B
Q³
¬
K
V A
H DL @
L
c
L
B
L
DH \
L
JB
W B
W
±
, e
C
L
Y
L
DE
L
D
q @
@
F
@
B
V
L
DL
M
Z S
F
m
B
@
F
H
V B
J
O@
C
CJ B
N
YO
M
5Mx
3}|
s
t} |
~x
{
u
~ z
r
.
]
P
H
HF
H
Y
OI H
A
C
CB
OY
F
CB
\
OTZ
U
¤
k
n
I
@
NH
'X$ ( ( = ' & ^ = $Z $% Y$ = ( ( $ ' = = $ ^ | ( '$ $$( &) ( ^ > $ # ' & #= ?> $ \ ="
S(t), c
E
D
CB \H
OZ
S
M Y
L
AB
CH Y
@
SH
H
B
] B
O
C\
Z
OD
IH
@
@
L@
S
cu
CU
H
CU
L
M
M
AD
G@
@
]
p=
MO
AF
R
SB
H IH
DZ
L
[
C
H
c = e−r∆t (cu p + cd q),
L
IH
SH
AB
G
NH
NH
AD @
CB
A@
C
B
qm
C
V D
L
J
E
-
, e
ZS
H SX @ JF
V H A
B
j
∆S
J
OTE @
t
L@
L
K
U
S(t)u
IH
-
,
H
[
G
\
M
IO
C
L@
@
] ¬
]
I
D
H
L V H A E
L
OT
A@
MH
IB
O
∆w
³
¬
M
L@
H
L
E
F
[H
YC
F
L
B
AF S
MH
U
X
GD BF
E
D
S(t) −
] B
O
W
Z
DH
NH
D
OL @
G@
YG
Y
B
JU
E@
HF
B
L@
P
H
H
E
B
M
Dm
L
B
B
CB
C
∂f = 0. ∂t
C\
K
cd H
B
MH
AY
YMO
L
M
K
Z
L@
B
n V
L
D
F
m
D
OF
\
C
F
F
B
@
B
U
OD
U
DL
B
MO H
t
W
@
NH I
¤
@
A
\S
B
\
YO
B
@
I
NH
D
OD
H
AD @
Y
CB M
L@
A@
E@
M
@
∆t + σ∆w.
P
[H
F
H
S CB
AD
ZS
JSH
H
B
A
A@
cB
σ2 2 ln S(t) + µ − (T − t), σ (T − τ ) . 2 ∆t
B
-
V
] AL B
C
M
L
AB
CH
SB
A
GL
I
P
O
MH
B
¤
M
L
Mn IB CB î
C
\
ln S(T ) − ln S(t) (µ−σ 2 /2)(T −t) ξ ∼ N (m, s2 ) ξ
N
@
X
GL
YJS
H
AL F \S F
B
@
L
A
U
M
J
N
O
] m
U
U
BF
B
L
ln S
C
\
lL
A
C
M
V
Dj (t)
JL
B
,
DN H
MH
LF
D
L@
@
IP
S
"!
]
H
@
AF
¦
f (S, t) = ln S
W
L
M
A@ D S
C
CH
i
E
B
L@
Dq m
Y
SH
)" !
l
E@
\
H
OF
L
L
B
SH \m S
YA
M
1 ∂2f = − 2, ∂S 2 S
D
OD
F
Si (t)
M
K
L
M
B
Y
F
F DSB
Z
I
D
Bî
B OF
L
O D
A
B
E
V
O
F IB
ZS U
B
F
A
E@
D
∂f ∂f ∂f 1 2 2 ∂ 2f ∆t + o¯(∆t). ∆f = σS + ∆w + µS + σ S ∂S ∂S 2 ∂S 2 ∂t
]
F
@
CH SB
L
J
E
A
MO H
I
B
F IB
ZS
@
σ 2
l
DH
NH
IH
Z
B
B
qD
D
L
D@
I
DSB
Z
2
J U V SB L
c
L
IH
D
OD
M
L
B
(
²
@
A
µ−
E@
L
W
D
D
G@
] L
DH
I\
*
2
DU
MP
H
H
JU V \
E@
B
IJ B
ZS
Y
L
D
I
c
\ FF
cD H
s
H
DZS
OD
A HF S
Z
H
NH
G BF
E
AD @
* 2
S
∆(ln S) =
\
L
L
B
H
NH
D
AU H
@a
S
∆t
e
F
A HF S
[
B
L@
W
] DA FH \U V X C O V D@ \j N H
L
L
S0 (t), S1 (t), . . . , Sn (t), D1 (t), . . . , Dm (t) S0 (t) . . . Sn (t) B
CB
A
∂f 1 = , ∂S S
OF
l
CJ
D
M
Z
B
G@
j
D P@
V
OD
IH
B CU
L
L
W
D
ln S(T ) ∼ N
B
P
H
H
_G
@
MH
W B
F
B
@
L@
W
NO
m
L
D
G
B oU
A HF
Dj (t)
OT
AU
DYq H
ZS
@
σ 2 (T − t)
B
U K
$= # ' $ ( Z $$ = ( & Y ( ( % "X ( Y \ * *
'
'
)
Z ( = #
'$
S ] MG H
)
SH
B
OD
C\
q
\
=
Y
f (S, t)
°
ef
AF
] e
]
l
k
ª
n
a
SB
OT
B
IH
YC
L@
OT @
CH
LD
DY
L@
IH
DZ
M\ p
e
¬
¡
X
¡
¶
ª¡
í
¥
£
p
¥
?¡
ª
n
a§
a
¥R
ª
¥
§
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
DE
B
B
B
B
MH
[H
SB
F
B
H
G@
AB
A
M\
A BF
q
\F
=
C8
:=
@AB
8=?
9>
=> <
w|
@
w z
}x
x z
~x
5
tx
}x
z
2 w|
z
w
}|
x
6
v
~x
t
t
s
u
{
wt x
yz
}~
5x
}x
y
ys
z
w
3s
~|
v
8 t
] e
°
-
,
V
Y
OK B
CH
M
B
@
F
B
B
AL F
M
YS
AY
O
H IB
ZS
] L
B
H
B
@ AB
I
MO
SH
A BF
LM
D
N
C
\
lL
JL
W
M
EA
@
B
H
BF
H DC
JSH
H
F
m
@
H
OD
YA
MO
AF
IB
ZS
@
SH
AB
GH U A
lL
J
IH
MP
DZS
l
D
D
V G
V
G
M
X
X
P F
B
@
SH
AB
G
YM
A
M
J
L@
&
%
$ #
SB
H
B
H
H
] A
M\
m
AF
A
\F
A
IP SB cGD
D
L
JL
W
D@
Z
ND H
D
O@ ª
¬
-
@
@
] mF Y
A
M
IJ
O
B GL @
L
SP
lL
D
L@
L
DE B
D@
NH S
U
]^J B L D Gª GD @ l
L
SH
@
B cD
V
@
B
L
N
B
D@
C
\B
L
LJ
W
AT @
M
IH
ZS
A HF
LT
cH
CP
H
IH
ZS
GD H
A
e¢
-
X
- M, S @ oB
E
D
D
OLD @
M
l
Y
DB
C
D
] B L NB
I
C\
C
\F
B
H
H
B
@
@
B
m
C\ B
O
cm
\m
LO @
M
AD
G@
Z
GD
\H
GZ
AT HF
L
cH
CP
M
A
M\
A BF
q
\
H
B
®
X
j
-
U
]
D
E
A
J
H
S @
@
L@
M
M
AD
G@
,
V
S CB
D
M
L
AB
CH
C
MH
E
A
C
Dc H
O
H
M
L
S
AB
CH
B
CU
qD
\B
G
MA
AT F
\ R
m L
\B
W
q\
\B
G
A
C
OD F
E
Dn
C
L
DL
M
S B
@
e Di
] Y
-
¦
,
-
,
Q@
IH
lF
DM @
H IH
G
OT
A
L
J
cH
CP
M
B
[
F
m
A
l SB
GD BF
E
DLC H
H
YM ó G
A
L
DL
M
S B
@
B
B \H
L
A
GF S
I
D
S @
@
J
P
L@
SB ] B
O@
M
L
A
CD H
O@
C
-
,
"!
&
[H
Z
SR IB
Y
A
M
'X$
F
B
@
B
@
H
H
AT @
C
L
DZ
L
cH
C
] B
L
DL
M
S
³
e
SB
H
Y
W
DL
G
\
F
] W BF
ZS
GD F
MH
L
M
DB
A
M\
AF
A
H
@ \F
A
M
B
@ AL F
M
S
AD F
H E@
M
U
U
`
+*
M
M
AD
G@
!
&
"
G@
G
B
L@
W
L
'
"
L@ ]
D
CB
- B, e "= e- ] ¦ D ZS
H SX ] @ e P D³ CQ H ] A L SB ZS AD B \S
C
Dq m
I\ B
OF
lL @
DC
H cD H
OZ
SR IB
F ] g
D ] S
OD
D
L
*
*
:
w
$
B
@
V
B
B
M
AD
G@ U
MJ H
GF
DB SP
AY
O
H IB
ZS
] B
L
DL
M
S
ei
¦
X
-
,
[
ZS
M
\
G
OT
@
H
G HF
Z
]
B
B
B
P@
@ [ \ OH
Oó G
OT
IH
Cm
L
\
M
JSB
ZS
L
U LD @
LoA
D X
] ³
e
-
,
dSi (t) = µi Si (t)dt + σi Si (t)dw.
D
OF
tx
ASH
Z
D
CB
JL
H AB
L
Z
CH
θ¯ = θ0 (t), θ1 (t), . . . , θn (t), θn+1 (t), θn+m (t) .
\H
L
L
9
S7
S
DC @
CH SB
L
L
W
D
Y
L
DL
OZ
F
CD B
L
B
CB
Bî
B AD HF
L
L
O
B
L@
W
S0
M
t
M
L
AB
CH
F
MH S
M
AD
V
H
IB
ZS
Y
IO HF
[
F
F
@
m
l
OT
@
F
HF
@
H
B
IH
C
L
\
OT @
DLC H
D
C
[\
[
S
DH S
A
C
J
a
a
¢
?
X
H @X ] DC
Z
F
] ei
-
,
dS0 (t) = r(t)S0 (t)dt
M
Πθ (t) > −K A BF
M
x
P
y
6w ~8 I | 7 S
wx
}x
5z
θ¯
L
D@
J
dx |9s B A A 6 | Is B H F 4 |9 6 wx ~5 x Y Z t 3 ASH '( s 1t| wIt| ~ 9 O K w| B us ys Y wv 5My| y~ V - ) x z | t} A V E :89 x M~dx OD BF H xs t ; Yq
L@
t
~x
w
~ z
x s
| t
x
}|
u
7
0 ... n t Πθ (t)
NH S
L
F
¯ θ(t) L
M
D P@
4 5 x s o y~
w|
Sr 7 ys o 8 ys
¢
ν
Ij O B @
OT K B
F
c(t + ∆t) c(t + ∆t) =
m
~
C
M
K
]
ASH
R
9
í
H
G@
A
A
L @ó
OH
O
H
LG @
W
Z
DH S @
S B
B
B
B
C
LD
JE
OD P
@
L@ S
O
H IB
ZS
TA X
J
A
cH
C
L
CB D
M
JL
AB
CH
C
B
JP L
M
B DLC @
E
D
OD
M
CD
L
JU
E@
\
OF
I\
P cB
C
Y
T
Mj
@
E L V H
NH
D
OL @
,
-
]
@
Q
I
M
D
YM
A
M
B
F
m
@
JF
Z
D P@
AD HF
L
L
MH
F
@
NO H
YF
MH
D
OF
\B
E@
\
OF
CH
ó An
Si
H
5M~x 4 5| F t t tI x
~
S(t)u + cu ν = S(t)d + cd ∆.
w
t
ν
LT
OF
t + ∆t
E
S(t)(u − d) . cd − c u
D
1
í
è ðÚ
æ
.
B
H
F
t + ∆t
Y
S(t)cu (u − d) S(t)(cu d − cd u) = . cu − c d cu − c d
ôè é3 Ú
t + ∆t S(t+∆t) S(t)u + νcu
ô ã êã ßì 0 ù Ý
ãø
Üë éÛÚ
ðè ù ßÛäè ô æêáìÚ
â
] Y
L
DL
OZ
D
S(t)
Û æÝ
Þç
ðßÛ
ÛÚ
.-
ÜÚ ßÜ Þ ßÞ ãìÚ çè Ü çÚ ç éÝÝ Þß æÝ çßàè ßáæ çÚ ðÚ Ý ùâ ðç ô ã éè - éåÛè ßÜ æèÚ ãè å Ú ãìÚ éã á ßÝä ç øÝè Ý æÚ çßàè ç éàè çÝ Ú î Þ æêì î ãè çè ßì ó Ý î è êÛ ðè çóÚ Þç çà è Ý Ý S(t+∆t) K
Ú
çè
ãøÝ
Þé
ß
ν
ßè
éÛÚ
ìÚ
-
è èåæÚ î é ÜÝ éãÝ ÞÚ éÛÚ øè çè ã . ç ÜÝ àè Ú î éÛÚ æÛ ãà . çÝ è Ý æ ðÚ ðç éè ô
ôã
Ú t
max S(t + ∆t) − K, 0
ðçè
éè æÛ
ðç
ðÚ
ßãè
óÚ
ν=
æ
ãà . è
ççè
S(t + ∆t) = S(t)u S(t + ∆t) = S(t)d t + ∆t S(t)d + νcd
îé
ñ
ðß ùÚ Þç ä ÜèÝ æå ÛÚ â é æ÷ ã . Ú à ðäÚ æè çê ð ðÚ ç÷óä çéè æ çßÝè Û ðè ë çßè
ã
ÜÚ ë ßÛ
Ð ÂÁÀ ¾¿ (ÈÉ Ê Á & ) ( ' Ï ÍÊ Å) ¿ * (Ê Þß ðç çÚ éè æ Ú æÛ ããáÚ ðè ç ô ßè ßã / î ìÚ éìÚ Ü àòìÚ Ú ß ßåÛ ù ß ë ä éàÝè éì áÞßÝ ß óî áÝ ß é êÚ ÜÚ ù í +,
Ï
t t + ∆t S(t)u S(t)d u>1 d<1
P (t)
e
Πt+∆t = S(t)u −
óâ
ãÚ
ÜÚ
î
áæÝÚ
çå
ßÚ
S(t)d, cd
öÜÚÚ éÛÚ
é
îó
öè 2 áÚ ìÚ ÛÚ ù ßéøäÝ
æÛ ã Þ ç 0 ðçè ßì èÝ ßè Þç éÝ è ßáæ Ý éßáÝ Ý Û æ æçÝ Ý Ý S(t), c
áæìÚ
ù
Πt+∆t
ßÛäè
çè 4 Ý
çó 1 ßéÝ S(t)u, cu
c(t)
A
Ý
ν
Û çÚä Ú ó æåÝ Ý éøÝ éóÝè ô î éÚ ÜÚ Þß çÚ æÚ îæ à÷Ú åò ß ÷æ ù ßÛäè æêáìÚ Ü ë éÛÚ ãø . è îç â ç Ú
Ü S(t + ∆t)
t
®
56
ç
] ¬
¡
¡
¡
]
B
OD
B
IH
CB
L
DH
A HF S
e h
R
n
] ª¡
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
§
r(t)
∆t
c
Πθ (t) = θ0 (t)S0 (t)+. . .+θn (t)Sn (t)+θn+1 (t)D1 (t)+. . .+θn+m (t)Dm (t).
dΠθ (t) = θ0 (t)dS0 (t)+. . .+θn (t)dSn (t)+θn+1 (t)dD1 (t)+. . .+θn+m (t)dDm (t).
¨
ì
R©
§¨
S(T ) N
H
¬ i
-
,
c(t) + Ke−r(T −t) = P (t) + S(t). ]
F
B
L
L
B
Z H
T H
B
D
CB
MJ H
P
U
U
@X
MJ H
LG @
IH
D
F
B
B
[
B
W
Z
DH
Y
L
DL
H
F
W
ZS
C
L
BF
H
JU
E@
YO F
AY
O
OZ
[
H
IB
ZS
\
D
L
AB
CH
C
MH
L
F
B
H
H L@
W
Z
DH
Y
L
DL
OZ
D
V
IO
F
B
B
F
BF
H CH
W
ZS
C
L
JU
E@
YO F \
AY
O
IB
ZS
H
OH
O
H LG @
W
Z
DH
B L@
í
c(t) Y
E
T
A
L
SB
m
H
JH
W
Z
DH
] F MH
Y
l
L
]
K
S
@
@
B
F
L@
M
lZ
\
A HF
LT
P@
H
X
P
F
B
@
P
A
B
E@
OD H
AH
L
OT @
Z
AT HF
C
IH
D
lH
B
W ] H L@ DW
L
Z D
OD X D
'$
Y
A
M
J
L@
B
ª X
@ L@
W
A
DH
CB î
MH
L@
¦
ZS
D
CH
L
W
Z
B DH
L
D@
I
@
OH
P@
m
DC B
ZS
TA
AD @
Z
O
B
F CH
lZ
CB ] DB L
S
[
F
B
B
H
P@
H
H
L
\L
O
m
Bî
B
IB
ZS
AT
AD @
Z
O
L
c
L
\
CH
L
W
DH Z
L
D@
I
@
OH
@
] W\ B L
MJ H
Z
Y
AY
YMO
Y
L
D@
ZA
J
F
X
¶
F
F
@
SH
A H
" Z
JH
W
DH
"
P@
@ H
@ U
P
B
SB
-
P@
L
H
F
L
JB
L
OZ
D
AT X
J
A
B \N H
C
JSH
A H
JH
W
Z
B DH
A
AT @
I
c
\F X
H
CB
I
\
CJ
V G
[H
Z
S
S
_
&
'
L
T
I
L
JL
G
&
'
U
"
B V B
O@ p
B
P@
@
XF
Y \
Y
] Y
[B
WM
IB
L
D
E
D
X
L
M
S
F
T@
O P@
@
H
X
@
DF
AT @
o
MJ H
Z
S
OX @
G
W
D
D
G@
B
L@
W
JH
Y
, $
L
IU T B
L
DI B
O HF
Z
MH
G
OT
] L
C\ B
lF
G@
G
AT @
M
A@ D S
C
L
cH
C
J
H L@
c
O
IH
L
B
X
¯®¬ °±A¬ D
H
]
F
B
B
S @
B
L@
M
A
I
\
W
D
D
G@
L@
W
Y
AF
Z
\
L
D
S CB
WM
L
D
IB
X
Bî
B
m
Bî
B
[
A
M
J
B
@
P
L@
L@
B
] W\ B L
L
\L
O
H
P@
IB
ZS
Y
L
IH
NH
L
L
O
H
H IB
ZS
I
M
AD
G@
í
F
B
@
B
]
SB
m
AT HF
L
AH Y
M
F
L@
Y
A
o
L
T
L
SB AH
@
Bî
@
MJ H
P
U
@X
AT @
B
IH
ZS
W
\
OT
I
@
OM H
M
ZS
A
I
AZ
\H
A HF
ZS
OD
D
( #
NH
A
A HF
LT
Y
V OD
m
AT HF
Y
B M
AH
SB
M
V H
X'
F
@ª D IU A
F
B
@
B
L@
Y
A
M
E
D
OD
M
\H X
OD
W
D
]
B
@
S @ L@
M
W
D
D
G@
L@
W
Y
q
\B
G
< â
G@ D
¨
R© ì
L@
B B
W
L
¤
¶
H
P@
m
Bî
B
]
P
=
F
B
@
Z [
CH
M
G@
) AD
Y
A
M
J
L@
M
AD
H G@
A
A
] W\ B L
L
\L
O
IB
ZS
Y
L
I
H
X
S
H
B
@
Bî
B
H
@
H
H
H
Bî
B NH
L
L
O
IB
ZS
I
M
AD
U
Bî
B
G@
L
JL
O
IB
ZS
TZA
D
\
G
W
\
OT
I
MO
M
ZS
[B
M
W
\
L
D
IB
Dm
I\
c
@
G@
OD
B
H
A
YI
A
N V G
C
\Y
Bî
@
@
SH
H
@
@
L@
L
W
_A
ó
OH
O
GH
A HF
ZS
D
'
&X
&
X
@ ¬
-
,
IH
M
J n
C
ZS
D
YM
A F
B
@
V
]
[
F
B
P@ E
SH
@
D@
I
CB
L
D@
c
IH SB
YF
AY
YMO
¬
g¬
K
M ] A
\ ± SH K C\ AF O @ ] L@ ¬ OMD D± M
O
C\
SB ] ® B
E ]
!
@
-
,
-
,
B
J
TO X AX ] S @ G¡ c @ J L@ P M JS @ A B G L F B V Y A L E \ BF H D q ZS A BF F \
DS
@ L@ JS
c
[
S @
SH
H
L
B
cH D
OZ
IB
ZS
Ll
JL
MH
OD
C\
K
H
@
B
m
M\ H
GF
DB SP
ATE
D
\H
OZ
\SH
A BF
LM
D
TA HF
L
cH
CP
M
A
I
L
G
] ±
¬
-
,
ϕ[τ,T ] = S(τ )e(r−q)(T −τ )
L
M
AF
A
S
@
F
L@
L
J
Z
D
V H A E
AZ
AB
CH
\
L@
C
H
W
MH
Z
V H A E
L
A BF
MP
H
DH
T
M
M
L
AB
CH
C
M
Y
F
m
H
MJ
L
A BF
MP
H
B L@
W
L V H
E
NH
D
DN H
SH
H
F
D@
AZ
J
D
M
B
F
l@
lZ
\
M
W
CD H
A@
A
W
D
D
G@
L
JB
B
B
OT
A P@
Y
H
L@
H
ª¬
ZS
D
]
GD @
C
E
¤
X
V
H
S
W
B
CB D
M
@
B
L
B
SB
V
m
A HF
LT
AH Y
M
F
B L@
OT
AW @
AB DH S
L
L@
AX V D ZS S @ [ [
DS
L
D@
MH
M
A BF
q
\F
SX @ P CH
C
DZ
Y
W
D
B
BF
H
G@
TA F
\
R
X
@
N
L
J
M
LIJ
I
M
D
I
S
B
@
B
DB
A
M
E
I
@
NH
] l
MJ H
IH
¤
q%
H
AH
AZ
\
D
OH
O
AY
L
OZ
F
AZ
\
D
OH
SH
H DN H
A
YG
IO
]
Y
SB AH
\
L
D
IB
D
k
V
B
Y
l
V
L
D
@
B
S CB
M
L
AB
CH
CU
Dq
\B
G
MA
L@
H
B
L
AB
CH
CU
qD
I\
\U
JSH
A
GL
M
X
V
] ¬g
-
,
ó
U
OT K B
ASH
Z
CJ
AD F
Z
\H
IH
_A
$
$D [
@
L
J
M ] C
[
M
S
H
SH
H NH
A
B LG @
W
S @
¤
I
@
NH
H
S
H
S L
B
CB D
M
L
AB
CH
C
Dq
\B
G
A
M
JSH
A
V G
@
H
B
B
IH
l
I
L
DE
^]JSH
A
IB
D
G
TA
AD F
Z
\H
I
A
N\ H
U
U
CU
JSH
A
G
Y
A
M
L
DW
F
B
@
B
H
τ
] \AZ
H MG H
L
W
Z
DH
DL
JH
W
Z
O
P (T ) = max K − S(T ), 0 .
D
¡
G
V
OL @
A
G
T c(T ) = max S(T ) − K, 0 .
OH
D
L
L
JH
W
Z
DH
TA F
R
F CH
H
ZS
M
E
D@
I
\B a OT P
U
W
L
DL
B
CP
] F Y
L
AT @
c
§¨
ϕ[τ,T ] = S(τ ) − I er(T −τ ) .
O
MG H
L
W
Z
DH
m
B
U
D
]
H
B
JB
W
P (t)
L
D
oH
L
H
AH
F
m
@
L
W
c(t)
$
'
F
B
@
P
-
¤
S(t)
Y
A
M
i
AY
I
NH
F
\
H L@
P (t)
J
¬
M SH
B
A
M
AD
G@
U
[L
W
m
H
U
P@
P@
m
] L
D@
ZA
J
F
DP
D
OD
OL
SB
D
C
I
L@
M ] H
AL F
] B
L
B
K
MO H
W
GD F
D
O
M
ó
C\ H
H
@
L
JB
W
B
B
MH
D
L
D
H
B
\
C
\
1
B
1 wt2 = wt ∆w + ∆t. 2 2
I
-
SB MH
L
_ DM
G@
D
L
M
F
B
\H R OF
c
L
DZ
OD
B
E
A
24
M
1 \
[
,
B
GB
A
B
@
L@
M
S
A HF
LT
YS
A
c
@
H
W ^]J B L
B
B
L D
D
C L
D
2/3
@a
,
σ = 20% 1/2 [H
f
®e
I
OD
IB
ZS
H
H
SB AH Y
M
G k
l
W
D
] OD
D
G@
L
D
o
MJ H
2
@
N
F
BF
H
k
j
OL
\
B
L@
W
@
D
o
6
MH
]
V
W
A
CJ
CB
Om
qT H
CH
Z
< å
SB
I
NH
Z
Πθ
@
IB
j
=1 O
∂f ∆t + ∆w. ∂w
N ] IH
]
∂2f ∂w 2
P
=w wt
B
∂f ∂w
SB
24
l
H
ZS
B
SH
B
1 2 2 wt
MJ H
=0 2/3 6 C64 · (2/3)4 (1/3)2 = 0,6584 4
IH
1 ∂2f ∂f + ∂t 2 ∂w2 20 2
E
S(0) = $40 µ = 16%
N
E
A
B
AY
M
A
MO H
I
¨ ì
CJ B
¤
24
@
V
V
A
k
B \m
L
DL
M
@
R©
lF
1/3
M
@
∂f ∂t
4
AD
I
I\ B
O HF
A
CJ
CB
OP k
F S
C\ H \
H
] W BF
§¨
t0
G@
L
∆ NH
1 2 2w
V
¤
f (t, w) = ]
∆ f (t, w) =
JB
f
®e
W
P
¯®¬ °±¬
ª¬
L
ZS
6 2
B
]
C
OR H Dc H θ tθ1 > t0 Π (t1 ) P Π (t1 ) > 0 > 0
L
B
LD V H
OT
A
6
@ª D < æ IU A \ R AT F B S @ F ZS IB O B D B
] l
MJ H
I
0 θ¯
V
V
[
H
e
B
H
] ¬
í
@
B
@
H
@
H
@
H
H
@
F
SB
m
F
B
@
\H
OZ
Y
A
I
cD H
E
T
MGD
A HF
Z
Y
L
I
I
E
A
C
Dc H
OZ
SR IB
N
O
I
CP SB
YS
IO
O
IB
ZS
TA @
¡
k
ª
n
a
¡
¶
ª¡
¥
£ MH
M
p
¥
?¡
ª
A BF
q
L\ F
c
O
I
n
a§ Q@
L
a
¥R
ª
¥
G
K
]
JSH
NH
MH
LI
K
¡
¡
¡
n
] ª¡
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
§
ei
e
e
§
CB
] S MH
S @
B
H
[B
V
AE
P@
SB
A HF
LT
AH
Y
M
E
D@
I
m
MH
D
\H R OF
B
F
DSB
Z
CB
L
D@
I
Dc H
C
GD BF
E
AD @
CB
A@
C
[
F
SH
F
B
Bî
B
OD
M
L B
H
E@
\
U
U
H OF U
L
L
O
IB
F
@
SH ZS
L
OT @
C
L
Y
AY
YMO
Y
AF
Z
\
MB
W
Y
F
ª¬
D@
S
[B
C
Y
AT
MB
I
CJ
D
O@
AZ
V G
]
j
ge
K
-
,
, e Dh
L V] H
F
@
NO H
¯®¬ °±¬
B
Y (0.75)
@ [B
I
\
F
Y
Z ] AF
X
V
[
G@
S
@
H
B
B
BF MB
W
Y
C
C
L
J
OT
AW @
DH S
AY
F
O P@
G
L
CJ
D
O@
AZ
K
B
O B
U
S
SB AH Y
M
F
JH
E
A
CJ
D
O@
AZ
GU D@
L
J
OTE @
L@
X
K
TA X
J
L
H
H
¢
[H
c D@
S CB D
M
L
AB
U
L
I G
A HF
LT
CH
H
C
M
B
m
\L
AB
CH
lL
LJ
O
C
L
JE
V
O
G P@
L
AB
CH
M
O
I
V
< L
¨
R©
§¨
L
DL
M
m
B
@ D
G
AZ
J \B S
AY
ì
B
H
H
cB L
B SH
M
A
MO H
I
MH V \
S
@
OH
S [B
M
A
G@
F
C
H
YG
B AH L S
A
E@
m
NH
L
L
E
m
@
G P@ O
I
NH
V G
U
è
]
p
H lL @
OD
O
DC
M
CJ B
Y
CP SB
V D
CJ
S
U
K
Y
A
= )
Z [
F
B
@
M
P
K
P F
B
@
OD
%
B
GZ
L@
o
OD
D
V
U
L
J
E
D
OD
H
B
MU
L
E@
\
B
H OY F
L
DL
O
GH
AH
NH
L
E
AD
\H =$ \$ G ZD@ X OD ZS & IH " c@ @ ' " > = V L@ M@ JP =" S MA@ CB B ^ Y YXF G AD GZ \H M V GD ' Z L@ * Y
A
M
J
L@
C
E
D
\H
OZ
H
m
Bî
B
V
L@
B
] W\ B L
L
\L
O
IB
ZS
P@
CB
q
I\
\
YM
CB
MH SB
F
H
Bî
í
X
R
H
P
@
[
B
S
F
I
M
L
S @
@
GB
IB
Y
D
L
D@
I
cD H
H
NH
G BF
E
AD @
CB
A@
C
Y
L
DE
L@
Y
Y
MO
[B
M
M
AD
G@
ATZ
D
\
G
OD
D
AT @
B IH
ZS
L
[H
AF
B
@
ZS
IH
D
o
H
A
GH
Z
F
H
ó
NO H
_A
$
\
F
]
oH
L
¨
-
,
Y
L
D
B
oH
SB
B
L
DE
F
B
L@
P
O@
M
L
A
D
H L
L
OT
A J
PF
MH D SB
AH
Y
DF
A
I
D
B CB
L
D
U
U
p
I
m
AT HF
L
cB î
L@
[H
lL @
0.95
IP
W
L
]
U
FU
H
m = 2 · 0.75 + Y (0) = 1.5 + Y (0) σ 2 = 9 · 0.75 = 6.75 P{Y (0.75) > 0} = 0.95 m Y (0.75) − m = 0.95. P >− σ σ L
E
D
A
cH
H
B
IB
ZS
W
V L
A
c\ B
CH
ZS
J
F
SB L@
B
V
E
m
CB
L
CB
TL
@
F
D=
L
L MB S
X
B
A
I
H
AH
F
D
L
D
B
S CB
M
JL
AB
CH
C
M
Y
OK B
ASH
Z
H
NH
MH
X DB ] SP GF
M
F
@ CH
I
c
G
H
@
@
B
D
L
JB
W
L
G
L@ JS
c
AX S @
DS
Y
M
D
AF
A
\F
A
\
O
Dn F
SH
H
B
H
@
SB
F
ZS
D
m
A HF
LT
AH Y
M
\
OF
E
D
C\ H
A
GL
L
M
S
AT BF
H
B
V AB
L
L
J
Y
P@
V
C
M
S L
B
CB D
M
L
AB
CH
)
=
"
$
Z
D
B L@
W
H
NB
OD BF
V
R
@
S
SH
ZS
±
e
R
@
K
, D
] B
O
C\
H
¯®¬ °±¬
ª¬
[
S
P@
L@
A BF
MP
B
]
*
Y
A
M
J
K
I
m
A HF
LT
cB î
L@
[
F
B
@
P
L@
L
CB D
M
L
AB
CH
C
MU
L
JL
MH
S
DCSH
F
OT K B
ASH
]
D
lX
Jm
YO
IO
B
L
DE B
P L@
M
SB O@
L
A
D
L
J
OT
A
B
@
MH D SB
I
A
cD
G
U
U
¦
N
E
P
B
SB
F
@
IH
1/2
O@
V
k
C\ H
AH
Zó
DL @
¶
F
-
,
9
AZ
GU D@
L
G@
A
Y
B
V
]
I
D
D
e
¬
C
YG
L@
L
IB
M
ZS
D
F
m SB
@
U
S
F
Bî
@
P@
Z
c
"
B
P ln S(T ) − M(ln S(T ))| 6 1.96σ = 0.95.
@
OH
J
D ] L
B
1 G
M
B
X
O
IB
F
@
B
L@
W
]
NH
E@ V GB
\
OF n
V
lL
J
E
MGD
H
B
A HF @
A
Z
G@
AH L S
L
W
YM
A
I
F
D
D@
IH
ZS
ln S(T )
O
W
@
S ZS
Y
W
DL
G
V
B
@
@
M
GH
NH
D
@
SB
MJ U
JSH
\U
H
O
I
S(T )
I
H AT BF
OD
A
M
L
JB
IB
M
ZS
K
]
B
B CB
L
D
O
]
MG
MGD
K
MGD
@
A HF
Z
F
SB
m
MGD
E
m
`
3.759 − 1.96 · 0.14 < ln S(T ) < 3.759 + 1.96 · 0.14
OTE @
L@
V A
m/σ = 1.645 MH
L V H
B
D
\
Y
L@
L@
B E
D
OD
E [B
\H
A
G@
AH L S
H
B
H
@
IB
ZS
L
L
I
H
AF U
L
L
T
TLA HF
P
@
L
E
K
A HF
Z
CB
="
S
S(T )
L
H
2.7738 OD
OT
H
SH OT @
C
L
YM
U
∼ N (0, 1) m m Y (0.75) − m =Φ >− , σ σ σ
Y (0.75)−m σ
O
A@
MH
IB
j
Y (0) = 2.7738 O
E
W
D
DL
K
_
L@
E@
\j
Y
L@
F
SB
F
GU
F
m
SB
K
L@
L
Y
H
A HF
Z
C
L
I
F
C
L
I
T
7
Y
DC
]
A
C
IH
D
G
\
Y
L
Φ
E
V B
o
V
B lL @
W
B
O
P
OT X H
√ 1.645 6.75
AT X
B
DE
-
,
OZ
L
DY
B
@
L
J
E
F
B
U
F
S
G@
F
0.95
LJ
X
Φ(x)
OD
L@
0.95
OD
@
AH
¶
B
¬
¬
m
K
A
@
H
m
H AB
W
ZS
H
D
YG
B
B L
t
c
H
D
R
I
N
]
E
@
N
OZ
F[τ,T ] (T ) = S(T )−ϕ[τ,T ]
MGD
TY
@
A
E@
@
@
A HF
Z
MG
MH
L
B
AY
D B
H
T
O
K
]
H
[P
-
,
T
I
NH
¤
B
IB
ZS
L
A HF
Z
L
AB
O
G P@
@
YA F
L@
X
D
CB
S
O
IB
T
OD
D
1 Π t1
I
S
@
t ] O
M
Z
¬
¬
B MH
] LI
I
M
D
@
F
SB
m
CH
I
B
GF
DB SP
H
M
B
A
G
t2
CJ
D
-
@
F
@
ϕ[τ,T ] = S(τ )er(T −τ ) .
SB
m
F[τ,T ] (t)
ϕ
h
F[τ,T ] (t) = S(t) − ϕ[τ,T ] e−r(T −t) .
t2
E
O
K
U
E
@
NH
Q
I
L
AB
W
Z
r
K ]
SH
B@
B
Q@
M
AD
G
L
W
B
I
DB
L
B
K
C
M
F[τ,T ] (t) = ϕ[t,T ] − ϕ[τ,T ] e−r(T −t) .
V G
M
H
B
L
ϕ = ϕ[τ,T ] F[τ,T ] (t)
T
C\
M\
AD
G@
C\ H
q
DE
\H
T
V
B
ZS
L
T
L
JB
W
Y
P
MH
I\
X
@
L
@ MD H
L
Y
D
T
e
@X
\U
H
OZ
L
MGD
D
CB
F
@
MH
CB
S
C
τ
t1
,
, H
@
CB
I
@
A HF
Z
Y
L
S
Z
D P@
ZS
H
M
Π t2 = er(t2 −t1 ) Π t1 .
IO
¬
Z
A@
Z
O
V D
M
AD
IH
M
L
AB
CH
τ F[τ,T ] (τ ) = 0
B
@
cD H
L@
G@
H
@
F
m SB
JL
AB
τ
L ] B
D
S
MJ
DM
D
A
E
CB
S
C
t
c
MJ
TA
@
R
GF
SP
M
T
t2
¬
Q
B
S
I
AT F
Cn
D
t2
L
M
@
IH
H
l
ϕ[τ,T ]
AF
A
H
SR \F
A
D
\
K
AT X
J
t
AD HF
B
IB
B
L@
L
T
τ
D
ZS
\
AY
H
W
D
Π(t2 )
t1 Π(t2 )
]
S CB
M
L
AB
CH
O
M
Π t2
t1
e¬
±
e
C
] ¬
¡
¡
¡
SR JSB
C
D
n
] ª¡
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
§
0.95 t = 0.5
ln S(T ) ∼ N (3.759, 0.142 ).
S(T ) ∈ [32.5, 56.6].
∆Y = 2∆t + 3∆w.
Φ(m/σ) =
1.5 + Y (0) =
H F ]
B
@ e
N
B
twt I
GD F
MH U
@
B
X
MG @
V
AD HF
L
AF
E@
h n
@
B
H OD
B
IH
CU
L
M
JSB
ZS
L
û
û
ÿ
û B
ù
÷
P@
L
J
OT
A
B
MH D SB
I
L
[J
AD
L@
E
D@
I
U
AD
G@
L
JB
W
U
U
P
B
SB
@
@
N ] IH
E
M
T = 0.3
[
H
\U
L
L
B
U
@
F
e
S l@
CG
YM
F
_
IH
N
A HF
LT
V
M
AD
cD H
@
G@
L@
B
@
H
@ª U
AT HF
L
A
Z
O
AD
] l
MJ H
@
H A HF
LT
IH
l
I
Y
@
CB
I
H
H
P@
] TB L
LI
IH
D
L
JB
W
L
E
L
U
N
H
H
S @ L@
M
L
JB
W
L
E
L
JS
TLA HF
OT
U
A
O B
@
] IH
NH
NH
A BF
oB
W
L
H
@ª
H
U
L
J
E
D
MU OD B
L
E@
\
OF U
B
A HF
L
Y
AH
DE
B
P L@
L MB SB
B
H
OTE @ L
B
]
D
B
D
O
F
@ IB
ZS
S
@ª AD U
] L
[
o
L
T
C
@
B
\H X
D
A
O
l
MJ
Z
L
DE
SB
B
B
B
An
B
B
W
ZS
@
F
BF
H
H
NH
L
E@
\
OF
L
U
B
B
B
l
Dq m
I\ B
OF
L
DE
A
M
S
B
B
L@
W
L
CB D
M
W
\
L
D
IB
Dm
I\
c
@
N
"
<
ì
ë
¨
7
¨6O
R L
PI
Q
\
W
ZS
L
F
BF
H
U
JU
E@
< ñ O \j
F
B
SH
B
L
D
m
B
oH
L
H
AH
AY
M
A
MO H
I
B
@
H
B
IH
N
W
\L
G
G
C
L
J
OT
AW @
DH S
A
A
I
\B
L
L
D
ZD @
CH
G
X
@
V
L
D
S
B
B
SB
F
B
I
G
AZ
L
J
H
H
cB
L
B
m
A HF
LT
AH Y
M
AZ
V
H
@
A HF
LT
OT
AD @
O
M
IH
N
P@
O@
H
@ L
M D@
G
CH
L
OTE @
L@
CH
BF qB
C
I
N G@
G
U
P@
D
SR
IH
] N
B
G
G
AZ
L
J
H
H
cB
L
U
< ð p
L
DL
M
m
B
@
SH
B
\B S
AY
M
A
MO H
I
\
L
D
DZ @
CH
]
@
B
H
I
F
m
A HF
LT
cB î
L@
@
O@
AS
M
WG
L
G
]
O@
AZ
GU D@
L
OTE @
@ª J M
@ ] O@
AS
M
G
@
O@
ZA
D@
GY
IO
O@
M
[
L
A
D
L
J
OT
A
SB
B
U
@ª U MH D SB
I
AD
L
D
F
B
S
B
B
qB
C
I
N
L
M H
@
V
@
P@
H
@
A HF
LT
OT
AD @
O
M
O@
AS
M
G
P@
B
G
G
AZ
L
J
H
H
cB
L
U
< ï
L
Qì ì
R
<
H
p
N M
6L
QK
6:
¨© J
8
RI6
:
6G
:
σ = 1.2
[H
M
@ª AD U
] l
MJ H
IH
H
M
W
D
D
G@
B L@
W
¶
(
σ = 1.0 0.95
W
H
MH
B ] B ]
@
U
Y
A
ù
OT
W
Y
JS
A HF
LT
36.5%
YA
S
H
IP O
M
L
L
B
IH
MH
H AD @
O
M
q
\B
G
S(0) = $40 20% ln S(T ) 99%
L@
L
AB
H
H
-
P cH
U
V
l
MJ H
IH
A
G
M
@
_
V OD
m
TLA HF
σ = 3 S(0)
GF
X
W
ZS
±
, e
B
CK @
NH D@ S
OY
N
On H AD @
S
L@
E
D
B
Y
G
@
C0 = 20
DB SP
V H W B L@
E
A
V Y
]
]
B
q
I\ ]
IJ B
ZS U
L
B
B
B
<
âò
[ U
E@
\
B
B OY F
L
D
O
IB
F
@
OD
YM
AH
SB
µ = 2
L
B
F
B
@
ZS
@
B
± û
DE
L
D
O
OT
AD @
O
[B
W
L
B
X
F
'
S(t)
OT
A
O
O
IP
@
û
DM H
D
S(T )
N
De
SH
NH
L@
P
IB
ZS
< â \ R AT F
â
H
X
D
CB
S
S ZS
W
\
U
M
]
@
L@
M
1.1
A@
-
,
M @ e V D
AD
G@
Y
L
û
ÿ
R
\n OF
L@
E
V OD
m
M
W
\
L
D
IB
@
Dm
I\
c
L@
OD
@
E@
AB
A
\j
O
M
F L@
Y
S
C
MH
@
H
] Y
B
@
ó
B
SB O@ M
F
@
F
@
@
m
W
D
D
G@
1.5
Q
O
OZ
IB
S
YA
L
DH
< å
S
[
S
G
¶
]
T@
A HF
j
B
@
B
@
L@
W
Y
q
\B
¤
C
IB
j
L
DL
M
\
L
H H SX @ P MH D@
L
B
H
B
@
A
M
E
B
3
O
H JB
L
Q
â
L
D
OD
M
Y
]
S L@
M
AT HF
L
D
OD
M
\H X
OD
G
L
DL
M
m
B
@
SH
B
\B S
AY
M
A
MO H
I
\
L
D
DZ @
CH
C
SH
H
ZS
JH
A
p
A
D
Y
AH
W
\
X
L@
A HF
LT U
E@
SB AH Y
M
F
P@
@
\j
O
S
F
B
@
] DB L
S CB
SB AH Y
M
F
σ = 4
C\
Z
F
B
S
A HF
ZS
Y
A HF S
R
[
M
D P@
B
SB
OD
m
A HF
LT
YM
A
W
D
D
G@
M
W
J
L
D
L@
W
D
< â
µ
Y
K
L
H
F
L
B
A HF
U
L
@
AB
j
]
SB
F
AH Y
M
M
E
B
IB
P
Y
D
G@
µ
CB ] DB L
M
A
S0 CP
A
D
OZ
W
[B
M
§
M
O
MJ
[
B
[B
E
LT
n
OX H B MB
A
M
T@
A HF
] DB L
µ = 18%
AY
B
H
IH
D
H
Y
AH U
SB
V
]
]V
]
B
B DB î L@
A
IU
D
L@
S CB
M
W
J
L
D
IB
P@
@
D
OD
S
3/5
µ = 3 S 5 S(6)
B
L
E@
F
B
D
P
B
SB
S
D P@
L
A
m
AB
V
L
B
B
D
E
B
L@
]
L
J
E
D
B
SB
F
B
dC = µCdt + σCdw,
Φ(t) 0.998650 0.999032 0.999313 0.999517 0.999663 0.999767 0.999841 0.999892 0.999928 0.999952
D
S
]
C
]
B
J
E
D
OD
M
L
J
E
D
H
U MU OD B
L
M
@
D
H MU OD B
A
−x2 /2 √1 dx −∞ 2π e
t 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9
N
D@
C
[
M
@
\
YO
L
D
B
YD
L
DE
P
@ L@
M
IH
L
JU
V6
\
C\ H
L
B
V DB TL E Y Z O HF
O
X
G
H
E@
F
OT
A
\
OH
E
A
AT @
C
L
V1 V2 1 −1
L
D
D
@ L
Y
IB
F
@
AH
S
@
j
E@
W
Y
q
\B
¤
E
S
H
m
F
MH
W
\
OD
A
AT
M
D@
A HF
ZS
H
]
U
V1 + V 2 + . . . + V 6
SB
M
A
] YG\L
H
C
OD
W
D
\H X
F
B
@
G
D
G@
YM
A
o
B
L
3/5
L SB
wt W
CJ
D
M
D P@
B
SX G HF
HF
m
@
V
W
D
D
G@
B
L@
W
L
T
< å
L@
Φ(t) 0.977250 0.982136 0.986097 0.989276 0.991802 0.993790 0.995339 0.996533 0.997445 0.998134
_ DM
H
ó
L
DC
T
S
OSH
M
A
N
D
ZS
D
\
_
C
\
2/5
D
O
M
V
V
]
MB
O HF
Z
Y
E
TL
D
L
B
B OY F
L
D
O
IB
ZS
S
V
OD
]V
]
B
U
]
H
cD H
OZ
B
L@
P
S R DE B IB ] L
æ
t 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9
C\ H
D
V
L
J
E
D
OD
@
<
50 1.04
F
V GB G@
B
B
B
B
OD
M
M
P
DB
MD H
A
N\ H
F
50 2
BF
H
A
OD
IB
L
D
O
IB
V1 V2 V8 P{Xi = −1} = 1/2 P{Xi = 1} = 1/2 X1 + X 2 + X 3 + X 4 = 0 X1 + . . . + X8 = 0 B
S
C
D
CJ
D
2/5
ZS
B
ZS
H
H
L
JU
E
D
OD
SX G HF
è Mp J L H B I \S V
<
Φ(t) 0.841345 0.864334 0.884930 0.903200 0.919243 0.933193 0.945201 0.955435 0.964070 0.971283
Rt
W
B
F
@
S
PB î
J
A
4
SH
ZS
S
E@
\j
O
ã
t 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9
AY
M
k
B
B
B
H
<
P{Vi = 1} = 1/2 X
A
L
MH
LG @
1.06
A
Y
AH
SB
Φ(t) 0.500000 0.539828 0.579260 0.617911 0.655422 0.691462 0.725747 0.758036 0.788145 0.815940
MO H
CJ
CB
Om
j
M
2
I
-
,
Tq H
CH
Z
ä
4
\e
I
M
Dm
L
<
S
] W BF
DL
Φ(t) =
e
h
ZS
t 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
] e
] ¬
¡
k
ª
n
a
¡
¶
ª¡
í
¥
£
p
¥
?¡
ª
n
a§
a
¥R
ª
¥
¡
¡
¡
n
] ª¡
¶
k¡
í
R
k
¥ª
n
a
ª¡
í
¥
§
g
e
§
0.95
dC = µCdt + σCdw,
∆S = µ∆t + σ∆w. 3
r = r(t)
S0 (t2 ) = er(t2 −t1 ) S0 (t1 )