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"5 > ($)W)+R2456? * normal
HJw K
dF = σN dW
?
J
H
w
F (0) = f,
? K
8j*=)4(*)+*11" σ D:-
%!)&>/: j%-ZY
)%=N& >
1FU5 Call [ ]"$ 7 "()+* "S
G*1S2456$*1
?
A
$%</F )N-/24$
N ormal 7
:9
@
|f − K|
V (t, f, a) = [f − K]+ +
4π
$- `]$24<)5 m) (` M~!2Q
*1M~!24"S5U !1FW5
A
V (t, f ) = (f − K) N (
Z
∞
(f −K)2
2σ2 τex
N
e−q
dq.
q 3/2
Call
√
f −K
f −K
√ ) + σN τex G( √ ),
σN τex
σN τex
†‡
8
N
$%<*1S5=%!1B U !29)+*:D
G
$%G*)45$
T!">f)+]1$ 2
1
G(q) = √ e−q /2 .
2π
€*D3-*)1GN& >*> !TFW5 >* !1j%
`*12456*
A
*24&3e*);(*=)*T!" normale
("$1{
j5
*4 O(2 ) 7
*=);(*=)!1*1"
normal
$%G"f&)+*h9-/$*1
5 ">
)+<*>245 6*Qvw3x`yk%X/
x2
θ
1
=
[1 − 22 2 τex ].
2
σN
(f − K)2
x
normale
24 *-T!>$%<5 "$>
)+
σN =
f −K
θ
[1 + 2 2 τex + . . .].
x
x
} P)3B]%1;)+*Z 5 .) !mF29)!1S-/!!$-E!245 *
* F
1l !"$-"$5 )&>A -/2;24
A
σN = (
8
2;24
O(2 ) ?,@
x = z[1+O()] 7
P]$,"$"$-/!
f −K
z
)(
)[1 + 2 (φ1 + φ2 + φ3 )τex + . . .],
z
x(z)
f −K
z
=
α(f − K)
R f df 0
Hw ‡ K
?
K C(f 0 )
f
Hw
df 0
)−1 .
αC(f 0 )
"
? K
:~J)-E! .! !"$&:*)P5 9- *1"M2;n :551a#
7 )%Z)+&.5R-/ ! f orward 5 )++CD Z 5 &Y f (Z*
A
A
@
XM~J)-E! :% ()+&<$%
Stike K ?
ζ
z
1
= (
f −K
x(z)
8
=
log(
√
Z
K
1−2ρζ+ζ 2 −ρ+ζ
)
1−ρ
,
ν f −K
df 0
=
[1 + O(2 )].
0
C(f
)
α
C(f
)
av
K
να f
ζ = νz = R
M!!2; f = √f K $%G*=)P24n Mf"$24"/!=N& >5 f ~ K
<-/ 0-1av$Z φ 7 φ φ &G5.!24$<5S-!-E!1
7 (? !T{])& 7 245 *19
1
8
2
2 φ1
=
2 φ2
=
2 φ3
=
3
O(2 )
1
αz p
2γ2 − γ12 2 2 2
C(f )C(K)) =
log(
α C (fav ) + . . .
2
z
f −K
24
1
x
2 − 3ρ2 2 2
log( [1 − 2ρνz + 2 ν 2 z 2 ]1/4 ) =
ν +...
2
z
z
24
1 2
B 0 (νz0 )
1
ραν
= 2 ρναγ1 C(fav ) + . . .
4
B(νz0 )
4
γ1
=
γ2
=
C 0 (fav )
,
C(fav
C 00 (fav )
.
C(fav
HJw ‰ „ K
?
HJw ‰
| K
?
HJw ‰ K
?
H w ‰ ‰ K
?
H w ‰+† K
?
BLACK − SCHOLES
9!"$ 24 7 uZ)+ #**UN& 7 4*U245 6*
~L !2Q
*1
V (t, f ) = (f − K) N (
$ "
*1U !TFp5 N ormal 7
A
Call
†
[ !#"$uD05 "U
)9*)
√
f −K
f −K
√ ) + σN τex G( √ ).
σN τex
σN τex
v
`*1>2456$*1Qvw:x`y -/>
1Fl$%35 "
)+<*>24c24~!2Q
*1 7 245 *4 (*)+*11" normale 124 7 *-T!
O(2 ) 7
# !(UN& U!1*>*=)
A
H w ‰ … K
?
ζ
α(f − K)
).
R f df 0 .(
x(ζ)
0
σN (K) =
"
K C(f )
Hw ‰ K
1
2γ2 − γ12 2 2
α C (fav ) + ρναγ1 C(fav )
?
24
4
HJw ‰ K
2 − 3ρ2 2 2
+
ν ) τex + . . .],
?
24
)m(p
1 − 2ρζ + ζ 2 − ρ + ζ
).
x(ζ) = log(
1−ρ
P5 Fi !24Z>~J)-/$ !;5 )
Q* /F !$!%t5
5"!241
&;s-24#24$;524
)+& 7 )m($-9
!"$5U5‹j)+W~J)-/$ 1 + [. . .]2 τ ]A < S-!-Eσ!N1W5* !5!5 1
A
} R]$<"/]51$3524c24*1>ex-$)<5 P ut )+`*=);
)+!T!" Call
[U/ab/ 7 *1 !1FW5 P ut $ ]"$
− P ut ?
A
`*2;56$*1Qv w3x`y $%35 "M])+
[1 + (
8R*=)4(*=)!1*1"
z V (t, f ) = (K − f ) N (
normale
=?2C/5>A=A>f/5
9
√
K −f
K −f
√ ) + σN τe x G( √ ),
σN τex
σN τex
24 *1=-/1>D35 "$
)+<*=);2;c$24~!2Q *>N
M*
Black − Scholes
')>Ay 2=A-6+v/5-
Call ?
€*Š$%: !"/~"$!)B *P5;N
+$:*$: !TFO$j$2445Q(*=)!1*T!" Black − Scholes } OZ)+ #**14N& Q*; 1FO5 A
5 Strike K /35M$24
G5 /FZ-/=-/ τ 7 %
`*M2456$*1 log − normal 7 ? D 35 "
)
Call
ex
A
VCall = f N (d1 ) − KN (d2 ),
)m(-
d1,2 =
2
τex
log(f /K) ± 21 2 σBS
,
√
σBS τex
8Rj);24=`*1~J)-E! G5 )-/
)*1=!) D(t )
?
X)4-$)+*1B Z)!1W5U245A 6* N ormal h4-/$*1set
log
2;
*1=N& SN& 7 245 *9 − norma
X);(*=)*T!"
σN (K) =
Black − Scholes
σBS (K) =
O(2 )
1
σBS (f − K)
2
(1 − 2 σBS
τex + . . .).
log(f /K)
24
2;
*1=-/1>D3)*1!`5 "
)
α log(f /K)
ζ
).
R f df 0 .(
x(ζ)
0
H w ‰ ‡ K
?
K C(f )
[1 + (
2
2γ2 − γ12 + 1/fav
1
2 − 3ρ2 2 2
α2 C 2 (fav ) + ρναγ1 C(fav ) +
ν ) τex + . . .].
24
4
24
$ %:*=)0(*)+*11"
5
)+& ):*=)9~2P *1
~!24"P5 ;h
5U @:!A TFR5 Call [
Black
]"$ −
*7 >Scholes
!TFl
"5TG])+<*7 M2456$*1Qvw:x`y standard
5 M* !1R$RN& D!1 Black
A
A
?
"
HJw ‰ K
?
Scholes
…+„
z
R8 6=A-
$%Š
;-$) )+=-/ *,5`* )+])+*n'5 "DC%h:~J)+1 [;/ab/ 7 *% 45.!24 *=)-/$
?
5@:"%A C%h4"/!)B *1 y<$24)!N& `A N& ?
ƒ:>2;c$24 7
f −K =
p
f K log(f /K)[1 +
Black − Scholes
σBS (K) =
2 2
z )+
fβ
C(f )
5
)+
*$ ~!2Q *$
(1 − β)4
(1 − β)2
log2 (f /K) +
log4 (f /K) + . . .]
24
1920
!*=)!1()W
1FRvw:x.y DG5 ">])+
α
(f K)( 1 − β)/2 1 +
[1 + (
1
1
log2 (f /K) +
log4 (f /K) + . . .]
24
1920
f 1−β − K 1−β = (1 − β)(f K)(1−β)/2 log(f /K)[1 +
X);(*=)*T!"M24 *-T!
1
(1−β)2
24
2
log (f /K) +
(1−β)4
1920
2
ζ
).
x(ζ)
log (f /K) + . . .
4
.(
(1 − β) α
ρανβ
2 − 3ρ 2 2
+
ν ) τex + dots].
+
24(f K)1−β
24
4(f K)(1−β)/2
-6'a{|$02@ (t2\B/D>A$&'=A>A-!@
H w † „ K
?
HJw †]| K
?
} I# >)1]%Š#
!%$:* )+ !mF129)+j245 * O(4 ) ] *1;245 6* normal stochastique (β = 0) M*
A
2456* log − normal stochastque
[Ua# 7 # <*>2456* stochastique normal 7 l! (N& (β = 1) ?
HJw † K
2 − 3ρ2 2 2
ν τex + . . .)
σN (K) = α(1 +
?
24
HJw †&‰ K
log(f /K)
ζ
α2
2 − 3ρ2 2 2
σBS (K) = α
.(
).(1 + [
+
ν ] τex + . . .)
?
f −K
x(ζ)
24f K
24
<*>2456* stochastique log − normal (β = 1) 7 *$`(*=)*T!$`2;
*1=-/1G%&<5 "$$`
)
L
f −K
ζ
−α2
ραν
(2 − 3ρ2 )ν 2 2
.(
).(1 + [
+
+
] τex + . . .)
log(f /K) x(ζ)
24
4
24
ραν
1
= α.(f racζx(ζ)).(1 + [
+ (2 − 3ρ2 )ν 2 ]2 τex + . . .).
4
24
σN (K) = α
σBS
HJw † † K
?
6 *)
5Sƒ:*=)+ n [,FZ-/=-/'$U*) f)f- !f!)2424)+0&!"BCD
HJI3`d` VAZ 7 | ?
?
?
S ? I)+**1 3/ LŠ?L !=)+ */ ?qL !5 TZ05UZ)+F 5 A &"$c$zO24"/!Y 5$l5n
)+24=N
$95 A "()+*
)+ /l5
-/ ($%! LJISQ I vc LM] 7 „„„
?
?
‰>€( 3 ? x'$f f9- !Y S-/24 *1*=)+ f
)+f ? HoI_COqHoVfZMZ 7 | ‡ ?
† G=-/*1[* >) ?Š@ ( !:5`=!N
$.5 )+
.*$`29)+Z-ZY "{
])+
-1$! ? I3NTHMZWYV
]SHfc Z 7 „„‰ ?
… 3)f)+ ? 29)+ ? X$ Go m&,)
5 V5 `)+Z5 ? O)
)+f1
fR241*9=om ? $-ZY =-)*Š$]% 7 x L )+!B
)M/
v-/"/!" d " "$!)*1
?
| @
…|
""
""