0 C
0
C
0
0
= P
0
P
0
0
+ (K
0
K)e
rT
0 P
0
0
P
0
= C
0
0
C
0
+ (K
0
K)e
rT
.
K
1
K
2
> K
1
F := K
2
K
1
(K
1
S
T
)
+
(K
2
S
T
)
+
+ F =
0 S
T
K
1
S
T
K
1
K
1
S
T
K
2
K
2
K
1
S
T
> K
2
= (S
T
K
1
)
+
(S
T
K
2
)
+
,
F
1
{(1, 2, 3), (1, 3, 2)} {(2, 1, 3), (2, 3, 1)}
{(3, 1, 2), (3, 2, 1)} F
2
= F
3
φ
n+1
= V
0
n
X
j=0
˜
S
j
θ
j
φ
n
= V
0
n1
X
j=0
˜
S
j
θ
j
.
φ
n
=
˜
S
n
θ
n
V
j1
= φ
j
B
j1
+ θ
j
S
j1
G
n
=
n
X
j=1
V
j1
= V
n
V
0
.
V
n
= V
0
+ G
n
n
V
n
= ∆G
n
=
n+1
X
j=1
(φ
j
B
j1
+ θ
j
S
j1
)
n
X
j=1
(φ
j
B
j1
+ θ
j
S
j1
)
= φ
n+1
B
n
+ θ
n+1
S
n
X
1
X
2
Y = X
1
+X
2
X
1
+ 1 X
2
+ 1 p = w/(r + w)
E Y = E X
1
+ E X
2
= (2/p) 2 = 2r/w
E X = 1/p
E[X(X 1)] =
X
n=2
n(n 1)q
n1
p = pq
X
n=2
d
2
dq
2
q
n
= pq
d
2
dq
2
q
2
1 q
=
2q
p
2
.
V X = E[X(X 1)] + E X E
2
X = (2q + p 1)/p
2
= q/p
2
.
N = r + w X
X = 2
P(X = 2) =
r
2
+ w
2
N
2
r(r 1) + w(w 1)
N(N 1)
{X = 3}
P(X = 3) =
2r
2
w + 2rw
2
N
3
2r(r 1)w + 2w(w 1)r
N(N 1)(N 2)
E X = 2 ·
r
2
+ w
2
N
2
+ 6 ·
r
2
w + rw
2
N
3
E X = 2 ·
r(r 1) + w(w 1)
N(N 1)
+ 6 ·
r(r 1)w + w(w 1)r
N(N 1)(N 2)
.
A F
V I
A
= E I
2
A
E
2
I
A
= P(A) P
2
(A) = P(A)P(A
0
)
f
X,Y
(x, y) = I
[0,1]
(x)I
[0,1]
(y)
E
4XY
X
2
+ Y
2
+ 1
=
Z
1
0
Z
1
0
4xy
x
2
+ y
2
+ 1
dy dx
=
Z
1
0
2x
ln (x
2
+ 2) ln (x
2
+ 1)
dx
=
Z
3
2
ln u du
Z
2
1
ln u du
= ln (27/16).
E (X + Y )
2
= E X
2
+ E Y
2
+ 2(E X)(E Y ) = E X
2
+ E Y
2
E (X+Y )
3
= E X
3
+E Y
3
+3(E X
2
)(E Y )+3(E X)(E Y
2
) = E X
3
+E Y
3
.
Z
b
a
e
αx
ϕ(x) dx = e
α
2
/2
Z
b
a
ϕ(x α) dx = e
α
2
/2
[Φ(b α) Φ(a α)] .
Z
b
a
e
αx
Φ(x) dx =
1
α
e
αx
Φ(x)
b
a
1
α
Z
b
a
e
αx
ϕ(x) dx
=
1
α
e
αx
Φ(x) e
α
2
/2
Φ(x α)
b
a
.
V X = E X
2
E
2
X E X =
1
2
(α + β)
E X
2
= (β α)
1
Z
β
α
x
2
dx =
1
3
(α
2
+ αβ + β
2
),
V X =
1
3
(α
2
+ αβ + β
2
)
1
4
(α + β)
2
=
1
12
(α β)
2
.
E
2
X = E X
2
V X E X
2
V X 0
X =
N
z
1
X
x
m
x

n
z x
x,
max(z n, 1) x min(z, m)
m
N
z
1
X
x
m 1
x 1

n
z x
= m
N
z
1
N 1
z 1
,
max(z 1 n, 0) x 1 min(z 1, m 1)
P(Y = 50) Φ(.1) Φ(.1) 0.07966.
100
50
2
100
= 0.07959
p = .5
P(40 < Y < 60) = P
2 <
2Y 100
10
< 2
Φ(2) Φ(2) .95.
A
P(A) =
X
ωA
P
1
(ω
1
)P
2
(ω
2
) ···P
N
(ω
N
)
=
X
ω
1
A
1
···
X
ω
N
A
N
P
1
(ω
1
) ···P
N
(ω
N
)
= P
1
(A
1
)P
2
(A
2
) ···P
N
(A
N
).
Z
n
X
n
S
n
(ω) = ω
n
S
n1
(ω)
X
n
F
S
n
F
X
n
F
X
n
F
S
n
S
n
F
X
n
F
S
n
F
X
n
S
0
d < K < S
0
u
C
0
= (1 + i)
1
(S
0
u K)p
=
(1 + i d)(S
0
u K)
(1 + i)(u d)
C
0
u
=
(1 + i d)(K S
0
d)
(1 + i)(u d)
2
> 0
C
0
d
=
(1 + i u)(S
0
u K)
(1 + i)(u d)
2
< 0.
d K/S
0
C
0
=
(S
0
u K)p
+ (S
0
d K)q
1 + i
= S
0
K
1 + i
.
k N/2 k A
k u A
j
j = 1, 2, . . . , N k + 1 k < N 1 A
j
A P(A
1
) = P(A
Nk+1
) = p
k
q
P(A
j
) = p
k
q
2
j = 2, . . . , N k P(A) = p
k
q(2+(N k 1)q)
k = N 1
f(x) = xI
(K,)
(x)
V
0
= (1 + i)
N
N
X
j=m
N
j
S
0
u
j
d
Nj
p
j
q
Nj
= S
0
N
X
j=m
N
j
ˆp
j
ˆq
Nj
.
(S
N
S
M
)
+
=
S
0
u
Y
N
d
NY
N
S
0
u
Y
M
d
MY
M
+
= S
0
u
Y
M
d
MY
M
u
Y
N
Y
M
d
L(Y
N
Y
M
)
1
+
,
(1 + i)
N
V
0
= E
(S
N
S
M
)
+
= E
S
0
u
Y
M
d
MY
M
E
u
Y
N
Y
M
d
L(Y
N
Y
M
)
1
+
= E
(S
M
)E
u
Y
N
Y
M
d
L(Y
N
Y
M
)
1
+
. α
E
(S
M
) = (1 + i)
M
S
0
. β
Y
N
Y
M
= X
M+1
+ . . . + X
N
B(p
, L)
E
u
Y
N
Y
M
d
L(Y
N
Y
M
)
1
+
= E
u
Y
L
d
LY
L
1
+
.
(1+i)
L
L
E
u
Y
N
Y
M
d
L(Y
N
Y
M
)
1
+
= (1 + i)
L
Ψ(k, L, ˆp) Ψ(k, L, p
), γ
k u
k
d
Lk
> 1
V
0
(α) (β) (γ)

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