C
0
P
0
S
0
= \$20.00 C
0
P
0
C
0
T K r σ C
0
P
0
T K r σ C
0
P
0
C
0
P
0
T K r σ
σ r T K
C = C(τ, s, K, σ, r) = sΦ(d
1
) Ke
Φ(d
2
), τ, s, K, σ, r > 0,
d
1,2
= d
1,2
(τ, s, K, σ, r) =
ln (s/K) + (r ±σ
2
/2)τ
σ
τ
.
C d
1,2
C(T, S
0
, K, σ, r)
C
0
K T
S
0
C(T t, S
t
, K, σ, r) = C
t
t
P = P (τ, s, K, σ, r) = C(τ, s, K, σ, r) s + Ke
, τ, s, K, σ, r > 0.
P (T, S
0
, K, σ, r)
P
t
:= P (T t, S
t
, K, σ, r) t
(i)
C
s
= Φ(d
1
) (iv)
C
σ
= s
τ ϕ(d
1
)
(ii)
2
C
s
2
=
1
τ
ϕ(d
1
) (v)
C
r
= Kτ e
Φ(d
2
)
(iii)
C
τ
=
σs
2
τ
ϕ(d
1
) + Kre
Φ(d
2
) (vi)
C
K
= e
Φ(d
2
)
C
s
,
2
C
s
2
,
C
τ
,
C
σ
,
C
r
C s
τ σ r K
S
0
K
(S
T
K)
+
T r Ke
rT
v(t, s) = C(T t, s, K, σ, r)
v
s
(t, s) = Φ (d
1
(T t, s, K, σ, r)) .
v
s
(t, S
t
) = θ
t
t
S
t
Φ
d
1
(T t, S
t
, K, σ, r)
t
v(t, S
t
) S
t
Φ
d
1
(T t, S
t
, K, σ, r)
= Ke
r(T t)
Φ
d
2
(T t, S
t
, K, σ, r)
t
Φ(d
1
)
Ke
r(T t)
Φ(d
2
)
C
(i) lim
s→∞
(C s) = Ke
(vi) lim
K0
+
C = s
(ii) lim
s0
+
C = 0 (vii) lim
σ→∞
C = s
(iii) lim
τ→∞
C = s (viii) lim
σ0
+
C = (s e
K)
+
(iv) lim
τ0
+
C = (s K)
+
(ix) lim
r→∞
C = s
(v) lim
K→∞
C = 0
S
0
C
0
S
0
Ke
rT
S
T
K S
0
T
T
Ke
rT
Ke
rT
S
0
> K T
S
0
K
S
0
σ S
0
> e
rT
K
S
0
e
rT
K
S T S
S S
0
= \$50.00 r = .10 σ = .2
K =
v(t, s) = αs+βe
rt
α β
P s
lim
s→∞
P lim
s0
+
P.
P
t
(s) = Ke
r(T t)
Φ
d
2
(T t, s)
sΦ
d
1
(T t, s)
.
A S
T
> K
t
V
t
= Ae
r(T t)
Φ
d
1
(T t, S
t
, K, σ, r)
.
S
T
S
T
> K
t
V
t
= S
t
Φ
d
1
(T t, S
t
, K, σ, r)
.
K t
V
0
V
T
=
S
T
I
(K
1
,K
2
)
(S
T
) 0 < K
1
< K
2
V
T
= min
max(S
T
, K
1
), K
2
, 0 < K
1
< K
2
.
t
V
t
= K
1
e
r(T t)
+ C(T t, S
t
, K
1
) C(T t, S
t
, K
2
).
V
T
= max(S
T
, F ) K = (S
T
F )
+
+ F K,
F = S
0
e
rT
K
V
t
t K
dS
k
dW dt
S
p q p > 0 x
1
x
2
Z
x
2
x
1
e
px
2
+qx
dx = e
q
2
/4p
r
π
p
Φ
q 2px
1
2p
Φ
q 2px
2
2p

,
Φ() := 1 Φ(−∞) := 0
C
0
= C(T, s, K, σ, r)
s
E
C
=
s
C
0
C
0
s
,
C
0
s
E
C
=
sΦ(d
1
)
sΦ(d
1
) Ke
rT
Φ(d
2
)
, d
1,2
:= d
1,2
(T, s, K, σ, r).
E
C
> 1
lim
s+
E
C
= 1
lim
s0
+
E
C
= +.
P
0
= P (T, s, K, σ, r)
s
E
P
=
s
P
0
P
0
s
,
P
0
s
E
P
=
sΦ(d
1
)
Ke
rT
Φ(d
2
) sΦ(d
1
)
lim
s+
E
P
= + lim
s0
+
E
P
= 0.
G(t, s) =
1
σ
p
2π(T t)
Z
0
f(z)e
d
2
2
/2
dz
z
,
d
2
= d
2
2
(T t, s, z, σ, r)

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