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
−rτ
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
−rτ
, 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
sσ
√
τ
ϕ(d
1
) (v)
∂C
∂r
= Kτ e
−rτ
Φ(d
2
)
(iii)
∂C
∂τ
=
σs
2
√
τ
ϕ(d
1
) + Kre
−rτ
Φ(d
2
) (vi)
∂C
∂K
= −e
−rτ
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
−rτ
(vi) lim
K→0
+
C = s
(ii) lim
s→0
+
C = 0 (vii) lim
σ→∞
C = s
(iii) lim
τ→∞
C = s (viii) lim
σ→0
+
C = (s − e
−rτ
K)
+
(iv) lim
τ→0
+
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
s→0
+
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
s→0
+
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
s→0
+
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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