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=
+
A
−
=
Proof
We show the result only for
t
0. Since
∂
t
V
V
rV
0, we have, by
e
−
0
r(X
s
)
d
s
V(t,X
t
)
is a martingale. Thus,
Proposition
4.1.3
, that the process
M
t
:=
V(
0
,x)
= E[
M
0
|
X
0
=
x
]
= E[
M
T
|
X
0
=
x
]
e
−
0
r(X
s
)
d
s
V(T,X
T
)
x
= E
|
X
0
=
e
−
0
r(X
s
)
d
s
g(X
T
)
x
.
= E
|
X
0
=
Remark 4.1.5
The converse
of
Theorem
4.1.4
is also true. Any
V(t,x)
as in (
4.1
),
which is
C
1
,
2
(J
C
0
(J
× R
)
∩
× R
)
with bounded derivatives in
x
, solves the PDE
(
4.3
).
We apply Theorem
4.1.4
to the Black-Scholes model [21]. In the Black-Scholes
market, the risky asset's spot-price is modeled by a geometric Brownian motion
S
t
,
i.e. the SDE for this model is as in (1.2), with coefficients
b(t, s)
=
=
rs
,
σ(t,s)
σs
,
where
σ>
0 and
r
0 denote the (constant) volatility and the (constant) interest
rate, respectively. Therefore, the SDE is given by
≥
d
S
t
=
rS
t
d
t
+
σS
t
d
W
t
.
We assume for simplicity that
no dividends are paid
. Based on Theorem
4.1.4
,we
get that the discounted price of a European contract with payoff
g(s)
,i.e.
V(
0
,s)
=
e
−
rT
g(S
T
)
E[
|
S
0
=
s
]
, is equal to a regular solution
V(
0
,s)
of the Black-Scholes
equation
1
2
σ
2
s
2
∂
ss
V
+
rs∂
s
V
−
rV
=
∂
t
V
+
0in
J
× R
+
.
(4.4)
The BS equation (
4.4
) needs to be completed by the
terminal condition
,
V(T,s)
=
g(s)
, depending on the type of option. Equation (
4.4
) is a parabolic PDE with the
second order “spatial” differential operator
1
2
σ
2
s
2
∂
ss
f(s)
+
rs∂
s
f(s),
(
A
f )(s)
=
(4.5)
which degenerates at
s
0. To obtain a non-degenerate operator with constant co-
efficients, we switch to the price process
X
t
=
=
log
(S
t
)
which solves the SDE
r
−
2
σ
2
d
t
+
σ
d
W
t
.
1
d
X
t
=
The infinitesimal generator for this process has constant coefficients and is given by
r
2
σ
2
∂
x
f(x).
1
2
σ
2
∂
xx
f(x)
1
BS
f )(x)
(
A
=
+
−
(4.6)
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