Information Technology Reference
In-Depth Information
→
−
We furthermore change to
time-to-maturity t
t
, to obtain a forward parabolic
problem. Thus, by setting
V(t,s)
=:
v(T
−
t,
log
s)
, the BS equation in real price
(
4.4
) satisfied by
V(t,s)
becomes the BS equation for
v(t,x)
in
log-price
T
BS
v
∂
t
v
−
A
+
rv
=
0 n
(
0
,T)
× R
,
(4.7)
g(e
x
)
in
with the initial condition
v(
0
,x)
=
R
.
Remark 4.1.6
For put and call contracts with strike
K>
0, it is convenient to intro-
duce the so-called
log-moneyness
variable
x
=
log
(s/K)
and setting the option price
V(t,s)
:=
Kw(T
−
t,
log
(s/K))
. Then, the function
w(t,x)
again solves (
4.7
), with
the initial condition
w(
0
,x)
g(Ke
x
)/K
. Thus, the initial condition becomes for
=
e
x
0
,e
x
a put
w(
0
,x)
=
max
{
0
,
1
−
}
and for a call
w(
0
,x)
=
max
{
−
1
}
where both
payoffs now do not depend on
K
.
4.2 Variational Formulation
We give the variational formulation of the Black-Scholes equation (
4.7
) which
reads:
L
2
(J
H
1
(
H
1
(J
L
2
(
Find
u
∈
;
R
))
∩
;
R
))
such that
a
BS
(u, v)
H
1
(
(∂
t
u, v)
+
=
0
,
∀
v
∈
R
),
a.e. in
J,
(4.8)
u(
0
)
=
u
0
,
where
u
0
(x)
:=
g(e
x
)
and the bilinear form
a
BS
(
·
,
·
)
:
H
1
(
R
)
×
H
1
(
R
)
→ R
is given
by
1
2
σ
2
(ϕ
,φ
)
a
BS
(ϕ, φ)
(σ
2
/
2
r)(ϕ
,φ)
:=
+
−
+
r(ϕ,φ).
(4.9)
We show that
a
BS
(
·
,
·
)
is continuous (3.8) and satisfies a Gårding inequality (3.9)on
H
1
(
V
=
R
)
.
Proposition 4.2.1
There exist constants C
i
=
C
i
(σ, r) >
0,
i
=
1
,
2
,
3,
such that for
all ϕ,φ
∈
H
1
(
R
)
a
BS
(ϕ, φ)
BS
(ϕ, ϕ)
2
H
1
(
2
L
2
(
|
|≤
C
1
ϕ
H
1
(
R
)
φ
H
1
(
R
)
,
≥
C
2
ϕ
)
−
C
3
ϕ
)
.
R
R
Proof
We first show continuity. By Hölder's inequality,
1
2
σ
2
a
BS
(ϕ, φ)
ϕ
L
2
(
R
)
φ
L
2
(
R
)
+|
σ
2
/
2
ϕ
L
2
(
R
)
|
|≤
−
r
|
φ
L
2
(
R
)
+
r
ϕ
L
2
(
R
)
φ
L
2
(
R
)
≤
C
1
(σ, r)
ϕ
H
1
(
R
)
φ
H
1
(
R
)
.
Search WWH ::
Custom Search