Information Technology Reference
In-Depth Information
13.4.1 Aggregated Black-Scholes Models
In the following, a full-rank Black-Scholes model and low-rank Black-Scholes
model are introduced. Option pricing under the low-rank Black-Scholes model
leads to lower dimensional PDEs compared to the full rank Black-Scholes model,
introducing an additional error due to the rank reduction.
Consider
d
assets
S
(S
1
,...,S
d
)
with spot price dynamics
Z
i
S
i
=
=
given by
d
Σ
ij
S
t
d
W
t
,i
=
d
S
t
=
μ
i
S
t
d
t
+
1
,...,d,
(13.19)
j
=
1
d
where
W
={
W
t
:
t
≥
0
}
is a standard Brownian motion in
R
and
:=
μ
i
1
≤
i
≤
d
∈ R
d
,
μ
(13.20)
Σ
:=
Σ
ij
1
≤
i,j
≤
d
∈ R
d
×
d
(13.21)
are the constant drift vector and volatility matrix, respectively, with the assumption
that rank
Σ
d
. Under the unique
=
d
. The state space domain is given by
G
= R
EMM, the log-price dynamics
X
i
log
S
i
:=
are given by
d
Σ
ij
d
W
t
,i
d
X
t
=
η
i
d
t
+
=
1
,...,d,
(13.22)
j
=
1
Q
:=
ΣΣ
∈ R
d
d
sym
.
×
where
η
i
:=
(r
−
1
/
2
Q
ii
)
,
i
=
1
,...,d
and
denotes the
volatility covariance matrix. Since
Q
is symmetric positive definite, there ex-
d
×
d
U
=
∈ R
Q
ists an orthogonal matrix
U
such that
U
D
with diagonal matrix
diag
(s
1
,...,s
d
)
,
s
1
≥···≥
:=
D
s
d
>
0. Without loss of generality, we rescale time
t
∗
=
s
1
t
, yielding
D
∗
:=
diag
(s
∗
2
1
,...,s
∗
2
d
in (
13.19
) such that
t
→
)
with normal-
ized
s
1
=
1,
s
i
2
,...,d
. In the remainder, we drop the
∗
=
s
i
/s
1
,
i
=
and define a
process
Y
:= {
U
X
t
:
t
≥
0
}
with dynamics
d
Y
t
s
i
d
W
t
,i
=
+
=
λ
i
d
t
1
,...,d,
(13.23)
U
η
. The components
Y
1
,...,Y
d
where
λ
:=
now satisfy the system of
d
decoupled
SDEs (
13.23
).
Let 1
≤
d<d
be a parameter and define
D
diag
(s
1
,...,s
d
)
∈ R
d
×
d
:=
with
s
i
,
≤
d,
≤
1
i
s
i
=
ˆ
(13.24)
d
0
,
+
1
≤
i
≤
d,
U
D
2
with
U
and
s
i
,
i
and
Σ
:=
=
1
,...,d
, as before. Consider the log-price
process
X
:= {
X
t
:
t
≥
0
}
with dynamics
d
1
Σ
ij
d
W
t
,i
d
X
t
=
η
i
d
t
+
=
1
,...,d,
(13.25)
j
=
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