Good–deal bounds are price bounds for a financial portfolio which depends on an individual trader's preferences. Mathematically, if
A
is a set of portfolios with future outcomes which are "acceptable" to the trader, then define the function
ρ
:
L
p
→
R
by
ρ
(
X
)
=
inf
{
t
∈
R
:
∃
V
T
∈
A
T
:
X
+
t
+
V
T
∈
A
}
=
inf
{
t
∈
R
:
X
+
t
∈
A
−
A
T
}
where
A
T
is the set of final values for self-financing trading strategies. Then any price in the range
(
−
ρ
(
X
)
,
ρ
(
−
X
)
)
does not provide a good deal for this trader, and this range is called the "no good-deal price bounds."
If
A
=
{
Z
∈
L
0
:
Z
≥
0
P
−
a
.
s
.
}
then the good-deal price bounds are the no-arbitrage price bounds, and correspond to the subhedging and superhedging prices. The no-arbitrage bounds are the greatest extremes that good-deal bounds can take.
If
A
=
{
Z
∈
L
0
:
E
[
u
(
Z
)
]
≥
E
[
u
(
0
)
]
}
where
u
is a utility function, then the good-deal price bounds correspond to the indifference price bounds.