No free lunch with vanishing risk (NFLVR) is a no-arbitrage argument. We have free lunch with vanishing risk if by utilizing a sequence of time self-financing portfolios which converge to an arbitrage strategy, we can approximate a self-financing portfolio (called the free lunch with vanishing risk).
For a semimartingale S, let
K
=
{
(
H
⋅
S
)
∞
:
H
admissible
,
(
H
⋅
S
)
∞
=
lim
t
→
∞
(
H
⋅
S
)
t
exists a.s.
}
where a strategy is admissible if it is permitted by the market. Then define
C
=
{
g
∈
L
∞
(
P
)
:
g
≤
f
∀
f
∈
K
}
. S is said to satisfy no free lunch with vanishing risk if
C
¯
∩
L
+
∞
(
P
)
=
{
0
}
such that
C
¯
is the closure of C in the norm topology of
L
+
∞
(
P
)
.
If
S
=
(
S
t
)
t
=
0
T
is a semimartingale with values in
R
d
then S does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure
Q
such that S is a sigma-martingale under
Q
.
