254 13 The Banach Space of Workable Contingent Claims in Arbitrage Theory
Definition 13.1.3.The locally bounded semi-martingaleSsatisfies the no-
arbitrage or (NA) property if
C∩L∞+={ 0 }.Definition 13.1.4.We say that the locally bounded semi-martingaleSsatis-
fies the no free lunch with vanishing risk or (NFLVR) property if
C∩L∞+={ 0 },where the bar denotes the closure in the sup-norm topology ofL∞.
The fundamental theorem of asset pricing, as in Chap. 9, can now be
formulated as follows:
Theorem 13.1.5.The locally bounded semi-martingaleSsatisfies the prop-
erty (NFLVR) if and only if there is an equivalent probability measureQsuch
thatSis aQ-local martingale. In this case the setCis already weak-star (i.e.
σ(L∞,L^1 ))closedinL∞.
Remark 13.1.6.IfQis an equivalent local martingale measure forSand if the
integrand or strategyHsatisfiesH·S≥−a, i.e.Hisa-admissible, then by
aresultofEmery [E 80] and Ansel-Stricker [AS 94], the process ́ H·Sis still
a local martingale and hence, being bounded below, is a super-martingale. It
follows that the limit (H·S)∞exists a.s. and thatEQ[(H·S)∞]≤0.
We also need the following equivalent reformulations of the property of no
free lunch with vanishing risk, see Chap. 9 for more details.
Theorem 13.1.7.The locally bounded semi-martingaleSsatisfies the no free
lunch with vanishing risk property or (NFLVR) if for any sequence ofS-
integrable strategies(Hn,δn)n≥ 1 such that eachHnis aδn-admissible strategy
and whereδntends to zero, we have that(H·S)∞tends to zero in probabilityP.
Theorem 13.1.8.The locally bounded semi-martingaleSsatisfies the prop-
erty (NFLVR) if and only if
(1)it satisfies the property (NA)
(2)K 1 is bounded inL^0 , for the topology of convergence in measure.
Theorem 13.1.9.The locally bounded semi-martingaleSsatisfies the prop-
erty (NFLVR) if and only if
(1)it satisfies the property (NA)
(2)There is a strictly positive local martingaleL,L 0 =1, such that at infinity
L∞> 0 ,P-a.s. and such thatLSis a local martingale.