190 9 Fundamental Theorem of Asset Pricing
the sequence (gn)n≥ 1 tends to 0 inL^1 (Q). Therefore the sequence (gn)n≥ 1
tends to 0 in probability for the measureQ. BecausePis absolutely contin-
uous with respect toQwe deduce thatgntends to 0 in probability for the
probabilityP.ThisimpliesthatSsatisfies the(NFLVR)property for inte-
grands with bounded support. BecauseQis the only martingale measure for
Sand becauseQis not absolutely continuous with respect toP, the process
Scannot satisfy the no free lunch with vanishing risk property for general in-
tegrands (Theorem 9.1.1). In fact, precisely as predicted in Proposition 9.6.4,
there is already arbitrage if general integrands are allowed! Take e.g.Hthe
predictable process identically one. BecauseS 0 =0,wehaveH·S=Sand
His therefore admissible. NowSntends toffor the probabilityQand hence
Sntends to (^1) Ω{a}for the measureP, i.e. tends to the constant function 1 for
the probabilityP. The processSdoes not satisfy(NA)for general integrands.
9.7 Simple Integrands .......................................
In this section we investigate the consequences of the no-free-lunch like prop-
erties when defined with simple integrands. It turns out that there is a relation
between the semi-martingale property and the no free lunch with vanishing
risk(NFLVR)property for simple integrands. For continuous processes we are
able to strengthen Theorem 9.1.1 and the main theorem of [D 92].
Definition 9.7.1.A simple predictable integrandis a linear combina-
tion of processes of the formH=f (^1) ]]T 1 ,T 2 ]] wherefisFT 1 -measurable and
T 1 ≤T 2 are finite stopping times with respect to the filtration(Ft)t∈R+(see
also [P 90]). The expression “elementary predictable integrand” is reserved for
processes of the same kind but with the restriction that the stopping times are
deterministic times.
Simple predictable integrands seem to be the easiest strategies an investor
can use. The integrandH=f (^1) ]]T 1 ,T 2 ]]corresponds to buyingfunits at time
T 1 and selling them at timeT 2. The requirement that only stopping times and
predictable integrands are used reflects the fact that only information available
from the past can be used. The interpretation of simple integrands is therefore
straightforward. The use of general integrands, however, seems more difficult
to interpret and their use can be questioned in economic models. It is therefore
reasonable to investigate how far one can go in requiring the integrands to be
simple.
As pointed out in Sect. 9.2, we can define the concepts such as no-arbitrage,
...with the extra restriction that the integrands are simple. In the case of
simple integrands, stochastic integrals are defined for adapted processes. In
this section we therefore suppose thatS is a cadlag adapted process. The
following theorem shows that the condition of no free lunch with vanishing
risk for simple integrands, already implies thatSis a semi-martingale. In