22 2 Models of Financial Markets on Finite Probability Spaces
In this casea=EQ[f], the stochastic integralH·Sis unique and we have
that
EQ[f|Ft]=EQ[f]+(H·S)t,t=0,...,T.
The Fundamental Theorem of Asset Pricing 2.2.7 allows us to prove the
following proposition, which we shall need soon.
Proposition 2.2.13.Assume thatSsatisfies (NA) and letH·Sbe the process
obtained fromSby means of a fixed strategyH∈H.Fixa∈Rand define
theR-valued processSd+1=(Std+1)Tt=0bySd+1=a+H·S. Then the process
S=(S^1 ,S^2 ,...,Sd,Sd+1)also satisfies the (NA) property and the setsMe(S)
andMe(S)(as well asMa(S)andMa(S))coincide.
Proof.IfQ∈Me(S)thenH·Sis aQ-martingale. ConsequentlySsatisfies
(NA).
2.3 Equivalence of Single-period with Multiperiod Arbitrage
with Multiperiod Arbitrage
The aim of this section is to describe the relation between one-period no-
arbitrage and multiperiod no-arbitrage. At the same time we will be able to
give somewhat more detailed information on the set of risk neutral measures
(this term is often used in the finance literature in a synonymous way for
martingale measures). We start off with the following observation. Recall that
we did not assume thatF 0 is trivial.
Proposition 2.3.1.IfSsatisfies the no-arbitrage condition,Q∈Me(S)is
an equivalent martingale measure, andZt=EP
[
dQ
dP∣
∣
∣Ft]
denotes the densityprocess associated withQ, then the processLt=ZZt 0 defines the density process
of an equivalent measureQ′such thatdQ
′
dP =LT,Q′∈Me(S)andQ′|F
0 =
P|F 0.
Proof.This is rather straightforward. SinceQ∈Me(S)wehavethatSZ
is aP-martingale. SinceZ 0 >0 and since it isF 0 -measurable the process
SZZ 0 is still aP-martingale. SinceSLis now aP-martingale and since the
densityLT >0, we necessarily haveQ′ ∈Me(S). AsL 0 =1weobtain
Q′|F 0 =P|F 0.
Theorem 2.3.2.LetS=(St)Tt=0be a price process. Then the following are
equivalent:
(i) Ssatisfies the no-arbitrage property.
(ii)For each 0 ≤t<T, we have that the one-period market(St,St+1)with
respect to the filtration(Ft,Ft+1)satisfies the no-arbitrage property.