2.4 Pricing by No-Arbitrage 23Proof.Obviously (i) implies (ii), since there are less strategies in each single
period market than in the multiperiod market. So let us show that (ii) implies
(i). By the fundamental theorem applied to (St,St+1), we have that for eacht
there is a probability measureQtonFt+1equivalent toP, so that underQt
the process (St,St+1)isaQt-martingale. This means thatEQt[St+1|Ft]=
St. By the previous proposition we may takeQt|Ft=P|Ft.Letft+1=ddQPt
and defineLt=f 1 ...ft− 1 ftandL 0 = 1. Clearly (Lt)Tt=0is the density process
of an equivalent measureQdefined byddQP=LT. One can easily check that,
for allt=0,...,T−1wehaveEQ[St+1|Ft]=St, i.e.,Q∈Me(S).
Remark 2.3.3.The equivalence between one-period no-arbitrage and multi-
period no-arbitrage can also be checked directly by the definition of no-
arbitrage. We invite the reader to give a direct proof of the following: ifH
isastrategysothat(H·S)T ≥0andP[(H·S)T >0]>0 then there is
a1≤t≤Tas well asA∈Ft− 1 ,P[A]>0sothat (^1) A(Ht,∆St)≥0and
P[ (^1) A(Ht,∆St)>0]>0 (compare Lemma 5.1.5 below).
Remark 2.3.4.We give one more indication, why there is little difference be-
tween the one-period and theTperiod situation; this discussion also reveals a
nice economic interpretation. GivenS=(St)Tt=0as above, we may associate a
one-period processS ̃=(S ̃t)t^1 =0, adapted to the filtration (F ̃ 0 ,F ̃ 1 ):=(F 0 ,FT)
in the following way: choose any collection (f 1 ,...,fm) in the finite dimen-
sional linear spaceKdefined in 2.2.1, which linearly spansK. Define the
Rm-valued processS ̃byS ̃ 0 =0,S ̃ 1 =(f 1 ,...,fm).
Obviously the processS ̃yieldsthesamespaceKof stochastic integrals as
S. Hence the set of equivalent martingale measures for the processesSandS ̃
coincide and therefore all assertions, depending only on the set of equivalent
martingale measures coincide forSandS ̃.InparticularSandS ̃yield the
same arbitrage-free prices for derivatives, as we shall see in the next section.
The economic interpretation of the transition fromStoS ̃reads as follows:
if we fix the trading strategiesHjyieldingfj=(Hj·S)T,wemaythinkoffj
as a contingent claim at timet=Twhich may be bought at price 0 at time
t= 0, by then applying the trading rules given byHj. By taking sufficiently
many of theseHj’s, in the sense that the correspondingfj’s linearly spanK,
we may represent the resultf=(H·S)Tofanytrading strategyHas a linear
combination of thefj’s.
The bottom line of this discussion is that in the present framework (i.e. Ω
is finite) — from a mathematical as well as from an economic point of view
—theTperiod situation can easily be reduced to the one-period situation.
2.4 Pricing by No-Arbitrage
The subsequent theorem will tell us what the principle of no-arbitrage implies
about the possible prices for a contingent claimf.Itgoesbacktothework
of D. Kreps [K 81].