12 2 Models of Financial Markets on Finite Probability Spaces
Readers who are not so enthusiastic about this mainly formal and elemen-
tary reduction might proceed directly to Definition 2.1.4. On the other hand,
we know from sad experience that often there is a lot of myth and confusion
arising in this operation of discounting; for this reason we decided to devote
this section to the clarification of this issue.
Definition 2.1.1.Amodelofafinancial marketis anRd+1-valued stochastic
processŜ=(Ŝt)Tt=0=(Ŝt^0 ,Ŝt^1 ,...,Ŝtd)Tt=0, based on and adapted to the filtered
stochastic base(Ω,F,(Ft)Tt=0,P). We shall assume that the zero coordinate
Ŝ^0 satisfiesŜt^0 > 0 for allt=0,...,TandŜ^00 =1.
The interpretation is the following. The prices of the assets 0,...,dare
measured in a fixed money unit, say Euros. For 1≤j ≤dthey are not
necessarily non-negative (think, e.g., of forward contracts). The asset 0 plays
a special role. It is supposed to be strictly positive and will be used as a nu-
m ́eraire. It allows us to compare money (e.g., Euros) at time 0 to money at
timet>0. In many elementary models,Ŝ^0 is simply a bank account which
in case of constant interest rateris then defined asŜt^0 =ert. However, it
might also be more complicated, e.g.Ŝt^0 =exp(r 0 h+r 1 h+···+rt− 1 h)where
h>0 is the length of the time interval betweent−1andt(here kept fixed)
and wherert− 1 is the stochastic interest rate valid betweent−1andt.Other
models are also possible and to prepare the reader for more general situations,
we only requireŜt^0 to be strictly positive. Notice that we only require that
Ŝt^0 to beFt-measurable and that it is not necessarilyFt− 1 -measurable. In
other words, we assume that the processŜ^0 =(Ŝt^0 )Tt=0is adapted, but not
necessarily predictable.
An economic agent is able to buy and sell financial assets. The decision
taken at timetcan only use information available at timetwhich is modelled
by theσ-algebraFt.
Definition 2.1.2.A trading strategy(Ĥt)Tt=1=(Ĥt^0 ,Ĥt^1 ,...,Ĥdt)Tt=1is an
Rd+1-valued process which is predictable, i.e.ĤtisFt− 1 -measurable.
The interpretation is that between timet−1andtimet, the agent holds
aquantityequaltoĤjt of assetj. The decision is taken at timet−1and
therefore,Ĥtis required to beFt− 1 -measurable.
Definition 2.1.3.Astrategy(Ĥt)Tt=1is called self financing if for everyt=
1 ,...,T− 1 , we have (
Ĥt,Ŝt)
=
(
Ĥt+1,Ŝt)
(2.1)
or, written more explicitly,
∑dj=0ĤtjŜtj=∑dj=0Ĥtj+1Ŝtj. (2.2)The initial investment required for a strategy isV̂ 0 =(Ĥ 1 ,Ŝ 0 )=
∑d
j=0
Ĥ 1 jŜj 0.