The Mathematics of Arbitrage

2.1 Description of the Model 13

The interpretation goes as follows. By changing the portfolio fromĤt− 1
toĤtthere is no input/outflow of money. We remark that we assume that
changing a portfolio does not trigger transaction costs. Also note thatĤtjmay
assume negative values, which corresponds to short selling assetjduring the
time interval ]tj− 1 ,tj].
TheFt-measurable random variable defined in (2.1) is interpreted as the
valueV̂tof the portfolio at timetdefined by the trading strategyĤ:

V̂t=(Ĥt,Ŝt)=(Ĥt+1,Ŝt).

The way in which the value (Ĥt,Ŝt) evolves can be described much easier
when we use discounted prices using the assetŜ^0 as num ́eraire. Discounting
allows us to compare money at timetto money at time 0. For instance we
could say thatŜt^0 units of money at timetare the “same” as 1 unit of money,
e.g., Euros, at time 0. So let us see what happens if we replace pricesŜby

discounted prices

(̂

S Ŝ^0

)

=

(̂

S^0 Ŝ^0 ,

Ŝ^1 Ŝ^0 ,...,

Ŝd Ŝ^0

)

. We will use the notation

Stj:=

Ŝjt Ŝ^0 t

,forj=1,...,dandt=0,...,T. (2.3)

There is no need to include the coordinate 0, since obviouslyS^0 t=1.Letus
now consider (Ĥt)Tt=1=(Ĥ^0 t,Ĥt^1 ,...,Ĥtd)Tt=1to be a self financing strategy
with initial investmentV̂ 0 ;wethenhave

V̂ 0 =

∑d

j=0

Ĥ 1 jŜj 0 =Ĥ 10 +

∑d

j=1

Ĥ 1 jŜ 0 j=Ĥ 10 +

∑d

j=1

Ĥ 1 jS 0 j,

since by definitionŜ^00 =1.

We now write (Ht)Tt=1=(H^1 t,...,Htd)Tt=1for theRd-valued process ob-
tained by discarding the 0’th coordinate of theRd+1-valued process (Ĥt)Tt=1=

(Ĥt^0 ,Ĥt^1 ,...,Ĥtd)Tt=1, i.e.,Htj=Ĥtjforj=1,...,d. The reason for dropping
the 0’th coordinate is, as we shall discover in a moment, that the holdings
Ĥt^0 in the num ́eraire assetSt^0 will be no longer of importance when we do the
book-keeping in terms of the num ́eraire asset, i.e., in discounted terms.

One can make the following easy, but crucial observation: foreveryRd-
valued, predictable process (Ht)Tt=1=(Ht^1 ,...,Htd)Tt=1there exists a unique
self financingRd+1-valued predictable process (Ĥt)Tt=1=(Ĥt^0 ,Ĥt^1 ,...,Ĥtd)Tt=1

such that (Ĥtj)tT=1=(Hjt)Tt=1forj=1,...,dandĤ 10 = 0. Indeed, one de-
termines the values ofĤ^0 t+1,fort=1,...,T−1, by inductively applying

(2.2). The strict positivity of (Ŝt^0 )Tt=0−^1 implies that there is precisely one func-
tionĤ^0 t+1such that equality (2.2) holds true. Clearly such a functionĤt^0 +1is

The Mathematics of Arbitrage

(̂

)

=

(̂

)

V̂ 0 =

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