108156.pdf

88 Mathematics for Finance

Definition 4.2a

An investment strategy is calledself-financingif the portfolio constructed at
timen≥1 to be held over the next time stepn+ 1 is financed entirely by the
current wealthV(n), that is,

∑m

j=1

xj(n+1)Sj(n)+y(n+1)A(n)+

∑k

i=1

zi(n+1)Di(n)=V(n).

Definition 4.3a

An investment strategy is calledpredictableif for eachn=0, 1 , 2 ,...the port-
folio
(x 1 (n+1),...,xm(n+1),y(n+1),z 1 (n+1),...,zk(n+1))

constructed at timendepends only on the nodes of the tree of market scenarios
reached up to and including timen.

Definition 4.4a

A strategy is calledadmissibleif it is self-financing, predictable, and for each
n=0, 1 , 2 ,...
V(n)≥ 0

with probability 1.

The No-Arbitrage Principle extends without any modifications.

Assumption 4.6a (No-Arbitrage Principle)

There is no admissible strategy such thatV(0) = 0 andV(n)>0 with positive
probability for somen=1, 2 ,....

Finally, the Fundamental Theorem of Asset Pricing takes the following form.

Theorem 4.4a (Fundamental Theorem of Asset Pricing)

The No-Arbitrage Principle is equivalent to the existence of a probabilityP∗
on the set of scenariosΩsuch thatP∗(ω)>0 for each scenarioω∈Ωand the
discounted prices of primary and derivative securitiesS ̃j(n)=Sj(n)/A(n)and
D ̃i(n)=Di(n)/A(n) form martingales with respect toP∗, that is, satisfy

E∗(S ̃j(n+1)|S(n)) =S ̃j(n),E∗(D ̃i(n+1)|S(n)) =D ̃i(n)