108156.pdf

4. Discrete Time Market Models 87

Assumption 4.2a (Positivity of Prices)

The prices of primary securities, including the money market account, are pos-
itive,
S 1 (n),...,Sm(n),A(n)>0forn=0, 1 , 2 ,....

Assumption 4.3a (Divisibility, Liquidity and Short Selling)

An investor may buy, sell and hold any number of assets, whether integer or
fractional, negative, positive or zero. In general,

x 1 ,...,xm,y,z 1 ,...,zk∈R.

Assumption 4.4a (Solvency)

The wealth of an investor must be non-negative at all times,

V(n)≥0forn=0, 1 , 2 ,....

Assumption 4.5a (Discrete Unit Prices)

For eachn=0, 1 , 2 ,...the pricesS 1 (n),...,Sm(n),A(n),D 1 (n),...,Dk(n)are
random variables taking only finitely many values.

Definitions 4.1 to 4.4 also extend immediately to the case in hand.

Definition 4.1a

Aportfoliois a vector

(x 1 (n),...,xm(n),y(n),z 1 (n),...,zk(n))

indicating the number of primary and derivative securities held by an investor
between timesn−1andn. A sequence of portfolios indexed byn=1, 2 ,...
is called aninvestment strategy.Thewealthof an investor or thevalue of the
strategyat timen≥1is

V(n)=

∑m

j=1

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

∑k

i=1

zi(n)Di(n).

At timen=0theinitial wealthis given by

V(0) =

∑m

j=1

xj(1)Sj(0) +y(1)A(0) +

∑k

i=1

zi(1)Di(0).