74 Mathematics for Finance
respectively, and the risk-free position byy. The wealth of an investor holding
such positions at timenwill be
V(n)=∑mj=1xjSj(n)+yA(n). (4.1)Assumptions 1.1 to 1.5 of Chapter 1 can readily be adapted to this general
setting. The motivation and interpretation of these assumptions are the same
as in Chapter 1, with the natural changes from one to several time steps and
from one to several risky assets.
Assumption 4.1 (Randomness)
The future stock pricesS 1 (n),...,Sm(n) are random variables for anyn=
1 , 2 ,.... The future pricesA(n) of the risk-free security for anyn=1, 2 ,...are
known numbers.
Assumption 4.2 (Positivity of Prices)
All stock and bond prices are strictly positive,
S(n)>0andA(n)>0forn=0, 1 , 2 ,....Assumption 4.3 (Divisibility, Liquidity and Short Selling)
An investor may buy, sell and hold any numberxkof stock shares of each kind
k=1,...,mand take any risk-free positiony, whether integer or fractional,
negative, positive or zero. In general,
x 1 ,...,xm,y∈R.Assumption 4.4 (Solvency)
The wealth of an investor must be non-negative at all times,
V(n)≥0forn=0, 1 , 2 ,....Assumption 4.5 (Discrete Unit Prices)
For eachn=0, 1 , 2 ,...the share pricesS 1 (n),...,Sm(n) are random variables
taking only finitely many values.