108156.pdf

76 Mathematics for Finance

Definition 4.1

Aportfoliois a vector (x 1 (n),...,xm(n),y(n)) indicating the number of shares
and bonds held by an investor between timesn−1andn. A sequence of
portfolios indexed byn=1, 2 ,...is called aninvestment strategy.Thewealth
of an investor or thevalue of the strategyat timen≥1is

V(n)=

∑m

j=1

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

At timen=0theinitial wealthis given by

V(0) =

∑m

j=1

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

We have seen in Example 4.1 that the contents of a portfolio can be adjusted
by buying or selling some assets at any time step, as long as the current value
of the portfolio remains unaltered.

Definition 4.2

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)=V(n). (4.2)

Example 4.2

Let the stock and bond prices be as in Example 4.1. Suppose that an initial
wealth ofV(0) = 3,000 dollars is invested by purchasingx 1 (1) = 18.22 shares
of the first stock, short sellingx 2 (1) =− 16 .81 shares of the second stock,
and buyingy(1) = 22.43 bonds. The time 1 value of this portfolio will be
V(1) = 18. 22 × 65 − 16. 81 ×15 + 22. 43 ×110 = 3, 399 .45 dollars. The investor
will benefit from the drop of the price of the shorted stock. This example
illustrates the fact that portfolios containing fractional or negative numbers of
assets are allowed.

We do not impose any restrictions on the numbersx 1 (n),...,xm(n),y(n).
The fact that they can take non-integer values is referred to asdivisibility.
Negativexj(n) means that stock numberjissold short (in other words, a