108156.pdf

78 Mathematics for Finance

always determined by the current wealth and the positions in risky assets.

Proposition 4.1

Given the initial wealthV(0) and a predictable sequence (x 1 (n),...,xm(n)),
n=1, 2 ,...of positions in risky assets, it is always possible to find a sequence
y(n) of risk-free positions such that (x 1 (n),...,xm(n),y(n)) is a predictable
self-financing investment strategy.

Proof

Put

y(1) =

V(0)−x 1 (1)S 1 (0)−···−xm(1)Sm(0)

A(0)

and then compute

V(1) =x 1 (1)S 1 (1) +···+xm(1)Sm(1) +y(1)A(1).

Next,

y(2) =

V(1)−x 1 (2)S 1 (1)−···−xm(2)Sm(1)

A(1)

,

V(2) =x 1 (2)S 1 (2) +···+xm(2)Sm(2) +y(2)A(2),

and so on. This clearly defines a self-financing strategy. The strategy is pre-
dictable becausey(n+ 1) can be expressed in terms of stock and bond prices
up to timen.

Exercise 4.1

Find the number of bondsy(1) andy(2) held by an investor during the

first and second steps of a predictable self-financing investment strategy

with initial valueV(0) = 200 dollars and risky asset positions

x 1 (1) = 35. 24 ,x 1 (2) =− 40. 50 ,

x 2 (1) = 24. 18 ,x 2 (2) = 10. 13 ,

if the prices of assets follow the scenario in Example 4.1. Also find the

time 1 valueV(1) and time 2 valueV(2) of this strategy.

Example 4.4

Once again, suppose that the stock and bond prices follow the scenario in
Example 4.1. If an amountV(0) = 100 dollars were invested in a portfolio with