Solutions.......................................................
Chapter 1
1.1The value of the portfolio at time 0 is
V(0) =xS(0) +yA(0) = 1, 600
dollars. The value of the portfolio at time 1 will be
V(1) =xS(1) +yA(1) =
{
1 ,800 if stock goes up,
1 ,700 if stock goes down.
Hence, the return on the portfolio will be
KV=V(1)V−(0)V(0)=
{
0 .1250 if stock goes up,
0 .0625 if stock goes down,
that is, either 12.5% or 6.25%.
1.2Given the same bond and stock prices as in Exercise 1.1, the value of a portfolio
(x, y) at time 1 will be
V(1) =
{
x30 +y100 if stock goes up,
x20 +y100 if stock goes down.
Thus, we obtain a system of equations
{
x30 +y100 = 1, 160 ,
x20 +y100 = 1, 040.
The solution isx= 12 andy= 8. A portfolio with 12 shares of stock and
8 bonds will produce the desired value at time 1. The time 0 value of this
portfolio is
V(0) = 12×25 + 8×90 = 1, 020
dollars.
1.3An arbitrage opportunity can be realised as follows:
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