- Introduction: A Simple Market Model 9
This way, you will be holding a portfolio (x, y) withx= 1 shares of stock
andy=−1 bonds.The time 0 value of this portfolio is
V(0) = 0.At time 1 the value will become
V(1) ={
Su−A(1) if stock goes up,
Sd−A(1) if stock goes down.IfA(1)≤Sd, then the first of these two possible values is strictly positive,
while the other one is non-negative, that is,V(1) is a non-negative random
variable such thatV(1)>0 with probabilityp>0. The portfolio provides an
arbitrage opportunity, violating the No-Arbitrage Principle.
Now suppose thatA(1)≥Su. If this is the case, then at time 0:
- Sell short one share for $100.
- Invest $100 risk-free.
As a result, you will be holding a portfolio (x, y) withx=−1andy= 1, again
of zero initial value,
V(0) = 0.
The final value of this portfolio will be
V(1) =
{
−Su+A(1) if stock goes up,
−Sd+A(1) if stock goes down,which is non-negative, with the second value being strictly positive, since
A(1)≥Su.Thus,V(1) is a non-negative random variable such thatV(1)> 0
with probability 1−p> 0 .Once again, this indicates an arbitrage opportunity,
violating the No-Arbitrage Principle.
The common sense reasoning behind the above argument is straightforward:
Buy cheap assets and sell (or sell short) expensive ones, pocketing the difference.
1.4 Risk and Return
LetA(0) = 100 andA(1) = 110 dollars, as before, butS(0) = 80 dollars and
S(1) =
{
100 with probability 0.8,
60 with probability 0.2.