Solutions 281
This defines a predictable self-financing strategy. Its time 0 and time 1 value
is 0. The value at time 2 will be
{
0ifV(1)≥ 0 ,
V(2)−V(1)>0ifV(1)< 0.
This is, therefore, an admissible strategy realising an arbitrage opportunity.
4.4If you happen to know that an increase in stock price will be followed by a fall
at the next step, then adopt this strategy:- At time 0 do not invest in either asset.
- At time 1 check whether the stock has gone up or down. If down, then
again do not invest in either asset. But if stock has gone up, then sell
short one share forS(0)u, investing the proceeds risk-free.
Clearly, the time 0 and time 1 value of this strategy isV(0) =V(1) = 0. If
stock goes down at the first step, then the value at time 2 will beV(2) = 0.
But if stock goes up at the first step, then it will go down at the second step
andV(2) =S(0)u(r−d) will be positive, as required, sinceu>r>d.(The
notation is as in Section 3.2.)
Clearly, this is not a predictable strategy, which means that no arbitrage
has been achieved.
4.5a) If there are no short-selling restrictions, then the following strategy will
realise an arbitrage opportunity: - At time 0 do not invest at all.
- At time 1 check the priceS(1). IfS(1) = 120 dollars, then once again do
notinvestatall.ButifS(1) = 90 dollars, then sell short one share of the
risky asset and invest the proceeds risk-free.
The time 0 and time 1 value of this admissible strategy is 0. The value at
time 2 will be {
0 in scenariosω 1 andω 2 ,
3 in scenarioω 3 ,
which means that arbitrage can be achieved.
b) In a) above arbitrage has been achieved by utilising the behaviour of stock
prices at the second step in scenarioω 2 : The return on the risky asset is
lower than the risk-free return. Thus, shorting the risky asset and investing
the proceeds risk-free creates arbitrage. However, when short selling of risky
assets is disallowed, then this arbitrage opportunity will be beyond the reach
of investors.
4.6The arbitrage strategy described in Solution 4.5 involves buying a fraction of
a bond. If 9 S(1) = 90 dollars, then one share of stock should be shorted and
11 of a bond purchased at time 1. To obtain an arbitrage strategy involving
an integer number of units of each asset, multiply these quantities by 11, that
is, short sell 11 shares of stock and buy 9 bonds.
4.7Suppose that transaction costs of 5% apply whenever stock is bought or sold.
An investor who tried to follow the strategy in Solution 4.5, short selling one
share of stock at time 1 ifS(1) = 90 dollars, would have to pay transaction
costs of 90×5% = 4.50 dollars. If the remaining amount of 90− 4 .50 = 85. 50
dollars were invested risk-free, it would be worth 85. 5 ×^121110 =94.05 dollars
at time 2. But closing the short position in stock would cost $96, making the
final wealth negative. As a result, there is no arbitrage strategy.