- Forward and Futures Contracts 127
Proof
We shall prove formula (6.1). Suppose thatF(0,T)>S(0)erT. In this case, at
time 0
- borrow the amountS(0) until timeT;
- buy one share forS(0);
- take a short forward position, that is, agree to sell one share forF(0,T)at
timeT.
Then, at timeT
- sell the stock forF(0,T);
- payS(0)erTto clear the loan with interest.
This will bring a risk-free profit of
F(0,T)−S(0)erT> 0 ,contrary to the No-Arbitrage Principle. Next, suppose thatF(0,T)<S(0)erT.
In this case we construct the opposite strategy to the one above. At time 0
- sell short one share forS(0);
- invest the proceeds at the risk-free rate;
- enter into a long forward contract with forward priceF(0,T).
Then, at timeT
- cash the risk-free investment with interest, collectingS(0)erTdollars;
- buy the stock forF(0,T) using the forward contract;
- close out the short position in stock by returning it to the owner.
Youwillendupwithapositiveamount
S(0)erT−F(0,T)> 0 ,again a contradiction with the No-Arbitrage Principle.
The proof of (6.2) is similar. Simply replace 0 byt, observing that the time
elapsed between exchanging the forward contract and delivery is nowT−t.
In a market with restrictions on short sales of stock the inequalityF(0,T)<
S(0)erTdoes not necessarily lead to arbitrage opportunities.
Exercise 6.1
Suppose thatS(0) = 17 dollars,F(0,1) = 18 dollars,r=8%,and short-
selling requires a 30% security deposit attracting interest atd=4%.Is
there an arbitrage opportunity? Find the highest ratedfor which there
is no arbitrage opportunity.