- Forward and Futures Contracts 129
Proof
Suppose that
F(0,T)>[S(0)−e−rtdiv]erT.
We shall construct an arbitrage strategy. At time 0
- enter into a short forward contract with forward priceF(0,T) and delivery
timeT; - borrowS(0) dollars and buy one share.
At timet
- cash the dividend div and invest it at the risk-free rate for the remaining
timeT−t.
At timeT
- sell the share forF(0,T);
- payS(0)erTto clear the loan with interest and collect er(T−t)div.
The final balance will be positive:
F(0,T)−S(0)erT+er(T−t)div> 0 ,a contradiction with the No-Arbitrage Principle. On the other hand, suppose
that
F(0,T)<[S(0)−e−rtdiv]erT.
In this case, at time 0
- enter into a long forward contract with forward priceF(0,T) and delivery
at timeT; - sell short one share and invest the proceedsS(0) at the risk-free rate.
At timet
- borrow div dollars and pay a dividend to the stock owner.
At timeT
- buy one share forF(0,T) and close out the short position in stock;
- cash the risk-free investment with interest, collecting the amountS(0)erT,
and pay er(T−t)div to clear the loan with interest.
The final balance will again be positive,
−F(0,T)+S(0)erT−er(T−t)div> 0 ,completing the proof.