- Forward and Futures Contracts 133
Proof
Suppose that
V(t)<[F(t, T)−F(0,T)]e−r(T−t).
If so, then at timet
- borrow the amountV(t) to enter into a long forward contract with forward
priceF(0,T) and delivery timeT; - initiate a short forward position with forward priceF(t, T), at no cost.
Next, at timeT
- close out the forward contracts collecting (or paying, if negative) the
amountsS(T)−F(0,T) for the long position and−S(T)+F(t, T)for
the short position; - pay back the loan with interest amounting toV(t)et(T−t)in total.
The final balanceF(t, T)−F(0,T)−V(t)et(T−t)>0 will be your arbitrage
profit.
We leave the case when
V(t)>[F(t, T)−F(0,T)]e−r(T−t)as an exercise.
Exercise 6.6
Show thatV(t)>[F(t, T)−F(0,T)]e−r(T−t)leads to an arbitrage op-
portunity.Observe thatV(0) = 0,which is the initial value of the forward contract,
andV(T)=S(T)−F(0,T)(sinceF(T,T)=S(T)), which is the terminal
payoff.
For a stock paying no dividends formula (6.8) gives
V(t)=[S(t)er(T−t)−S(0)erT]e−r(T−t)=S(t)−S(0)ert. (6.9)The message is: If the stock price grows at the same rate as a risk-free invest-
ment, then the value of the forward contract will be zero. Growth above the
risk-free rate results in a gain for the holder of a long forward position.
Remark 6.3
Consider a contract with delivery priceXrather thanF(0,T). The value of
this contract at timetwill be given by (6.8) withF(0,T) replaced byX,
VX(t)=[F(t, T)−X]e−r(T−t).