- Forward and Futures Contracts 137
Proof
Suppose for simplicity that marking to market is performed at just two inter-
mediate time instants 0<t 1 <t 2 <T. The argument below can readily be
extended to cover more frequent marking to market.
Take a long forward position with forward priceF(0,T) and invest the
amount of e−rTF(0,T) risk free. At timeTclose the risk-free investment, col-
lecting the amountF(0,T), purchase one share forF(0,T) using the forward
contract, and sell the share for the market priceS(T).Your final wealth will
beS(T).
Our goal is to replicate this payoff by a suitable strategy using futures
contracts. At time 0
- we open a fraction e−r(T−t^1 )of a long futures position (at no cost);
- we invest the amount e−rTf(0,T) risk free (this investment will grow to
v 0 =f(0,T) at timeT).
At timet 1
- we receive (or pay) the amount e−r(T−t^1 )[f(t 1 ,T)−f(0,T)] as a result of
marking to market; - we invest (or borrow, depending on the sign) e−r(T−t^1 )[f(t 1 ,T)−f(0,T)]
(this investment will grow tov 1 =f(t 1 ,T)−f(0,T) at timeT); - we increase our long futures position to e−r(T−t^2 )of a contract (at no cost).
At timet 2
- we cash (or pay) e−r(T−t^2 )[f(t 2 ,T)−f(t 1 ,T)] as a result of marking to
market; - we invest (or borrow, depending on the sign) e−r(T−t^2 )[f(t 2 ,T)−f(t 1 ,T)]
(this investment will grow tov 2 =f(t 2 ,T)−f(t 1 ,T)attimeT); - we increase the long futures position to 1 (at no cost).
At timeT
- we close the risk-free investment, collecting the amountv 0 +v 1 +v 2 =
f(t 2 ,T); - we close the futures position, receiving (or paying) the amountS(T)−
f(t 2 ,T).
The final wealth will beS(T), as before. Therefore, to avoid arbitrage, the
initial investments initiating both strategies have to be the same, that is,
e−rTF(0,T)=e−rTf(0,T),which proves the claim.
This construction cannot be performed if the interest rate changes unpre-
dictably. However if interest rate changes are known in advance, the argument