Forward contracts and futures are much alike, except that the former are traded over- the-counter and the latter are traded in exchanges. Since the exchanges would like to organize trading such that contract defaults are minimized, an investor who buy a futures in an exchange is requested to deposit funds in a margin account to safeguard against the possibility of default (the futures agreement is not honored at maturity). At the end of each trading day, the futures holder will pay to or receive from the writer the full amount of the change in the futures price from the previous day through the margin account. This process is called marking to market the account. Therefore, the payment required on the maturity date to buy the underlying asset is simply the spot price at that time. However, for a forward contract traded outside the exchanges, no money changes hands initially or during the life-time of the contract. Cash transactions occur only on the maturity date. Such difference in the payment schedules may lead to differences in the prices of a forward contract and a futures on the same underlying asset and date of maturity. This is attributed to the possibility of different interest rates applied on the intermediate payments.
Here, let us argue how the equality of the two prices when the interest rate is constant contours. First, consider one forward contract and one futures which both last for n days. Let Fi and Gi represent the forward price and the futures price at the end of the ith day. We aim to show F0 = G0. Let Sn denote the asset price at maturity. Let the constant interest rate per day be δ. Suppose we initiate the long position of one unit of the futures on day 0. The gain/loss of the futures on the ith day is (Gi – Gi-1), and this amount grows to the currency value (Gi – Gi-1)eδ(n-i) at maturity, which is the end of the nth day (n ≥ i). Therefore, the value of this one long futures position at the end of the nth day is the summation of (Gi – Gi-1)eδ(n-i), where i runs from 1 to n. The sum can be expressed as,
∑i=1n (Gi – Gi-1)eδ(n-i)
The summation of gain/loss of each day reflects the daily settlement nature of a futures. Instead of holding one unit of futures throughout the whole period, the investor now keeps changing the amount of futures to be held on each day. Suppose he holds αi units at the end of the (i − 1)th day, i = 1,2,···,n, αi to be determined. Since there is no cost incurred when a futures is transacted, the investor’s portfolio value at the end of the nth day becomes
∑i=1n αi (Gi – Gi-1)eδ(n-i)
On the other hand, since the holder of one unit of the forward contract initiated on day 0 can purchase the underlying asset which is worth Sn using F0 currency units at maturity, the value of the long position of one forward at maturity is Sn − F0. Now, we consider the following two portfolios:
Portfolio A :
long position of a bond with par value F0 maturing on the nth day
long position of one unit of forward contract
Portfolio B : long position of a bond with par value G0 maturing on the nth day
long position of e-δ(n-i) units of futures held at the end of the (i − 1)th day, i = 1, 2, · · · n
At maturity (end of the nth day), the values of the bond and the forward contract in Portfolio A become F0 and Sn − F0, respectively, so that the total value of the portfolio is Sn. For Portfolio B, the bond value is G0 at maturity. The value of the long position of the futures (number of units of futures held is kept changing on each day) is obtained by setting αi = e-δ(n-i). This gives
∑i=1n e-δ(n-i)(Gi – Gi-1)eδ(n-i) = ∑i=1n (Gi – Gi-1) = Gi – G0
Hence, the total value of Portfolio B at maturity is G0 + (Gn − G0) = Gn. Since the futures price must be equal to the asset price Sn at maturity, we have Gn = Sn. The two portfolios have the same value at maturity, while Portfolio A and Portfolio B require an initial investment of F0e−δn and G0e−δn currency units, respectively. In the absence of arbitrage opportunities, the initial values of the two portfolios must be the same. We then obtain F0 = G0, that is, the current forward and futures prices are equal.