Financial Derivatives/Basic Derivatives Contracts – Futures

Futures contracts, like forward contracts, specify the delivery of an asset at some future date. Futures contracts, unlike forward contracts,

1. Require the buyer or the seller of the futures contract to post margin.
2. Have minimum margin requirements; these requirements are achieved through a margin call.
3. Use the process of mark-to-market.

There three requirements in practice are not unique to futures contracts. The best way to understand them is by looking at a specific futures contract.

The corn futures contract trades at the Chicago Board of Trade (CBOT). The specifications of the contract are very strict and require the delivery of "no. 2 yellow" corn; if other grades of corn are delivered instead the price paid is adjusted [1]. The contract size can be in multiples of 5,000 bushels of corn. Futures can be purchased for delivery of corn in months December, March, May, July and September only. Trading this contract ceases on the business day nearest to the 15th calendar date of the delivery month. Delivery takes place two business days after the 15th calendar date of the delivery month.

Assume that one lot (5,000 bushels) of the Jul-07 contract was bought at 418 cents/bushel on 24th January 2007. The exchange would require the buyer to post initial margin of $900. If the buyer does not post this amount of money in her account with the exchange, her order cannot be executed. For this contract the maintenance margin is the same; during the life of this futures contract the balance of the account cannot go below this level; if for any reason the balance of the account falls below the maintenance margin, the buyer of this contract will receive a margin call.

On the date on which the trade was executed the mark-to-market of the futures contract is zero. Assume that on the next trading date, the settlement price of the futures contract is 418 3/4 cents/bushel (settlement price is the price traded for a futures contract at the close of the trading session). The mark-to-market of the Jul-07 corn futures is,

MtM = 5,000 * (4183 / 4 − 418) = $37.50

The balance on the buyer’s account will now be $937.50. The account is like a normal deposit account and earns interest on its balance.

If the market price of the Jul-07 corn contract drops in the following day, the mark-to-market could drop from $37.50 to $12.50. In this case $25 are withdrawn from the buyer’s account and the balance is now $912.50.

If on the last trading date of this contract (13th July 2007) the settlement price is 420 1/4 cents/bushel then the mark-to-market is $112.50. The final balance of the buyer’s account is $1012.50 plus interest earned. Since the corn that will be delivered on the 17th July 2007 is worth $21,012.50, the buyer will pay this amount to the clearing house. The clearing house acts as counterparty in the transaction between the corn producer and the buyer and makes sure payments are made and corn is delivered to the warehouse nominated by the buyer.

Since the trader has earned $112.50 (plus interest) in effect the net payment for delivery of corn is $20,900. This is equivalent to paying 418 cents/bushel on the corn delivered. The futures contract has therefore enabled the buyer to purchase corn at the original price of 418 cents/bushel and hedge against price changes.

In order to compare the price of a forward contract F₀ and the price of a futures contract Φ₀ we look at the following set of trades:

• We sell a forward contract to deliver a specific quantity of corn at some future date for price F₀.
• For dates i = 0,1,2,…,N − 1 we purchase a quantity q_i of the futures contract so that the following conditions are satisfied:
  • q₀(1 + r₁)^{N − 1} = 1
  • (q₀ + q₁)(1 + r₂)^{N − 2} = 1
  • …
  • (q₀ + q₁ + … + q_{N − 1})(1 + r_{N − 1}) = 1

where r_i is the daily interest rate applicable for period starting on date i and ending on date N. This set of equations can be solved recursively. The value of the margin account on date N will be,

To undestand the last equation, we know that on date i the total quantity of futures purchased is q₀ + q₁ + … + q_{i − 1}. By the end of date i the mark-to-market change is equal to (q₀ + q₁ + … + q_{i − 1})(Φ_i − Φ_{i − 1}). Depending on the direction of the change Φ_i − Φ_{i − 1} this is a gain or a loss and earns or requires the payment of principal plus interest at the expiry date of the contract.

Given the conditions that give rise to the solutions for q_i, the last equation is equal to Φ_N − Φ₀. Since at expiry the price of the futures is equal to the spot price of the asset and therefore F_N = Φ_N, if Φ₀ is different from F₀ a risk-free profit can be generated. Note that there is no cost in entering into the series of futures contracts and depending on the sign of the difference F₀ − Φ₀ the strategy can be reversed. Therefore the forward price must be equal to the futures price.

The strategy used in this analysis assumes that when we purchase an additional quantity q_i of the futures contract we know the interest rate for the period i + 1 to N. Since in practise the actual value of the interest rate is not known assume that we can lock in a forward rate. However since we cannot predict the change in the mark-to-market of the futures contract in the period i to i + 1 we do not know the notional amount we must purchase.

Assume that on date i we make the assumption that there will be no change in the mark-to-market of the futures and therefore there is no need to lock in a forward rate for the period i + 1 to N. Since the most likely scenario is that we will be wrong, we will have to borrow or deposit the actual change in the mark-to-market at the spot rate for the period r_{i + 1}.

As long as the error in our estimate of the mark-to-market change is independent of the spot rate we can expect that the costs/benefits will balance to zero. But if the mark-to-market change of the futures contract is a function of spot rate the costs/benefits will not balance to zero and the futures strategy described above will not be able to replicate the payoff of the forward. We conclude that when the futures contract is a function of the interest rate the futures price will not be equal to the forward price.

Another exception occurs when the futures price can change by large amounts from one date to the next. The term "large amounts" here means that a one day move accounts for a large percentage of the difference between Φ₀ and Φ_N. In this case on the date when this large price change occurs the error in the notional locked in at the forward rate is large enough to magnify the error in our estimate of the change in the mark-to-market. Furthermore, all subsequent mark-to-market changes are much smaller and cannot balance this cost/benefit. Fortunately, most exchanges limit the maximum change in the futures price that can occur from one date to the next. But if such large price moves are possible then, even if the futures price is not a function of the interest rate, the assumption that it is equal to the forward price is wrong.

In general, the relation between the futures and the forward price cannot be derived through a static arbitarge strategy unless interest rates have a deterministic term-structure. The derivation of the relation between the futures and the forward price of an asset is one of the first applications of dynamic hedging [Black 1976].