Futures Markets and Contracts

The steps in a cash-and- carry arbitrage are as follows:

*At the initiation of the contract:* • Borrow money for the term of the contract at market interest rates. • Buy the underlying asset at the spot price. • Sell (go short) a futures contract at the current futures price. *At contract expiration:* • Deliver the asset and receive the futures contract price. • Repay the loan plus interest. *If the futures contract is overpriced, this 5-step transaction will generate a riskless profit. The futures contract is overpriced if the actual market price is greater than the noarbitrage price.*

If the futures price is too low (which presents a profitable arbitrage opportunity), the opposite of each step should be executed to earn a riskless profit. This is reverse cash-and-carry arbitrage. The steps in reverse cash-and-carry arbitrage are as follows:

*At the initiation of the contract:* • Sell the asset short. • Lend the short sale proceeds at market interest rates. • Buy (go long) the futures contract at the market price. *At contract expiration:* • Collect the loan proceeds. • Take delivery of the asset for the futures price and cover the short sale commitment. *It may help to remember "buy low, sell high." If the futures price is "too high," sell the future and buy the spot. If the futures price is "too low," buy the future and sell the spot.

Treasury bond (T-bond) futures are traded for T-bonds with a maturity of

15 years or more. The contract is deliverable with a face value of $100,000. T-bond futures are quoted as a percent and fractions of 1% (measured in l/32nds) of face value.

A Eurodollar futures contract is similar to a forward rate agreement to lend US$1,000,000 for three months beginning on the contract settlement date. Eurodollar futures are based on

90-day LIBOR, which is an add-on yield. By convention, however, the price quotes are calculated as (100 - annualized LIBOR in percent). These contracts settle in cash, and the minimum price change is one "tick," which is a price change of 0.0001 = 0.01%, or $25 per $1 million contract.

Like forward contracts, futures contracts have no value at contract initiation. Unlike forward contracts, futures contracts do not:

Accumulate value changes over the term of the contract. Since futures accounts are marked to market daily, the value after the margin deposit has been adjusted for the day's gains and losses in contract value is always zero. The futures price at any point in time is the price that makes the value of a new contract equal to zero.

Marking to market is the process of:

Adjusting the margin balance in a futures account each day for the change in the value of the contract from the previous trading day, based on the settlement price. The futures exchanges can require a mark to market more frequently (than daily) under extraordinary circumstances.

A cash-and-carry arbitrage consists of:

Buying the asset, storing/holding the asset, and selling the asset at the futures price when the contract expires.

The opposite result will occur when the correlation between the price of the underlying asset and interest rates is negative:

Consider forwards and futures contracts on fixed income prices. Fixed income values fall when interest rates rise, so rates and values are negatively correlated. Borrowing costs are higher when funds are needed and reinvestment rates are lower when funds are generated by the mark to market of the futures contracts.

The Eurodollar futures are priced as a discount yield and LIBOR is subtracted from 100 to get the quote.

Every basis point (0.01%) move in (annualized) 90-day LIBOR represents a $25 gain or loss on the contract, just as with the T-bill contract. LIBOR, however, is actually an add-on yield, the rate you would earn on the face amount of a deposit. An add-on yield account for 167 days that pays $1 at maturity can be valued at expiration (77 days later) using 90-day LIBOR 77 days from now (L90t=77) as:

There may also be non-monetary benefits from holding an asset in short supply:

For a manufacturing firm, for example, this may be the benefit of having a ready supply so that a temporary shortage of their primary input will not disrupt their operations. The return from these non-monetary benefits is called the *convenience yield*.

The no-arbitrage price of a futures contract should be the same as that of a forward contract that was presented in the previous topic review:

However, there are a number of "real-world" complications that will cause futures and forward prices to be different. If investors prefer the mark-to-market feature of futures, futures prices will be higher than forward prices. If investors would rather hold a forward contract to avoid the marking to market of a futures contract, the forward price would be higher than the futures price.

The value of a futures contract strays from zero only during the trading periods between the times at which the account is marked to market:

If the futures price increases, the value of the long position increases. The value is set back to zero by the mark to market at the end of the mark-to-market period.

Any positive costs associated with storing or holding the asset in a cash and carry arbitrage will *increase* the no-arbitrage futures price because:

It is costly to buy, store, and deliver the asset. Many commodities have storage costs (e.g., corn, live cattle, and gold). There also is risk of loss from spoilage (corn), disease (cattle), and fire (oil or gas). Insuring or bearing these risks adds to the cost of holding these assets.

If we view a futures contract as a transfer of risk from an asset holder to the buyer of the contract, we would expect the futures price to be

Lower than the *expected* price in the future to compensate the future buyer for accepting asset price risk. This situation is called *normal backwardation*. If the futures price is greater than the expected spot price, it is called *normal contango*.

The generalized no-arbitrage futures price is now:

Positive net costs of holding the asset increase the futures price. When the asset generates cash flows, the net costs are negative (a net benefit), and the futures price is decreased.

The price of a currency future is derived exactly as we did for forwards:

Recall that R(FC) and R(DC) are the risk-free returns in the two different currencies, S0 is the spot exchange rate, and F(T )is the price of a futures contract of T years duration, with both the spot and futures price quoted in units of domestic currency per one unit of foreign currency.

Because a T-bill is a pure discount security, there are no cash flow benefits to consider, and we have the familiar (by now) relation:

Remember, for an asset without cash flows or storage costs, the right side of this equation is the cost of buying and holding the asset for a period of Tyears. The holding costs are simply the interest costs at Rf over the term of the futures contract.

At expiration, the spot price must equal the futures price because:

The futures price has become the price today for delivery today, which is the same as the spot. Arbitrage will force the prices to be the same at contract expiration.

In the case of a net benefit, the no-arbitrage futures price is:

The futures price will be lower when the dividend or coupon yield on the underlying asset is higher or when the non-monetary benefits of holding the asset are higher.

With financial assets there may be a significant benefit to holding the underlying asset, but there are no storage costs other than:

The opportunity cost of the funds. For example, holders of dividend-paying stocks, coupon bonds, and currencies will earn dividends, coupon payments, and interest, respectively. A monetary benefit from holding the asset will *decrease* the no-arbitrage futures price because the net cost of holding the asset is reduced.

Prior to contract expiration, the Eurodollar will be worth:

The present value of the expectation of this value. This is important because the value of the deposit will not change $25 for every one basis point change in expected 90-day LIBOR in 77 days as does the value of the futures contract. The asset value is not perfectly hedged by the contract value as it is with the T-bill contract. While no riskless arbitrage relation exists for the Eurodollar futures contract, it is still a very useful, and widely used, hedging instrument for exposure to LIBOR.

Treasury bill (T-bill) futures contracts are based on a $1 million face value 90-day (13-week) T-bill, and they settle in cash.

The price quotes are 100 minus the annualized discount in percent on the T-bills. For example, a price quote of 98.52 represents an annualized discount of 1.48%, an actual discount from face of 0.0148(90 / 360) = 0.0037, and a "delivery" price of (1 - 0.0037) x $1 million = $996,300. Each change of 0.01 in the price of a T-bill futures contract is worth $25. If you took a long position at 98.52 and the price fell to 98.50, your loss is $50 per contract.

The futures price is:

The price today for delivery at some future point in time (the maturity date).

Stock index futures are based on the level of an equity index. The most popular stock index future is the S&P 500 Index Future that trades in Chicago. Settlement is in cash and is based on a multiplier of 250.

The value of a contract is 250 times the level of the index stated in the contract. With an index level of 1000, the value of each contract is $250,000. Each index point in the futures price represents a gain or loss of $250 per contract. A smaller contract on the same index has a multiplier of 50.

Each exchange has a clearinghouse. The clearinghouse guarantees that:

Traders in the futures market will honor their obligations. The clearinghouse does this by splitting each trade once it is made and acting as the opposite side of each position. To safeguard the clearinghouse, the exchange requires both sides of the trade to post margin and settle their accounts on a daily basis. Thus, the margin in the futures markets is a performance guarantee

A preference for the mark-to-market feature will arise from a positive correlation between interest rates and the price of the contract asset:

When the value of the underlying asset increases and the mark to market generates cash, reinvestment opportunities tend to be better due to the positive correlation of asset values with higher interest rates. When the value of the underlying asset decreases and the mark to market requires cash, borrowing costs tend to be lower due to the positive correlation.

The short on the T-bond contract has the option of delivering any one of a number of different bonds. Each bond is assigned a conversion factor (CF)

a multiplier for the futures price on the "contract" bond, to adjust the settlement payment for delivery for higher or lower coupon bonds (with identical face value). The conversion factor is used to adjust the no-arbitrage price for the "cheapest to deliver" of all the permitted bonds. An arbitrage involving the cheapest-to-deliver bond may not be risk-free because the cheapest-to-deliver bond may change during the term of the contract. This offers an advantage to an arbitrageur who is short the future because the short holds the delivery option, not the long.

Contango refers to

a situation where the futures price is above the spot price. If there are no benefits to holding the asset (e.g., dividends, coupons, or convenience yield), the futures price will be FP = SQ(1 + Rf)T + FV (NC), and contango will occur because the futures price will be greater than the spot price.

Backwardation refers to

a situation where the futures price is below the spot price. For this to occur, there must be a significant benefit to holding the asset, either monetary or non-monetary. Backwardation might occur if there are benefits to holding the asset that offset the opportunity cost of holding the asset (the risk-free rate) and additional net holding costs.

For an index contract, rather than take the present value of each dividend on (possibly) hundreds of stocks, we can make the calculation as if the dividends are paid:

continuously (rather than at discrete times) at the continuous time equivalent of the dividend yield rate on the index.

The short in a T-bond futures contract has the option to deliver any of several bonds, which will satisfy the delivery terms of the contract. This is called a

delivery option and is valuable to the short. Each bond is given a conversion factor that is used to adjust the long's payment at delivery so the more valuable bonds receive a larger payment. These factors are multipliers for the futures price at settlement. The long pays the futures price at expiration multiplied by the conversion factor.

If we define the net cost of holding the asset as the costs net of any non-monetary benefits, we have:

net costs (NC) = storage costs — convenience yield

The most likely situation in financial markets is one in which futures prices are biased predictors of

spot rates (i.e., futures prices do not equal *expected* spot prices) and, more specifically, futures prices are less than *expected* spot prices (normal backwardation).

The currency futures market is smaller in volume than the forward market described in the previous topic review. In the United States, currency contracts trade on the euro, Mexican peso, and yen, among others. Contracts are set in units of

the foreign currency, and the price is stated in U.S. dollars per unit of foreign currency.

If both parties to a futures transaction are hedging existing risk

the futures price may be equal to *expected* future spot prices. In markets of hedgers, the futures price might be temporarily above or below expected future spot prices, but it would be an unbiased predictor of future spot rates.

The spot (cash) price of a commodity or financial asset is:

the price for immediate delivery.

T-bill futures are priced using the no-arbitrage principle. The key to understanding the pricing of T-bill futures is to recognize that

the underlying asset is a 90-day T-bill at the maturity of the futures contract. For example, suppose we want to price a 77-day T-bill future. If we bought a 167-day T-bill, we could deliver it in 77 days to satisfy a short position in a 90-day T-bill futures contract. In other words, in 77 days the 167-day T-bill will be a 90-day T-bill. If the borrowing cost of the 77-day loan to finance the purchase is equal to the gains from selling the T-bill in 77 days at the futures price, there is no arbitrage opportunity.

To calculate the no-arbitrage futures price for a T-bond contract, we must take account of:

the value of the coupon payments in constructing an arbitrage relation. To adjust the futures price for the expected coupon payments, subtract the future value of the coupon payments (FVC) from the no-arbitrage futures price on a bond with no coupon payments. Because the cost to hold the asset is reduced by the asset cash flows, the futures price that insures that a cash-and-carry arbitrage would provide no profit is lower than without the cash flows. It is cheaper to buy, hold, and deliver the asset because of the coupon payments.

A futures contract on an individual stock may have expected dividend payments over the life of the futures contract. To price such a contract:

we must adjust for the future value of the expected dividends. The no-arbitrage futures price adjusted for the future value of the dividends (FVD) or present value of the dividends (PVD) can be written as:

Futures contracts are very much like the forward contracts we learned about in the previous topic review. They are similar in that:

• Deliverable contracts obligate the long to buy and the short to sell a certain quantity of an asset for a certain price on a specified future date. • Cash settlement contracts are settled by paying the contract value in cash on the expiration date. • Both forwards and futures are priced to have zero value at the time the investor enters into the contract.

There are important differences, including:

• Futures are marked to market at the end of every trading day. Forward contracts are not marked to market. • Forwards are private contracts and do not trade on organized exchanges. Futures contracts trade on organized exchanges. • Forwards are customized contracts satisfying the needs of the parties involved. Futures contracts are highly standardized. • Forwards are contracts with the originating counterparty; a specialized entity called a clearinghouse is the counterparty to all futures contracts. • Forward contracts are usually not regulated. The government having legal jurisdiction regulates futures markets.

From a technical standpoint, the differences between the theoretical (no-arbitrage) prices of futures and forwards center on the correlation between interest rates and the mark-to-market cash flows of futures:

• Higher reinvestment rates for gains and lower borrowing costs to fund losses lead to a preference for the mark-to-market feature of futures, and higher prices for futures than forwards, when interest rates and asset values are positively correlated. • A preference to avoid the mark-to-market cash flows will lead to a higher price for the forward relative to the future if interest rates and asset values are negatively correlated.