FMP 10: Pricing Financial Forwards and Futures
A forward contract fixes today the price for a trade that settles later, and no-arbitrage reasoning, not a forecast of the delivery-date price, decides where that fixed price has to sit. Covered interest parity answers the same question for a foreign currency, and the arguments below extend it to stocks, bonds and stock indices.
A financial asset owes its value to a claim of some kind rather than to physical usefulness: shares claim company earnings and net assets, bonds a promised stream of payments, deposits a bank. Whatever market participants hold for investment rather than for use is called an investment asset, a group covering every financial asset plus a handful of non-financial ones, gold and silver being the usual pair. The remainder are consumption assets, where crude oil and copper belong. That split matters because each arbitrage trade below needs somebody prepared to give up the asset, by selling it or lending it, against a certain sum later, and a refinery holding crude for next month’s production run will not.
Three categories of financial asset
Pricing divides according to what the asset pays its holder while the contract runs. Some pay nothing: a non-dividend-paying stock, a Treasury bill, a zero-coupon bond. Some pay a known amount of cash, such as a bond with a known coupon or a stock whose dividend has been declared. The rest pay a known percentage of their own value, which is how a stock index and a foreign currency get treated.
Several trades ahead involve selling an asset the trader does not own, with a repurchase to come later. A short sale of that kind pays when the price falls and loses when the price rises.
An investor wanting to short 100 shares of Company X instructs a broker, who borrows those shares from another client and sells them in the ordinary way. Closing out later is called covering: the broker buys 100 shares back and returns them to the account they came from. Any dividend falling due while the position is open is owed by the short seller and passed to that account. A broker in the United States can lend out shares another client has bought on margin without asking that client, so small investors are sometimes blocked altogether while large investors borrow from institutions such as State Street Corporation, paying a fee normally below 50 basis points per year that climbs when the security is scarce.
An investor shorts 100 shares in May at a share price of USD 50, covering in September at USD 30. A USD 2 per share dividend lands in July. Brokerage and borrowing fees are ignored.
Several countries banned investors from shorting financial stocks during the global financial crisis of 2007-2008, on the view that such trades were making matters worse. Many analysts hold the opposite position, treating short selling as part of how prices come to reflect information.
An asset paying its holder nothing trades at USD 70 a unit, and a large bank funds itself, or places cash, for one year at 5%. Testing two candidate forward prices, one high and one low, shows why a single figure survives.
Scenario A: the one-year forward price is USD 80
Here the trade runs long the asset and short the forward contract. A trader borrows USD 70, buys one unit today, and agrees to sell it in a year at USD 80. At maturity the unit is delivered, USD 80 arrives, and the loan is settled: a year of interest at 5% on USD 70 comes to USD 3.50. Taking USD 80.00, less the USD 70.00 outlay and less USD 3.50 of interest, leaves USD 6.50, with no price exposure at any moment. Any forward price above USD 73.50 leaves a riskless margin, so arbitrage drives the forward price down and caps it at USD 73.50.
Scenario B: the one-year forward price is USD 65
Now reverse the trade, running short the asset and long the forward contract. The trader borrows a unit, sells it for USD 70, and agrees to repurchase it a year later at USD 65, setting aside any borrowing fee. Reacquiring for USD 65 what was sold for USD 70 is worth USD 5, and the USD 70 raised by the sale earns a further USD 3.50 at the risk-free rate, giving USD 8.50 in total. Every forward price below USD 73.50 supports this trade, so arbitrage drives the forward price up and the floor is USD 73.50 as well. Ceiling and floor meet, leaving one forward price consistent with the absence of arbitrage.
Two rates rather than one
Banks borrow and lend at slightly different rates. What matters to a bank trader is sometimes called the opportunity cost of capital, in practice the interbank rate. Put the borrowing rate at 5.1% and the lending rate at 5.0%: Scenario A funds itself at the higher figure and breaks even at 70 x 1.051 = 73.57, while Scenario B is untouched because it invests rather than borrows, so the argument delivers a band from USD 73.50 up to 73.57. Everything below uses one risk-free rate. A borrowing fee would likewise make Scenario B dearer, which is answered by handing that trade to an investor who already owns the asset.
Write F for the forward price and S for the spot price of the asset, T for the number of years the contract still has to run, and R for the annually compounded risk-free rate applying to maturity T. That rate is the relevant opportunity cost of funds, usually an interbank borrowing or lending rate.
Two inequalities follow from this expression. Where F sits above S(1+R)T, an arbitrageur buys the asset and shorts the forward contract, locking in F – S(1+R)T. Where F sits below it, an arbitrageur shorts the asset, or sells it if it is already owned, goes long the forward contract, and locks in S(1+R)T – F.
A non-dividend-paying stock trades at USD 50. A three-month forward contract to sell it is being priced, and the annually compounded rate out to three months is 4% per year.
What happens when financing cost is booked as profit
Joseph Jett, a trader at Kidder Peabody in the early 1990s, ran precisely the Scenario A strategy, buying Treasury strips, which are zero-coupon bonds, and selling them forward. A flaw in the systems there let him book the gap between the forward price and the spot price as profit immediately: buying at USD 70 and selling a one-year forward at USD 73.50 showed USD 3.50 of profit on day one. That USD 3.50 is the cost of financing the position for a year, recognised only on settlement, so the appearance of profit demanded a book that kept growing. Bonuses of over USD 13 million reached Jett between 1992 and 1994, and once the scheme was understood the firm wrote its profits down by over USD 300 million.
Let the asset now pay something while the contract runs: a bond with a known coupon, or a stock whose dividend is known ahead of time, either because it was declared before the ex-dividend date or because management has signalled that the level will hold. Keep the asset at USD 70 and add a cash flow of USD 5 arriving in six months. Rates are no longer flat: 5% per year out to one year, and 4% per year out to six months, both annually compounded.
At a forward price of USD 80 a trader buys the asset for USD 70, sells the one-year forward, and knows USD 5 will arrive halfway through. Funding splits in two. One piece, 5 discounted over half a year at 4%, is USD 4.903, repaid exactly by the income when it lands. The other, 70 – 4.903 = 65.097, is borrowed for the year at 5% and grows to USD 65.097 x 1.05 = USD 68.352. Delivering the asset brings USD 80, so the trader keeps USD 80 – USD 68.352 = USD 11.648, and every forward price above USD 68.352 works this way.
Run in reverse, at a forward price of USD 65, the trader shorts the asset for USD 70 and buys the one-year forward. The short position now owes the USD 5 of income to the owner of the asset in six months. Splitting the proceeds covers it: USD 4.903 placed for six months at 4% grows to exactly USD 5, and the remaining USD 65.097 placed for a year at 5% grows to USD 68.352, against which the asset is bought back for USD 65, leaving USD 68.352 – USD 65 = USD 3.352. Every forward price below USD 68.352 works, so USD 68.352 is the only price standing, well under the USD 73.50 of the previous section because whoever holds the contract will not collect the USD 5.
Generalization
Let I stand for the present value of income due over the life of the contract, with S, F, T and R unchanged.
Above, I is 4.903 and F = (70 – 4.903) x 1.05, or 68.352. Buying the asset and selling it forward pays whenever F exceeds (S – I)(1+R)T, and shorting it while buying forward pays whenever F falls short.
A ten-month forward contract is written on a bond that pays coupons of USD 4 at the three-month and nine-month points, with a flat 6% per year risk-free rate across maturities and a cash price of USD 105.
In the third case the income arrives as a percentage of what the asset is worth rather than as a fixed sum. A stock index behaves that way, and so does a holding of foreign currency. Call the yield Q per year, annually compounded.
Reinvestment makes this case tractable. Ploughing every payment back into the asset makes the holding grow at rate Q, so one unit today becomes (1 + Q)T units by time T. Two trades exploit that: buy one unit for S now, and sell forward (1 + Q)T units at price F per unit for delivery at time T. Reinvestment turns the single unit into precisely the quantity already sold, so a certain S today buys a certain F(1 + Q)T later.
Which side of the spot price the forward price lands on depends on which rate is larger. With R above Q the ratio exceeds one and the forward price climbs with maturity, financing costing more than holding pays. With Q above R it falls away instead, and where the two match, spot and forward coincide at every horizon. An asset yielding 3% per year over three years, priced at USD 30 with a 4% risk-free rate, gives a three-year forward price of 30 x 1.02941 = USD 30.88, barely a dollar of carry. One caution on conventions: 4% with semi-annual compounding is the same thing as 4.04% annually compounded, since 1.02 x 1.02 – 1 = 0.0404, and Equation (10.3) wants the annually compounded figure.
Why a foreign interest rate behaves like a yield
Covered interest parity produced a forward rate formula for the GBP/USD exchange rate, which means the number of US dollars per British pound.
Seen from the United States, a holding of British pounds is an asset priced in dollars that throws off interest at the GBP rate. Where that interest is paid makes it a yield: it arrives in pounds, so a US investor values it at the GBP rate applied to the dollar price of a pound, and each unit generates income equal to a fixed percentage of its own price. Putting R equal to RUSD and Q equal to RGBP in Equation (10.3) reproduces Equation (10.9), so the currency case is the known yield case relabelled. A currency paying more interest than the domestic rate trades at a forward discount, one paying less at a forward premium.
Two quantities carry similar names and behave nothing alike. The forward price is what the delivery price would be if a contract were negotiated now. The value of a forward contract is what an existing contract is worth to the party holding it.
When such a contract is first written on a financial asset, the forward price comes from the sections above and the contract itself is worth zero, or near enough. Anything else would oblige one side to pay the other at the outset, in the way an option buyer pays a premium. Time passes, the price of the asset moves, and the contract drifts into positive or negative territory while the delivery price stays where it was written.
Take a long forward contract to buy an asset for price K, agreed at some earlier date. Write F for the forward price a fresh contract of the same remaining maturity would carry today, and T for the time left. Both deliver the same asset on the same day, differing only in the amount handed over at maturity, K under the old contract against F under a new one. Holding the old contract means paying F – K less at time T, and that saving is certain. A contract struck today is worth zero, which pins the old contract at the present value of that difference.
Substituting each forward price formula for F restates the same value in the spot price.
Return to the asset at USD 70 with no income and a one-year rate of 5%, so the current no-arbitrage forward price is USD 73.50. A long forward contract to buy at USD 78 in one year was agreed some time ago.
Futures contracts settle daily while forward contracts settle at maturity. Place two contracts side by side that agree in every other respect. Were interest rates constant, or moving predictably, their theoretical no-arbitrage prices would coincide, a result established by Cox, Ingersoll and Ross in 1981. Rates move unpredictably, so the two come apart, the size and sign of the gap following from the correlation between interest rates and the return on the underlying asset.
Suppose the price of an asset tends to move in step with interest rates. A rise hands the long futures position an immediate cash gain through daily settlement, reinvested at what is on average a relatively high rate, and a fall triggers an immediate cash loss whose funding on average comes cheap. Both branches favour the long futures position over the long forward position, so the futures price is bid slightly above the forward price. Reverse the correlation and the argument reverses: gains then arrive when reinvestment is poor and losses when funding is dear, putting the theoretical futures price slightly below the forward price.
Gaps of this kind are small enough to ignore in most work, with Eurodollar futures contracts the notable exception, so Equations (10.1) to (10.3) serve for futures prices too. Equations (10.4) to (10.8) have no futures counterpart, since daily settlement resets a futures position to zero value each day.
Delivery dates and the equivalent forward maturity
A futures contract usually permits delivery across a range of dates while a forward contract names one, and the choice belongs to the party holding the short position. For a financial asset it comes down to the interest rate against the income the asset generates. When the rate is larger, holding costs more than it pays and delivery comes as early as possible; when the income is larger, delivery is pushed as late as the contract allows. Whichever end gets chosen supplies the maturity to use when a futures contract is treated as a forward contract.
Equal prices do not mean equal profits
The one-year forward price and the one-year futures price for an asset both stand at USD 2.00, and across a single day both climb to USD 2.10. Trader A, long a futures contract on 1,000 units, collects USD 100 straight away through daily settlement. Trader B, long a forward contract, gains USD 100 as well, but only in a year, so the books show Trader A up by USD 100 and Trader B up by the present value of it. A fall of USD 0.10 works the same way in reverse, costing Trader A USD 100 at once and Trader B slightly less on paper.
A stock index follows a hypothetical portfolio of shares, so a rise of X% in that portfolio shows up as a rise of X% in the index. The weight of a stock is the percentage of the portfolio invested in it, set in most indices by market capitalization, meaning share price multiplied by shares outstanding.
| Index | Stocks in the portfolio | Weighting basis | CME contract sizes |
|---|---|---|---|
| Dow Jones Index | 30 large stocks | Share price | USD 5 and USD 10 times the index |
| NASDAQ-100 | 100 stocks on the Nasdaq Stock Market | Market capitalization | USD 20 and USD 100 times the index |
| S&P 500 | 500 stocks | Market capitalization | USD 50 and USD 250 times the index |
Source: the index futures contracts described in the chapter, ordered from the smallest portfolio to the largest.
Settlement is in cash, and the S&P 500 contract settles against the opening index level on the delivery month’s third Friday. Such contracts trade elsewhere too: the CSI 300 Index, built from 300 Chinese stocks and weighted by market capitalization, has futures on the China Financial Futures Exchange (CFFEX).
An index can be handled as a financial asset that pays dividends, the dividends being whatever a holder of the underlying portfolio would collect. Estimating every constituent’s dividend would give a total cash income for Equation (10.2), but practice assumes a known yield instead, so Equation (10.3) applies with Q set to the average dividend yield over the life of the contract. Dividends are normally left out of an index calculation, so the index understates total return. A total return index is the exception, built by reinvesting them into the hypothetical portfolio.
An index stands at 2,500, with dividends yielding 3% per year against a risk-free rate of 5% per year, both flat across maturities.
Whenever the futures price of an index sits above its theoretical value, an arbitrageur buys the index constituents and sells the futures contract; whenever it sits below, the arbitrageur shorts those constituents and buys the futures. Both go by the name of index arbitrage, and either means firing hundreds of stock orders at an exchange in the same instant as the futures order, done by computer and called program trading.
The mechanism normally holds that relationship in place, but it can be swamped. On October 19, 1987, remembered as Black Monday, the market fell by more than 20% and share volume on the New York Stock Exchange beat every previous record. Orders backed up, index arbitrage could not be executed efficiently, and the futures price dropped well below the theoretical price. Equation (10.3) also needs a portfolio whose price always equals the index, and that can fail. The Chicago Mercantile Exchange lists a Nikkei 225 Index futures contract settled in dollars instead of yen: a trader can hold a portfolio always worth the index in yen, but exchange rates move, so none is always worth that in dollars.
Three categories of financial asset, one pricing idea, and a value formula falling out of each price. The table sets the categories across the top, so every row asks one question of all three.
| Provides no income | Known income, present value I | Known yield at rate Q | |
|---|---|---|---|
| Forward or futures price | |||
| Value of a long forward with agreed price K | |||
| What happens to the spot price first | Nothing is removed | The present value of the income is deducted | The spot price is discounted at the yield |
| Typical underlying | Treasury bill, non-dividend-paying stock | Coupon bond, stock with a declared dividend | Stock index, foreign currency |
Source: the summary of results in the chapter, transposed so the categories run across the columns, with the bottom two rows added here. S is the current asset price, T the maturity, R the interest rate.
Reading across a row shows the only thing that changes between the cases, which is how income gets handled: ignored, deducted as a cash present value, or divided out as a rate. Futures and forward prices are approximately equal on one asset for one maturity, so the top row does duty for both, while the second row belongs to forward contracts alone. Interest rate futures yield to no-arbitrage arguments of the same family, but they carry features of their own and are handled separately.