DER 6 Pricing and Valuation of Futures Contracts

A futures contract is a standardized, exchange-traded derivative. Like a forward, it is a forward commitment: at inception no cash changes hands, and neither party holds an asset or owes a liability, so its value begins at zero. The futures price agreed at the start, written f0(T), is the spot price grown at the risk-free rate across the whole life of the contract, exactly the relationship that governs a forward price. Both prices are built from the cost of carry, meaning the benefits earned and the costs paid for holding the underlying asset until the derivative expires.

So why devote a separate treatment to futures? Because of one mechanical feature that a plain forward lacks: daily settlement through a margin account. Each trading day the exchange marks the position to the current futures price, ft(T), pays or collects the day’s gain or loss in cash, and resets the contract value back to zero. This repeats every day until the contract matures, at which point the futures price and the spot price converge to the same number, the terminal spot price ST.

Why the difference matters

Daily settlement does not change where the position ends up: by maturity the cumulative realized gain or loss comes out almost identical on the futures and on a comparable forward. What it changes is the timing and pattern of the cash flows. A forward accumulates an unsettled gain or loss that is paid only once, at maturity. A futures position turns that same profit into a stream of daily cash receipts and payments. Because those interim cash flows are received or paid at different points in time, they can be reinvested or financed at prevailing interest rates, and that is enough to open a small gap between the two prices. The size of the gap depends on how volatile interest rates are and on how futures prices and interest rates move together.

This module works through five ideas: pricing a futures contract at inception, tracking its mark-to-market value against a matching forward, the mechanics of short-term interest rate futures (whose price follows a 100 minus yield convention), the reasons forward and futures prices can differ (including the convexity bias that separates interest rate futures from forward rate agreements), and how central clearing of over-the-counter derivatives has narrowed those differences.

When the contract is struck, nothing is owed by either party, so the initial value is zero:

For an underlying that carries neither a cost nor a benefit, no arbitrage forces the futures price to equal the spot price grown at the risk-free rate. With discrete compounding, used for futures on a single underlying asset:

When the underlying is a portfolio, for instance a credit, commodity, fixed-income, or equity index, or when the trade involves foreign exchange with interest rates in two currencies, continuous compounding is the preferred convention:

Many underlyings do pay a benefit or impose a cost during the contract’s life. Writing the present value of any income or ownership benefit as PV0(I) and the present value of any carrying cost as PV0(C), the relationship becomes:

At inception the two contracts price the same way. Once time passes, their mark-to-market (MTM) values diverge, driven entirely by daily settlement. A forward price stays fixed at F0(T) for the life of the contract, and its running value is the difference between the current spot price and the present value of that fixed forward price. From the buyer’s side:

That value is not settled in cash until maturity, so it accumulates as an unrealized position and leaves the counterparties exposed to each other’s credit the whole time. A futures position works the opposite way: each day the gain or loss is paid across in cash (variation margin), the contract value snaps back to zero, and the counterparty credit exposure is limited to a single day’s move.

Futures on short-term interest rates give traders a liquid, standardized alternative to a forward rate agreement (FRA). Contracts exist for monthly or quarterly market reference rates (MRR) over successive periods, out to maturities as long as ten years. The underlying is the MRR on a hypothetical future deposit, just as with an FRA, but interest rate futures are quoted on a price basis rather than as a rate:

Suppose a contract on the three-month rate three months forward (A = 3m, B = 6m, B − A = 3m) trades at 98.25. Inverting the quote gives the implied rate:

This 100 minus yield convention produces an inverse relationship between price and yield, but the quoted number does not equal what a zero-coupon bond would cost at that rate. A long position effectively receives the MRR in A periods, and a short position pays it, which fixes the direction of the exposures:

• Long futures (the lender): gains as the price rises, which happens as the future MRR falls.
• Short futures (the borrower): gains as the price falls, which happens as the future MRR rises.

Mapping this onto forward rate agreements, a fixed-rate payer on an FRA gains when rates rise, which matches a short interest rate futures position. The table below lines up the two instruments.

Basis point value and daily settlement

Daily settlement on an interest rate future tracks price changes, which translate into a basis point value (BPV), the cash change per one basis point move:

Take a $1,000,000 notional tied to a 2.21% three-month MRR running one quarter (90/360). The underlying deposit value is $1,005,525 (= $1,000,000 × [1 + 2.21% / 4]). A one basis point rise to 2.22% lifts it to $1,005,550, and a one basis point fall to 2.20% drops it to $1,005,500. Either way the value moves by $25, so the BPV is $25. Short-term interest rate futures have this fixed, linear link between price and yield changes, a point that matters when we reach the convexity bias.

Although a forward and a futures on the same underlying share the same symmetric payoff at maturity, their cash flow paths over the intervening life are not the same, and that is what can drive their prices apart. The defining features of a futures contract are the initial margin, the daily mark-to-market, and the daily settlement of gains and losses. These features cap the futures value at just the one day’s move since the last settlement; once that amount is paid through the margin account, the price resets to the settlement level and the value returns to zero. A forward, by contrast, rests on privately negotiated credit terms (sometimes backed by cash or securities collateral), requires no daily cash settlement, and is squared up only at maturity in a single payment of the accumulated change in value.

Those different cash flow patterns are the source of any price gap. Forward and futures prices are identical in two cases:

• when interest rates are constant, or
• when futures prices and interest rates are uncorrelated.

Break either condition and a difference can appear. Suppose the futures price tends to rise and fall together with interest rates (a positive correlation). A long futures holder then earns gains precisely when rates are rising, so those cash inflows are reinvested at higher rates, while losses land when rates are falling and can be financed cheaply. That timing makes a long futures more attractive than an otherwise identical long forward, and the more attractive contract commands the higher price. The differential also widens as interest rate volatility increases. If instead the price moves opposite to rates, the logic reverses: the long forward becomes the more desirable, and higher-priced, contract.

For most futures the maturity is short and participants can borrow near the risk-free rate over that horizon, so in practice the futures and forward prices differ little. The notable exception among interest rate instruments is the convexity bias, which comes from the different way an interest rate future and an FRA respond to a rate change.

Recall the $1,000,000 notional on a 2.21% three-month MRR that runs one quarter (90/360), whose underlying deposit value was $1,005,525 and whose contract BPV was $25. Now compare it with a $1,000,000 FRA on the three-month MRR three months forward at the same 2.21% rate. The FRA net payment is the rate difference applied to the notional over the period:

If the observed MRR at settlement is 2.22% (a rise of one basis point), the raw net payment is $25 (= $1,000,000 × 0.01% × 1/4). The FRA settles on the present value of that final cash flow, though, discounting at the MRR:

Widen the rate move and compare the FRA present-value settlement against the linear futures settlement, and the asymmetry shows.

The futures settlement is a straight line: every basis point is worth exactly $25 either way. The FRA is not linear. Its percentage price change is larger in absolute value when the MRR falls than when it rises, the same convexity that characterizes fixed-income instruments. The discounting step, present in the FRA but absent from the daily-settled futures, is what creates the bias, and it grows with the length of the discounting period.

In periods of market stress, a large price move combined with a counterparty that cannot meet a margin call may force a futures position to be closed out before maturity. An over-the-counter forward, with its more flexible credit terms, may instead stay open through the same episode. That contrast used to be one more way futures and forwards differed.

The spread of central clearing has narrowed it. Dealers who buy and sell forwards to end users are now often obliged to lodge cash or eligible high-quality collateral with a central counterparty, a margining arrangement that closely mirrors futures. Facing those requirements themselves, dealers tend to pass similar margin terms on to their own clients. The result is that over-the-counter forwards increasingly carry futures-like daily collateral demands, which shrinks the difference in cash flow impact between exchange-traded and over-the-counter derivatives, and with it the price difference between futures and forwards.

The practical takeaway for an investor using either instrument is the same: hold enough cash or eligible collateral to meet margin and collateral calls, and factor in the financing, transaction, and administrative costs of carrying the positions when derivatives are used in a portfolio.