Forward and Futures Pricing: Cost of Carry

A forward price is the price for a future delivery agreed today. A futures price uses the same basic idea, with daily settlement added by the exchange. Neither price is simply a prediction of where the asset will trade later.

The useful model is:

spot + financing + storage − income ≈ fair delivery price

The forward contract article shows the basic trade. This article turns it into a small pricing-engine model.

Reproduce the delivery exposure

Suppose an asset costs $100 today. The delivery date is six months away, and financing the purchase costs about $2 over that period. A trader can:

borrow $100;
buy the asset today;
agree to sell the asset at the delivery date.

The asset is available for delivery later, but the borrowing has grown to about $102. If the forward price is $105, the trade locks in a $3 difference before fees. If the forward price is $99, an owner of the asset can sell it today, invest the proceeds, and use a long forward to buy it back for $99 later.

These are cash-and-carry trades. They connect the delivery price to the cost of creating the same exposure with the spot asset. The exact equality depends on executable financing, trading costs, taxes, short-sale rules, and the relevant interest-rate convention.

The pricing engine’s inputs are therefore more useful than the formula alone:

spot price
funding curve and maturity
income or dividends
storage and other holding costs
currency and settlement conventions

Its output is a fair delivery price under those assumptions. A risk engine may then compare that fair price with the traded price and calculate the value of an existing contract.

Choose the income branch

The pricing branch depends on what the underlying pays. In these formulas, $F_0$ is today’s fair delivery price, $S_0$ is today’s spot price, $r$ is the funding rate, and $T$ is the time to delivery:

no income: $F_0 = S_0 e^{rT}$ ;
known cash income with present value $I$ : $F_0 = (S_0 - I)e^{rT}$ ;
proportional yield $q$ : $F_0 = S_0 e^{(r-q)T}$ .

Set $I$ or $q$ to zero when the asset has no income. For an FX quote such as EUR/USD, where the quote is US dollars per euro, $S_0$ is dollars per euro, $r$ is the dollar funding rate, and $q$ is the euro interest rate. The formula therefore becomes $F_0 = S_0 e^{(r_{USD}-r_{EUR})T}$ . An inverse quote uses the opposite units and rate direction. In these exponential formulas, $r$ and $q$ are continuously compounded rates. If a curve supplies simple or nominal rates, the engine must convert them first or use discount factors; it must not insert them into $e^{rT}$ unchanged.

Income changes the input set

If the asset pays income before delivery, the owner receives something that a forward holder does not. That income reduces the cost of carrying the asset.

For a share or index, the income may be dividends. For a currency, the foreign interest rate acts like income from holding that currency. For a commodity, storage is usually a cost rather than income. An investment commodity, such as gold, is held by many people mainly as an investment. A consumption commodity, such as oil, is held mainly to be used in production.

For example, with a stock index at 4,000, a five percent annual funding rate, and a two percent annual dividend yield, the approximate six-month carry is three percent per year:

4{,}000 \times (1 + 0.03 \times 0.5) = 4{,}060

The numbers are an approximation using simple interest. A production engine may use continuously compounded rates, a discount curve, dividend dates, and day-count conventions. Those details change the result slightly; they do not change the cash-and-carry relationship.

A currency pair follows the same idea. If the dollar rate is five percent and the euro rate is three percent, the dollar price of one euro tends to include the two-percent rate difference over the delivery period. The quote direction matters: EUR/USD = 1.10 means $1.10 per euro, while the inverse quote has different units.

Storage can add a cost, but physical inventory can add a benefit

A commodity price also depends on whether someone needs the physical item now. Financing a barrel of oil and storing it until delivery adds to its carry. A refinery may still value the barrel itself because it can use it in production. A futures contract gives price exposure but cannot be put into the refinery.

That practical benefit is called convenience yield. It can make the futures price lower than a simple spot-plus-financing-plus-storage calculation.

This is why the same pricing model has different branches:

Underlying	Main adjustment to carry
Share or index	Dividends reduce carry
Foreign currency	Foreign interest acts like income
Investment commodity	Storage adds to carry; income may reduce it
Consumption commodity	Storage adds to carry; physical availability can add convenience yield

The system should store these as different inputs and conventions, not as one universal “interest rate” field.

Mark an existing forward

A new forward is priced using today’s spot and market inputs. An existing forward has a fixed delivery price $K$ , so its value changes when the fair delivery price changes.

Suppose the old contract allows a long position to buy at $100, while a new comparable forward would cost $102. The old contract is useful because it lets the holder buy below the current fair delivery price. The exact mark-to-market value also discounts the difference back to today.

This distinction matters in a pricing service:

new contract → fair forward price
existing contract → value using fair price versus fixed K

The traded futures price can also differ slightly from a forward price when interest rates are uncertain. Futures settle gains and losses each day, so the timing of funding and reinvestment matters. The futures settlement article explains those daily cash movements.

Delivery price and future spot price

The fair delivery price is not a promise about the future spot price. The asset can be more or less valuable at expiry because demand, risk, and expected returns change. The delivery price is constrained by the cost of creating the exposure today; it does not remove the future market risk.

For exact compounding, curve construction, and product-specific conventions, a pricing engine needs a larger model. For the basic explanation, keep the central rule: financing and holding costs push the delivery price up, while income and physical benefits push it down.

For the basic futures contract and exchange mechanics, read Futures: The Standardized Exchange Contract. The next instrument in the series is Swaps: Interest Rates, Currencies, and Credit Protection.