The price of a forward contract is the spot price of the underlying asset, compounded forward at the risk-free rate, then adjusted for any income the asset throws off or any cost of holding it until delivery. That single idea, known as cost-of-carry pricing, runs through every asset class. What changes from stocks to commodities to currencies is which adjustments you plug in.
A forward contract is a private agreement to buy or sell an asset at a fixed price on a future date. The pricing question is simple: what should that fixed price be, given that you could just buy the asset today and hold it? The answer has to be the number that leaves no room for a risk-free profit.
The Core Cost-of-Carry Formula
Start with an asset that pays no income and costs nothing to store. Let S be the spot price, r the risk-free rate (commonly approximated by the yield on short-term U.S. Treasury bills), and T the time to delivery in years. Under continuous compounding, the forward price F is:
F = S × erT
The forward price is just the future value of today’s spot price. If you buy the asset now and finance the purchase at the risk-free rate, you need to receive at least F at delivery to break even. Any higher or lower and someone can lock in a guaranteed profit.1University of Leicester. EC3070 Financial Derivatives – Present Values
How Arbitrage Forces the Price
The formula holds because two trading strategies punish any deviation.
If the forward price is too high (F > S × erT), a trader borrows cash at the risk-free rate, buys the asset at the spot price, and simultaneously sells a forward at the inflated price. At maturity, they deliver the asset, collect the forward price, repay the loan, and keep the difference. This is cash-and-carry arbitrage. Traders piling into that trade push the forward price down and the spot price up until the gap disappears.
If the forward price is too low, the reverse trade works. A trader short-sells the asset, invests the proceeds at the risk-free rate, and buys the cheap forward. At maturity, they take delivery, return the borrowed asset, and pocket the profit. Reverse cash-and-carry pushes the forward price back up.
Both trades require zero initial capital and produce guaranteed profits when the price is out of line, which is why the market self-corrects.2University of Texas at Austin. Lecture 9 An Introduction to Pricing Forward Contracts
Adjusting for Storage and Carrying Costs
Physical commodities cost money to hold. Crude oil needs tank storage. Grain needs insurance against spoilage. These are real out-of-pocket costs for anyone who buys the commodity today and holds it until delivery, so the forward price has to reimburse them.
When carrying costs are expressed as a continuous rate c, add them straight to the financing rate:
F = S × e(r + c)T
The combined exponent is the full cost of carry: financing plus storage, insurance, and deterioration risk.2University of Texas at Austin. Lecture 9 An Introduction to Pricing Forward Contracts
When the costs are known fixed dollar amounts instead of a continuous rate, add the present value of those payments to the spot price before compounding:
F = (S + PV(C)) × erT
The logic is the same either way.
Adjusting for Dividends and Income
Many financial assets pay their holders. Stocks pay dividends. Bonds pay coupons. Broad equity indices generate a steady stream of dividend income. That income reduces the effective cost of holding the asset because the buyer collects it while waiting for delivery.
When income is modeled as a continuous yield q (standard for broad equity indices), subtract it from the risk-free rate:
F = S × e(r − q)T
The quantity (r − q) is the net cost of carry. When financing costs exceed the income yield, the forward price sits above the spot. When the dividend yield exceeds the risk-free rate, the forward price sits below the spot.3National Taiwan University. The No-Arbitrage Pricing of Forward Contracts
For individual stocks with specific dividend payment dates, subtract the present value of expected dividends from the spot price before compounding:
F = (S − PV(D)) × erT
Here PV(D) is the sum of each expected dividend discounted back to today at the risk-free rate. This version is more precise when you know the ex-dividend dates.4University of Texas at Austin. Lecture 10 An Introduction to Pricing Forward Contracts
Pricing Foreign Exchange Forwards
An FX forward locks in the rate at which you will exchange one currency for another on a future date. The pricing follows from covered interest rate parity: the forward exchange rate must offset the interest rate gap between the two currencies, or traders could earn risk-free profits by borrowing in one and lending in the other.
The trade works like this. Borrow domestic currency, convert it to foreign currency at the spot rate, invest at the foreign risk-free rate, and sell the foreign currency forward to lock in the conversion back. If the forward rate is off, this round-trip produces a guaranteed profit. Covered interest rate parity forces the forward rate to make that profit exactly zero.
Under continuous compounding, the FX forward rate is:
F = S × e(rdomestic − rforeign)T
Structurally, this is identical to the dividend yield formula. The foreign interest rate plays the same role as a continuous dividend yield: holding the foreign currency earns you the foreign rate, which offsets financing costs.
The interest rate differential decides whether the foreign currency trades at a forward premium or discount. When the domestic rate is higher, the foreign currency trades at a forward premium (costs more forward than spot). When the foreign rate is higher, the foreign currency trades at a forward discount. Either way, the premium or discount exactly offsets the interest rate advantage.4University of Texas at Austin. Lecture 10 An Introduction to Pricing Forward Contracts
Valuing a Forward After Inception
At inception, a forward contract has zero value to both parties. The delivery price K is set equal to the no-arbitrage forward price, so neither side has an edge. As time passes and the spot moves, the contract gains value for one side and loses it for the other. Calculating that mid-life value matters for risk management, collateral, and financial reporting.
For a long position (the party who agreed to buy), the value at time t is:
Vt = St − K × e−r(T − t)
The first term is what the asset is worth today. The second is the present value of the delivery price, discounted at the risk-free rate over the remaining life (T − t). If the spot has risen since inception, the long position has positive value. If it has fallen, the value is negative.
For the short position, the value is the mirror image:
Vt = K × e−r(T − t) − St
Forwards are zero-sum at every point: the long’s gain is the short’s loss, and the two values always sum to zero. At expiration, when t = T, the discounting drops out and the value simplifies to ST − K, the terminal payoff.
For assets with dividends, storage costs, or convenience yields, the same principle applies, but the spot price is adjusted for the present value of any remaining costs or benefits between t and T.
Contango, Backwardation, and the Convenience Yield
The shape of the forward curve across delivery dates reveals the cost-of-carry dynamics in a market.
Contango is an upward-sloping curve: longer-dated forwards cost more than shorter-dated ones. This is the natural state for most commodities when storage is available. The forward price exceeds the spot because someone buying today and holding must pay financing and storage, and the forward reflects those accumulated costs.
Backwardation is the opposite. Forward prices sit below the current spot, and the curve slopes down. This happens when holding the physical commodity provides a benefit that a forward contract cannot deliver. That benefit is the convenience yield.
The convenience yield is the value of physical inventory on hand. A refinery with crude in its tanks can keep running through a supply disruption. A grain elevator with wheat in storage can fill orders immediately instead of waiting on delivery. These operational advantages belong to the holder of the physical commodity, not the holder of a forward.
Incorporating the convenience yield as a continuous rate y, the full formula becomes:
F = S × e(r + c − y)T
The convenience yield acts like a negative carrying cost. When inventories are abundant, the marginal value of one more barrel or bushel is near zero, the convenience yield is small, and the market sits in contango driven by storage and financing. When inventories tighten, the convenience yield rises, overwhelms the other carrying costs, and the market flips into backwardation.5ScienceDirect. Theory of Storage Implications in the European Natural Gas Market
This is where the clean no-arbitrage framework hits a practical limit. In contango, reverse cash-and-carry works cleanly: short the asset, invest the proceeds, buy a forward. In backwardation driven by a high convenience yield, that trade is hard to replicate because you cannot easily borrow the operational benefits of a physical commodity. The convenience yield opens a wedge that ordinary arbitrage strategies cannot fully close.
Why Real Prices Sit in a Band Around the Formula
The formulas above assume a frictionless world with no bid-ask spreads, brokerage fees, or margin requirements. In practice, every trade costs something. Those costs do not change the theoretical forward price, but they change how tightly the market enforces it.
In a frictionless market, any deviation from the no-arbitrage price triggers immediate arbitrage. With transaction costs, small deviations are not worth exploiting because trading costs eat the potential profit. The result is a band around the theoretical price. Inside that band, the forward can float without triggering corrective trades.
The band’s width depends on the round-trip cost of the arbitrage strategy. For liquid financial assets like major equity indices or G-10 currencies, the band is tight. For illiquid or physically settled commodities, the band widens considerably because of larger bid-ask spreads, transportation costs, and the difficulty of short-selling physical goods.
The practical takeaway: observed forward prices often differ slightly from the theoretical value, and that gap is not a trading opportunity unless it exceeds the total cost of executing the arbitrage.