The price of a forward contract is set so that no trader can earn a riskless profit by buying the underlying asset today and holding it to the delivery date. In its simplest form, the forward price equals the spot price of the asset compounded at the risk-free rate over the life of the contract. That base formula is then adjusted for anything the holder of the physical asset either pays out (storage, insurance) or receives (dividends, coupons, foreign interest). The pricing of forward contracts across equities, commodities, and currencies is really one idea applied with different adjustments.
The Core No-Arbitrage Formula
A forward contract is a private agreement between two parties to buy or sell an asset at a fixed price on a future date. The pricing question follows naturally: what should that fixed future price be, given that you could just buy the asset today and hold it until the delivery date?
Start with the spot price S, the cost of acquiring the asset now. Anyone buying today must finance the purchase, and the relevant rate is the risk-free rate r, commonly approximated by the yield on short-term U.S. Treasury bills. Under continuous compounding, the forward price F for an asset that pays no income and has no storage cost is:
F = S × erT
T is the time to maturity in years. The formula says the forward price is simply the future value of today’s spot price. If you buy the asset now and finance it at r, you need at least F at delivery to break even.1University of Leicester. EC3070 Financial Derivatives – Present Values
How Arbitrage Enforces the Price
The formula is not just theoretical. Two trades keep the market honest.
If the forward price rises above S × erT, a trader borrows at the risk-free rate, buys the asset spot, and 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, and the pressure of many traders doing it pushes the forward price down and the spot price up until the gap closes.
If the forward price falls below S × erT, the reverse works. The 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. Both strategies require zero initial capital and yield guaranteed profit when the price is out of line, which is precisely why the deviation cannot last.2University of Texas at Austin. Lecture 9 An Introduction to Pricing Forward Contracts
Storage and Other Carrying Costs
Physical commodities cost money to hold. Crude oil needs tank storage. Grain needs insurance against spoilage. These expenses raise the break-even price for someone who buys today and holds to delivery, and the forward price rises with them.
When carrying costs run as a continuous rate c (a percentage of spot per year), they are added directly to the financing rate:
F = S × e(r + c)T
The combined exponent (r + c) 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 carrying costs are known fixed dollar amounts instead of a continuous rate, take their present value and add it to the spot price before compounding. If PV(C) is the present value of all storage and insurance payments over the life of the contract:
F = (S + PV(C)) × erT
The logic is identical. Whoever buys and stores the commodity bears real out-of-pocket costs, and the forward price must reimburse them.
Dividends and Income Yields
Financial assets often pay their holders. Stocks pay dividends, bonds pay coupons, and broad equity indices generate a near-continuous 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, common for 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 exceeds the yield, the forward sits above the spot. When the dividend yield exceeds the risk-free rate, the forward falls below the spot.3National Taiwan University. The No-Arbitrage Pricing of Forward Contracts
For individual stocks, where dividends come as specific cash payments on known ex-dividend dates, subtract the present value of expected dividends from the spot price before compounding:
F = (S − PV(D)) × erT
PV(D) is the sum of each expected dividend discounted back to today at the risk-free rate. This form is more precise for single stocks with a small number of known payments.4University of Texas at Austin. Lecture 10 An Introduction to Pricing Forward Contracts
Foreign Exchange Forwards
A foreign exchange forward locks in the rate at which you will exchange one currency for another on a future date. Its no-arbitrage price 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 mechanics: borrow domestic currency, convert to foreign at the spot rate, invest at the foreign risk-free rate, and simultaneously sell the foreign currency forward to lock in the conversion back. If the forward rate is set wrong, this round-trip prints money. Parity forces the forward rate to make the profit exactly zero.
Under continuous compounding:
F = S × e(rdomestic − rforeign)T
Structurally, this is the dividend-yield formula. The foreign interest rate plays the role of a continuous yield: holding foreign currency earns the foreign rate, which cuts the cost of carry.
The differential (rdomestic − rforeign) determines 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. When the foreign rate is higher, it trades at a forward discount. The premium or discount exactly offsets the interest rate advantage, so parking cash in the higher-yielding currency produces no arbitrage.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 sides. The delivery price K is set equal to the no-arbitrage forward price, so neither party has an advantage. As time passes and the spot price moves, one side gains and the other loses. Mid-life valuation 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 what you have committed to pay, discounted at the risk-free rate over the remaining time (T − t). If the spot has risen since inception, the long position is worth something positive. If it has fallen, negative.
For the short position, the value is the mirror image:
Vt = K × e−r(T − t) − St
Forward contracts are zero-sum at every point in time. The long’s gain is the short’s loss, and the two values sum to zero. At expiration, when t = T, the discounting drops out and the value collapses to ST − K, the terminal payoff.
For assets with dividends, storage costs, or convenience yields, the same principle applies. Adjust the spot leg for the present value of any remaining costs or benefits between t and T.
Contango, Backwardation, and Convenience Yield
The shape of the forward curve across delivery dates tells you about the market’s cost-of-carry dynamics.
Contango describes an upward-sloping curve: longer-dated forwards trade above shorter-dated ones. This is the natural state for most commodities when storage is available. Someone buying today and holding pays financing and storage, and the forward reflects those costs.
Backwardation is the opposite. Forward prices sit below the spot price and the curve slopes down. This happens when holding the physical commodity provides a benefit that a forward contract cannot replicate. That benefit is the convenience yield.
Convenience yield is the value of having 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. These operational advantages belong only to the holder of the physical commodity, never the holder of a forward.
Written as a continuous rate y, convenience yield enters the full formula as a negative carrying cost:
F = S × e(r + c − y)T
When inventories are plentiful, the marginal benefit of one more unit is small, y is near zero, and storage and financing drive the market into contango. When inventories tighten, y spikes, overwhelming the other carrying costs, and the market flips into backwardation.5ScienceDirect. Theory of Storage Implications in the European Natural Gas Market
This is also where the no-arbitrage framework hits a practical limit. In contango, reverse cash-and-carry works cleanly: short the asset, invest the proceeds, buy the forward. In backwardation driven by a high convenience yield, that trade is harder to replicate, because you cannot easily borrow a physical commodity’s operational benefits. The convenience yield opens a wedge that the usual arbitrage cannot fully close.
Transaction Costs and the No-Arbitrage Band
The formulas above assume a frictionless world with no bid-ask spreads, no brokerage fees, and no margin requirements. Real trading has all of these. They do not change the theoretical forward price, but they do change how tightly the market enforces it.
In a frictionless market, any deviation from the no-arbitrage price, however small, invites an arbitrage trade. With transaction costs, small deviations are not worth exploiting because the trading costs eat the profit. What you get is a band around the theoretical price rather than a single point. Inside that band, the forward price can float without triggering corrective trades.
The width of the band depends on the round-trip cost of executing the arbitrage. For liquid financial assets like major equity indices or G-10 currencies, the band is tight. For illiquid or physically settled commodities, the band can be substantially wider, reflecting higher 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 difference is not a trading opportunity unless it exceeds the total cost of the arbitrage strategy.
One further boundary is worth naming. The formulas here price the contract itself. They do not address counterparty credit risk on over-the-counter forwards, the collateral and netting mechanics governed by an ISDA Master Agreement, or the U.S. tax treatment of gains and losses under Internal Revenue Code Sections 1256 and 988. Those are separate questions with their own answers.