Futures Pricing & Valuation: Cost-of-Carry, Contango & Backwardation

Futures pricing is built on one foundational idea: the price of a futures contract is determined by the cost of holding the underlying asset until delivery — not by forecasting where the spot price will be in the future. This cost-of-carry model explains why futures contracts trade at a premium or discount to the current spot price, why commodity markets behave differently from financial markets, and why futures and spot prices always converge at expiration. Whether you are pricing stock index futures, commodity contracts, or currency forwards, the same no-arbitrage framework applies. This guide builds on the contract mechanics covered in our Futures Contracts guide and the forward pricing foundation to explain how futures prices are determined, interpreted, and applied.

What Determines Futures Prices?

Futures prices are set by arbitrage, not by consensus forecasts. If the futures price deviates from its theoretical fair value, traders can exploit the mispricing through riskless strategies that push the price back toward equilibrium.

Two arbitrage strategies enforce this relationship:

• Cash-and-carry arbitrage — if the futures price is too high relative to fair value, an arbitrageur borrows money, buys the underlying asset in the spot market, and simultaneously sells the futures contract. At expiry, they deliver the asset and repay the loan, locking in a riskless profit.
• Reverse cash-and-carry arbitrage — if the futures price is too low, the arbitrageur short-sells the underlying asset, invests the proceeds at the risk-free rate, and buys the futures contract. At expiry, they take delivery and close the short position.

These competing forces keep the futures price tethered to the cost-of-carry fair value. All formulas in this article use continuously compounded rates and yields — a standard convention in derivatives pricing that ensures mathematical consistency.

Futures Pricing Formula: The Cost-of-Carry Model

The cost-of-carry model takes different forms depending on whether the underlying asset is a financial instrument or a physical commodity.

Financial Assets (Stocks, Indices, Currencies)

Where:

• F₀ — theoretical futures price today
• S₀ — current spot price of the underlying asset
• r — continuously compounded risk-free rate
• q — continuously compounded dividend yield (or foreign interest rate for currencies)
• T — time to maturity in years

When the risk-free rate exceeds the dividend yield (r > q), futures trade above spot. This is because holding the asset requires financing that costs more than the income it generates.

Physical Commodities

Where:

• u — continuously compounded storage cost rate (warehousing, insurance, spoilage)
• y — continuously compounded convenience yield — the non-monetary benefit of holding physical inventory (e.g., the ability to meet unexpected demand or avoid production shutdowns)

Convenience yield is what makes commodity futures fundamentally different from financial futures. A refinery holding crude oil inventory can keep production running during a supply disruption — this operational benefit has real economic value that reduces the effective cost of carry. The relationship between spot and forward rates in fixed income markets follows an analogous no-arbitrage logic.

Contango vs Backwardation

The relationship between futures prices and spot prices falls into two market states that describe the shape of the futures term structure — the curve of futures prices across different expiration dates.

Futures Fair Value Examples

Futures Price vs Expected Future Spot Price

A common question in futures pricing is whether the futures price is a forecast of where the spot price will be at expiration. Two complementary frameworks address this:

These views are not mutually exclusive. The cost-of-carry model explains the mechanical pricing — how arbitrage pins the futures price to the spot price plus carry costs. The expectations hypothesis addresses whether the futures price is a biased or unbiased estimator of the future spot price. For financial assets like stock indices, where arbitrage is efficient, the cost-of-carry view dominates. For commodities with significant storage frictions and delivery constraints, risk premia and expectations play a more meaningful role. The same cost-of-carry logic extends to currency markets through interest rate parity.

Why Futures Prices Converge to Spot at Expiry

Regardless of whether a market is in contango or backwardation, the futures price and the spot price must equal each other at the contract’s expiration. This is called convergence.

The basis — defined as Basis = S_t – F_t — measures the gap between spot and futures at any point in time. As expiration approaches, the basis shrinks toward zero because the time remaining (T) approaches zero, and the cost-of-carry component vanishes.

Arbitrage enforces convergence:

• If F > S at expiry — sell the futures contract and buy the underlying in the spot market for riskless profit. This selling pressure pushes the futures price down.
• If F < S at expiry — buy the futures contract and sell the underlying. This buying pressure pushes the futures price up.

How convergence works mechanically depends on the settlement type. Cash-settled contracts (e.g., E-mini S&P 500) converge automatically because the final settlement price is set equal to the underlying settlement index. Physically-delivered contracts (e.g., crude oil, wheat) converge through the delivery mechanism — the futures price converges to the spot price of the contract’s specified deliverable grade at the designated delivery location. If the futures price diverged from this deliverable spot, arbitrageurs would take or make delivery to capture the difference.

How to Calculate Futures Prices

Follow these steps to compute the theoretical fair value of a futures contract:

1. Identify the spot price (S₀) of the underlying asset from a reliable market data source
2. Determine the risk-free rate (r) — use a Treasury rate or OIS rate that matches the contract’s time to expiry, and convert to continuous compounding if needed: r_continuous = ln(1 + r_{effective annual}). If your quoted rate is a nominal APR with non-annual compounding, first convert to an effective annual rate before applying the log transformation.
3. Estimate any income yield (q) — for equity indices, use the annualized dividend yield; for bond futures, the income component reflects the coupon income of the deliverable bond net of financing cost
4. For commodities, estimate storage costs (u) and convenience yield (y) — these are often inferred from observed market prices rather than measured directly
5. Apply the appropriate cost-of-carry formula — financial assets: F₀ = S₀ × e^{(r – q)T}; commodities: F₀ = S₀ × e^{(r + u – y)T}
6. Compare to the market price — if the observed futures price differs significantly from your fair value, re-examine your input assumptions before concluding a mispricing exists

Common Mistakes

Futures pricing is conceptually elegant but several common errors can lead to incorrect valuations:

1. Confusing contango/backwardation with price direction — Contango means the futures price exceeds the spot price (F > S); it does not mean prices are rising. Backwardation means F < S; it does not mean prices are falling. These terms describe the price structure, not the trend. A market can be in steep contango while spot and futures prices are both declining.

2. Ignoring convenience yield for commodities — The commodity pricing formula requires storage costs and convenience yield. Applying the simpler financial-asset formula (F = S × e^{(r – q)T}) to crude oil, natural gas, or agricultural commodities produces incorrect results. Commodities with tight supply often exhibit high convenience yields that push futures into backwardation.

3. Assuming futures always converge smoothly — While F_T must equal S_T at expiration, the convergence path can be volatile. Illiquid contracts, delivery logistics for physical commodities, and short squeezes can all cause temporary basis widening before final convergence.

4. Treating the futures price as a spot price forecast — The futures price reflects the cost of carry and arbitrage forces — not a consensus view of where spot will trade in the future. Using futures prices as directional predictions ignores the mechanical pricing that drives them.

5. Mismatching compounding conventions — The continuous-compounding formulas (e^rT) require continuously compounded rates. Plugging in simple annualized rates or discretely compounded rates without converting produces pricing errors. Always verify that your rate inputs match the formula’s compounding convention.

Limitations of the Cost-of-Carry Model

1. Stochastic interest rates — The model assumes constant, known interest rates over the life of the contract. In practice, rates fluctuate, which creates a small pricing difference between futures (with daily settlement) and forwards (settled at maturity). This difference — called the convexity adjustment — is usually small but can matter for long-dated contracts or in volatile rate environments.

2. Uncertain storage costs — For physical commodities, storage costs (warehousing, insurance, spoilage) can vary significantly over the contract’s life. The model treats these as known and constant, which introduces estimation error.

3. Unobservable convenience yield — Convenience yield cannot be directly measured. It is typically inferred from market prices by backing it out of the cost-of-carry formula — making the reasoning somewhat circular for pricing purposes. Different market participants may estimate different convenience yields for the same commodity.

4. No-arbitrage bands, not exact prices — Transaction costs, bid-ask spreads, and borrowing constraints mean that arbitrage is not truly costless. Instead of enforcing a single fair value, these frictions create a band within which the futures price can trade without triggering arbitrage. The wider the frictions, the wider the band.

5. Short-selling constraints — The reverse cash-and-carry arbitrage requires short-selling the underlying asset. If short-selling is restricted, costly, or impossible (as with many physical commodities), the lower bound of the no-arbitrage band becomes less binding, and futures can trade persistently below the theoretical floor.