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A futures contract is a standardized, exchange-traded agreement to buy or sell an underlying asset at a predetermined price on a specified future date. The fair value of a futures contract is determined by the cost-of-carry model, which prices the futures as the current spot price compounded at the risk-free rate, adjusted for any income generated by holding the asset (dividends, coupons, convenience yield) and costs incurred (storage, insurance, transportation). The fundamental pricing formula is: F = S × e^(r-q)T for continuously compounded rates, or F = S × (1+r-q)^T for discrete compounding, where S is the spot price, r is the risk-free rate, q is the continuous dividend yield or net convenience yield, and T is the time to expiration in years. For physically delivered commodities like oil and gold, the formula incorporates storage costs (u) and convenience yield (c): F = S × e^(r+u-c)T. The convenience yield is the implicit benefit of holding physical inventory — refiners value having crude oil on hand to avoid costly shutdowns — and explains why commodity futures markets can exhibit backwardation (futures below spot) even in a risk-free world. If futures prices deviate from fair value, cash-and-carry arbitrage or reverse cash-and-carry arbitrage restores equilibrium. CME Group and ICE publish daily settlement prices for futures contracts across equity indexes, interest rates, foreign exchange, energy, metals, and agricultural commodities. Understanding fair value is essential for basis traders who exploit the gap between futures and fair value, for risk managers computing hedge ratios, and for investors assessing roll costs when maintaining long-term commodity exposure through rolling futures positions.
F = S × e^(r-q)T for continuously compounded rates, or F = S × ( Where each variable represents a specific measurable quantity in the finance and lending domain. Substitute known values and solve for the unknown. For multi-step calculations, evaluate inner expressions first, then combine results using the standard order of operations.
- 1Identify the spot price (S) of the underlying asset and the time to expiration (T) in years.
- 2Determine the continuously compounded risk-free rate (r) from Treasury bill or OIS rates of matching maturity.
- 3For equities/indexes, determine the continuous dividend yield (q); for commodities, estimate storage costs (u) and convenience yield (c).
- 4Apply the cost-of-carry formula: F = S × e^(r + u − c − q)T.
- 5Compare the calculated fair value with the actual market futures price to identify the basis: Basis = F_market − F_fair.
- 6If F_market > F_fair, cash-and-carry arbitrage is possible (buy spot, sell futures).
- 7If F_market < F_fair, reverse cash-and-carry arbitrage is possible (sell spot, buy futures) — limited to those who can sell short.
Fair value premium = 42 points; ES futures trading near 5242-5248 is normal
The S&P 500 futures (ES) fair value at 5242 reflects the net carry: the risk-free rate of 5.30% must be paid to finance the spot position, partially offset by the 1.35% dividend yield collected. The net cost of carry is 3.95% annualized, translated to approximately 42 index points for the 75-day period. If ES trades at 5248, only 6 points above fair value, this is within normal bid-ask spreads and does not represent a meaningful arbitrage opportunity.
Oil market in moderate contango; futures above spot by $1.88
With financing cost (5.3%) and storage (6%) exceeding the convenience yield (2%), the 3-month WTI futures fair value of $81.88 exceeds the spot price of $80.00. This $1.88 contango reflects the cost of carrying physical oil inventory. Investors who maintain oil exposure by rolling long futures positions pay approximately $1.88 per barrel per 3-month roll — a significant drag in steep contango environments.
Gold is nearly pure financing cost; always in contango except in extreme stress
Gold pays no income and has very low storage costs, so gold futures are priced almost entirely by the cost of financing the spot position. At a 5.3% risk-free rate plus 0.25% storage over 6 months, the fair value is $2,261.3. Gold futures are virtually always in contango, with the slope of the forward curve directly reflecting the prevailing interest rate level. When rates were near zero (2010-2022), gold contango was tiny; rising rates steepened it significantly.
20-point richness exceeds typical transaction cost of ~5 pts; arbitrage potentially viable
The FTSE 100 December futures fair value is 7,875, but the contract trades at 7,895 — 20 points above fair value. An arbitrageur could buy the basket of FTSE 100 shares at 7,850, sell December futures at 7,895, collect dividends of approximately 19 points (3.8% × 75/365 × 7850), and finance the spot purchase at 5.1%. After costs, the strategy locks in approximately 15 points of risk-free profit, motivating institutional desks to execute the trade and close the mispricing.
Mortgage lenders and loan officers use Futures Pricing Calc to structure repayment schedules, compare fixed versus adjustable rate options, and calculate total borrowing costs for residential and commercial real estate transactions across different term lengths.
Personal finance advisors apply Futures Pricing Calc when counseling clients on debt reduction strategies, comparing the mathematical benefit of accelerated payments against alternative investment returns to determine the optimal allocation of surplus cash flow.
Credit unions and community banks rely on Futures Pricing Calc to generate accurate Truth in Lending disclosures, ensure regulatory compliance with TILA and RESPA requirements, and provide borrowers with standardized cost comparisons across competing loan products.
Corporate treasury departments use Futures Pricing Calc to model the cost of revolving credit facilities, term loans, and commercial paper programs, optimizing the company's capital structure and minimizing weighted average cost of debt financing.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in futures fair value calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in futures fair value calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in futures fair value calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
| Contract | Ticker | Exchange | Contract Size | Settlement Type | Typical Margin % |
|---|---|---|---|---|---|
| S&P 500 E-mini | ES | CME | $50 × Index | Cash | 5-8% |
| WTI Crude Oil | CL | NYMEX | 1,000 bbls | Physical | 4-8% |
| Gold 100oz | GC | COMEX | 100 troy oz | Physical | 3-5% |
| Corn | ZC | CBOT | 5,000 bushels | Physical | 3-6% |
| 10Y Treasury Note | ZN | CBOT | $100,000 face | Physical (bond) | 1-3% |
| EUR/USD | 6E | CME | EUR 125,000 | Physical (FX) | 2-4% |
| Henry Hub Nat Gas | NG | NYMEX | 10,000 MMBtu | Physical | 5-12% |
What is the cost-of-carry model?
The cost-of-carry model prices futures contracts based on the idea that the futures price must equal the spot price plus all costs of holding the physical asset (financing, storage, insurance) minus any benefits (dividends, convenience yield) over the life of the contract. If futures were priced differently, riskless arbitrage profits would be available by simultaneously trading futures and the underlying spot. Competition among arbitrageurs enforces this pricing relationship in liquid markets with low transaction costs.
What is basis and why does it matter?
The basis is the difference between the spot price and the futures price (or sometimes futures minus spot, depending on convention). For financial futures, the basis converges to zero at expiration — cash and futures prices must be equal at delivery. The basis also reflects market sentiment: in commodity markets, a widening basis (futures rising relative to spot) indicates improving forward demand, while narrowing basis signals near-term physical tightness. Basis risk — the uncertainty about the basis at the time a hedge is closed — is a key remaining risk for hedgers using futures.
What is convergence and why is it guaranteed?
As a futures contract approaches its expiration date, the futures price must converge to the spot price because at expiration, the futures buyer takes delivery (or cash settles) at the futures price and must receive goods worth the spot price. If futures were above spot at expiration, everyone would sell futures and buy spot for delivery, forcing prices together. This guaranteed convergence makes futures an effective hedging instrument: hedgers know their hedge will function perfectly at expiration even if the basis fluctuates during the hedge period.
What is roll return and how does it affect long-term commodity investment?
Roll return is the gain or loss experienced when a near-month futures contract is sold and a further-dated contract is bought to maintain continuous exposure (rolling). In a contango market, the far contract is more expensive than the near contract, generating a negative roll yield as investors buy high and sell low repeatedly. In backwardation, the opposite occurs — investors sell the expensive near contract and buy the cheaper far contract, earning a positive roll yield. The roll return significantly affects long-term commodity investment returns: S&P GSCI commodity index loses approximately 3-5% annually to negative roll in normal contango markets.
How does the convenience yield affect commodity futures pricing?
The convenience yield represents the implicit benefit of holding physical inventory — the ability to maintain production continuity, respond to demand spikes, and avoid costly shutdowns. A high convenience yield occurs when physical supplies are scarce relative to demand, as holders of inventory command a premium for providing liquidity to the market. High convenience yields drive commodity futures into backwardation (futures below spot). For oil, the convenience yield fluctuates with inventory levels and geopolitical supply risks.
What is the difference between financial futures and commodity futures settlement?
Most financial futures (equity index, interest rate) settle in cash — the gain or loss versus the initial futures price is paid in cash at expiration, with no physical delivery of the underlying. Commodity futures can settle either by physical delivery (the holder of a long position at expiration receives actual barrels of oil, bushels of wheat, or ounces of gold) or by cash settlement (for some contracts like VIX, natural gas, or certain agricultural contracts). Physical delivery creates logistical obligations that most financial investors avoid by rolling or offsetting positions before expiration.
What is daily mark-to-market and variation margin?
Unlike forward contracts, exchange-traded futures are marked to market daily — gains and losses are settled in cash every business day. If your futures position loses value, you receive a margin call requiring you to deposit additional variation margin by the next morning. If it gains, excess margin is returned. This daily settlement eliminates accumulated credit risk between counterparties — neither party can accumulate large unrealized losses that they might be unable to pay. The exchange's clearinghouse stands between all buyers and sellers, guaranteeing contract performance.
プロのヒント
For equity index futures, calculate fair value each morning before the market opens using overnight interest rate changes and estimated dividends going ex-date during the contract period. Programs trading desks buy or sell futures versus the basket when the market price deviates from fair value by more than the bid-ask spread plus transaction costs.
ご存知でしたか？
The WTI crude oil futures contract traded on NYMEX (now CME) was introduced in 1983 and within 10 years became the world's most actively traded commodity futures contract. A single standard contract covers 1,000 barrels of oil — enough to fill about 42,000 gallons or run a US household's heating for roughly 3 years.