Forward and Futures Contracts

Introduction and context

Forward and futures contracts are agreements whose value depends on an underlying asset, reference rate, or index. They allow firms, investors, and financial institutions to transfer price risk, obtain economic exposure, or pursue arbitrage. Although both instruments involve a future transaction at a price agreed today, their contractual design differs materially. A forward is generally a privately negotiated over-the-counter (OTC) agreement, whereas a futures contract is standardized and traded on an organized exchange.

This distinction affects credit risk, liquidity, valuation, collateral, cash flows, and the effectiveness of hedging strategies. A graduate-level understanding therefore requires more than memorizing pricing formulas. It requires connecting no-arbitrage valuation with funding costs, income from the underlying asset, basis behavior, margining, delivery procedures, and the institutional environment in which contracts trade.

Learning objectives

By the end of this chapter, the reader should be able to:

1. Distinguish forwards from futures in terms of contract design, trading, settlement, and credit exposure.
2. Derive and apply no-arbitrage prices for forwards on equities, currencies, commodities, and interest rates.
3. Explain forward value, futures marking-to-market, basis, and convergence.
4. Analyze daily settlement, margin requirements, delivery options, and the influence of interest rates on futures prices.
5. Construct and evaluate hedges using the minimum-variance hedge ratio.
6. Identify practical, legal, regulatory, operational, and model-risk issues in derivative markets.

1. Contract architecture

A forward contract specifies that one party will buy and the other will sell an underlying asset at a future date T for a delivery price K. At initiation, the contract is normally entered at zero value, so K is selected to make the present value of expected contractual cash flows equal on both sides. At maturity, the payoff to the long position is:

where S_T is the spot price at maturity. The short position has payoff K − S_T. For a physically settled contract, delivery occurs in exchange for K; for a cash-settled contract, the difference is paid.

A futures contract has the same basic economic payoff but is standardized by an exchange. Contract size, maturity months, price quotation, tick size, delivery locations, eligible grades, and settlement procedures are specified in advance. Positions are marked to market daily, and gains or losses are transferred through the clearinghouse. The exchange clearinghouse becomes the buyer to every seller and seller to every buyer, substantially reducing bilateral counterparty exposure but creating the need for margin and default-management systems.

2. Forward pricing by no-arbitrage

The central principle is that two strategies with identical future cash flows should have the same price today. Let S₀ be the current spot price, r the continuously compounded risk-free interest rate, and T the time to maturity in years. If holding the asset generates known income, or incurs known costs, the forward price must reflect those cash flows.

2.1 Non-income-producing asset

Consider an asset that can be purchased today for S₀ and held until T. The cash-and-carry strategy is:

• borrow S₀ today;
• purchase one unit of the asset;
• deliver the asset into the forward at time T;
• repay the loan.

The amount owed at maturity is S₀ e^rT. Therefore the no-arbitrage forward price is:

This formula assumes continuous compounding, a known risk-free rate, no income or storage cost, no taxes or transaction costs, and the ability to borrow, lend, and trade the underlying without restriction. With annual compounding at rate i, the equivalent expression is F₀ = S₀(1+i)^T when T is measured in compounding periods.

2.2 Assets with known income

Suppose the asset provides known cash income with present value I. The investor needs to finance only the net cost S₀ − I. Thus:

For an equity paying a known discrete dividend, I is the present value of dividends received before delivery. If the asset has a continuously compounded dividend yield q, the formula becomes:

Here q is a proportional yield per year. The yield formulation is appropriate when income is modeled as proportional to the asset value, while the discrete-income formulation is appropriate when cash dividends are known in amount and timing.

2.3 Foreign currency

A foreign currency is an asset that provides a foreign risk-free return. Let S₀ be the domestic-currency price of one unit of foreign currency, r_d the domestic continuously compounded rate, and r_f the foreign rate. Holding foreign currency produces foreign-currency income, so:

This is covered interest parity under idealized assumptions. If the forward rate differed materially, borrowing in one currency, converting, investing in the other, and using a forward contract could produce an arbitrage, subject to funding access, credit limits, capital controls, and transaction costs.

2.4 Commodities

Commodities introduce storage costs and convenience yield. Let U be the present value of known storage costs and y the continuously compounded convenience yield. Then a useful theoretical expression is:

Storage costs increase the cost of carrying the physical commodity and therefore tend to increase the forward price. Convenience yield represents the non-monetary benefit of physically holding inventory—for example, production continuity or protection against shortages—and tends to reduce the forward price. Convenience yield is generally not directly observable and is often inferred from prices and inventory conditions.

2.5 Equity-index forwards

For an equity index with continuous dividend yield q, the theoretical forward price is F₀ = S₀e^(r−q)T. For a portfolio paying discrete dividends, use the present value of expected dividends. The formula is theoretical: taxes, index-rebalancing effects, stock-lending constraints, funding spreads, and dividend uncertainty can cause quoted prices to depart from the frictionless benchmark.

3. Forward value after initiation

The forward price is the delivery price that makes a newly initiated contract worth zero. The value of an existing long forward with delivery price K is different. For a non-income-producing asset under continuous compounding:

where S_t is the spot price at valuation time t, and r is the continuously compounded rate applicable from t to T. Equivalently, if F_t is the current forward price:

For an asset with known income present value I_t over the remaining life, the value becomes:

A short forward has the negative of the long value. These equations assume deterministic rates and known income. Stochastic interest rates require more careful valuation because the timing of gains and losses can interact with interest-rate movements.

Worked example 1: equity-index forward

An index is at 4,000. The continuously compounded annual risk-free rate is 5%, the continuous dividend yield is 2%, and maturity is six months. The theoretical forward price is:

F₀ = 4,000e^{(0.05−0.02)(0.5)} = 4,000e^0.015 ≈ 4,060.45

Thus, the no-arbitrage forward level is approximately 4,060.45 index points. If the market forward were materially above this level, a cash-and-carry strategy might be attractive; if materially below, a reverse cash-and-carry strategy might be considered. In practice, index replication, dividend uncertainty, financing spreads, and trading costs determine whether an apparent discrepancy is exploitable.

4. Futures pricing and daily settlement

At first approximation, a futures price equals the corresponding forward price when interest rates are deterministic and the contract is otherwise identical. With stochastic interest rates, however, daily settlement matters. A futures gain received today can be reinvested, while a loss must be funded today. If futures gains tend to occur when interest rates are high, reinvestment is valuable; if gains tend to occur when rates are low, the benefit is smaller. Consequently, futures and forward prices can differ.

The futures account is adjusted each trading day. If the futures price changes from F_t−1 to F_t, the daily gain for a long position with contract multiplier m is:

For a short position, the gain is G_t = m(F_t−1−F_t). The multiplier converts a quoted price movement into currency. Daily settlement means that the original futures price is not simply used to calculate one final payoff; interim gains and losses are transferred through the margin system.

Margining

Initial margin is collateral required to open a position. Maintenance margin is the minimum account balance that must be maintained. If the balance falls below maintenance margin, the trader receives a variation-margin call and must restore the account, commonly to the initial-margin level. Margin is not a down payment on the underlying; it is performance collateral.

Margining reduces credit exposure but does not eliminate risk. A participant may fail to meet a margin call, markets may move faster than collateral can be collected, and liquidation may occur during stressed conditions. Clearing members also face default-fund, liquidity, and concentration risks.

5. Basis and convergence

Basis is commonly defined as:

where S_t is the spot price and F_t is the futures price for the relevant maturity. Some markets use the reverse convention, so the definition must always be stated. As maturity approaches, the futures price and the appropriately deliverable spot value should converge. At expiration, absent market frictions:

Under the convention above, basis converges to zero. Basis risk is the risk that the relationship between spot and futures prices changes unexpectedly. It is especially important when the hedged asset is not identical to the futures deliverable, when the hedge date differs from contract maturity, or when local prices differ from the exchange reference price.

Contango describes a futures curve that generally slopes upward with maturity; backwardation describes a generally downward-sloping curve. These descriptions concern the curve, not necessarily whether an investor earns a profit. A futures investor’s total return can be decomposed conceptually into spot return, roll return, and collateral return. Rolling a long position in contango may impose a negative roll return, while backwardation may produce a positive roll return, although realized outcomes depend on price changes.

6. Hedging with futures

A short hedge is appropriate when a firm expects to sell an asset or has an economic exposure that loses value when prices fall. A long hedge is appropriate when a firm expects to buy an asset and fears price increases. The hedge should match the underlying asset, quantity, timing, and exposure as closely as possible.

For a one-to-one hedge, the number of contracts is approximately:

where Q_A is the quantity or exposure to hedge and Q_F is the quantity represented by one futures contract. If the hedge ratio is estimated statistically, the minimum-variance hedge ratio is:

and the number of contracts is:

Here ∆S and ∆F are changes in the spot and futures prices over the chosen hedge interval. The ratio assumes a linear hedge and is estimated from historical or model-based data. It minimizes the variance of hedged price changes under the selected sample and specification; it does not guarantee a profit or eliminate all risk.

If the spot exposure is measured in currency value and the futures contract has a currency value per one-point movement, units must be converted before calculating N. Regression estimates should also be examined for structural breaks, nonstationarity, seasonality, liquidity effects, and instability across market regimes.

Hedge effectiveness

A simple measure of variance-based hedge effectiveness is:

A value closer to 1 indicates a larger variance reduction in the sample. This measure is not the same as accounting effectiveness, economic profitability, or protection against extreme losses. Professional evaluation should include scenario analysis, stress testing, tail loss, liquidity requirements, and collateral costs.

7. Delivery and embedded options

Although many contracts are closed before maturity, delivery provisions can materially affect prices. A commodity futures contract may specify several acceptable grades, delivery locations, and delivery months. If the short can choose among assets or dates, the short holds a delivery option. The cheapest-to-deliver (CTD) instrument is the eligible asset that minimizes the short’s delivery cost after adjusting for conversion factors, accrued interest, financing, and invoice terms.

Interest-rate futures commonly incorporate delivery options. For a Treasury bond futures contract, the short may select an eligible bond and delivery date. The invoice amount is linked to the futures price and a conversion factor. Because the option has value, the futures price is not determined solely by a simple forward formula based on one particular bond.

Cash settlement avoids physical transfer but introduces a final reference-price methodology. The reference price must be robust against manipulation, transparent, and consistent with contract rules. Market participants should read the contract specifications rather than assume that “futures price” has a universal settlement meaning.

8. Worked example 2: currency forward

A firm must pay €10 million in one year. The spot exchange rate is 1.0800 U.S. dollars per euro, the U.S. continuously compounded rate is 4%, and the euro rate is 2.5%. The theoretical one-year forward rate is:

F₀ = 1.0800e^{(0.04−0.025)} = 1.0800e^0.015 ≈ 1.0963

The firm can lock in approximately $1.0963 per euro, producing a dollar payment of:

€10,000,000 × $1.0963/€ ≈ $10,963,000

This removes uncertainty about the exchange rate, but it also removes the benefit of a favorable currency movement. The firm must consider counterparty credit, collateral terms, accounting treatment, forecast accuracy, and whether the payment amount or date may change.

9. Applied case: an airline fuel hedge

Background

Illustrative case: Meridian Air expects to consume 20 million gallons of jet fuel over the next six months. Jet fuel prices are volatile, while the most liquid exchange-traded hedge is a crude-oil futures contract. One futures contract represents 1,000 barrels, and one barrel equals 42 gallons. The airline therefore faces a cross-hedge rather than a perfectly matched hedge.

Problem

The airline wants protection against rising fuel costs but cannot know the exact monthly consumption, and jet fuel does not move identically with crude oil. A full one-to-one hedge could over-hedge the exposure if travel demand falls or fuel efficiency improves.

Analysis and approach

Management estimates the historical regression of changes in jet-fuel prices on changes in crude-oil futures prices. Suppose the estimated minimum-variance hedge ratio is 0.80. The futures quantity represented by one contract is:

1,000 × 42 = 42,000 gallons.

The approximate number of contracts is:

N* = 0.80 × (20,000,000 ÷ 42,000) ≈ 381 contracts.

Because the airline will buy fuel, it takes a long futures position. If fuel prices rise, the cost of physical fuel increases, but gains on the futures position partially offset the increase. The hedge is rolled as contracts approach expiration, creating roll and basis risk.

Outcome

The hedge reduces variance in fuel expense but does not lock in an exact price. Residual risks include the jet-fuel/crude basis, timing mismatch, volume uncertainty, contract liquidity, margin calls, and changes in the estimated hedge ratio. Treasury should maintain liquidity sufficient to meet adverse daily settlements even when the economic hedge is working.

Learning points

The case illustrates that hedge design is an optimization problem rather than a mechanical matching exercise. A statistically efficient hedge may produce substantial cash-flow pressure, and a physically perfect hedge may be unavailable. Governance should therefore evaluate both economic exposure and funding capacity.

10. Risks, governance, and regulation

Forward and futures activity should be governed by clearly documented mandates. Essential controls include approved counterparties and exchanges, position limits, independent valuation, confirmation procedures, collateral monitoring, segregation of duties, model validation, and escalation rules for margin calls.

OTC forwards expose parties to counterparty default and replacement-cost risk. Netting agreements and collateral reduce—but do not eliminate—this exposure. Futures reduce bilateral counterparty risk through central clearing, but clearing members and customers remain exposed to liquidity, operational, legal, and systemic risks. Regulations in major jurisdictions commonly address clearing, reporting, margin, business conduct, and market-abuse controls, but precise requirements vary by product, entity, and jurisdiction.

Ethically, firms should not describe a hedge as risk-free when it merely changes the form of risk. Boards and senior managers should understand leverage, collateral calls, basis risk, delivery obligations, and model limitations. Valuation models should use observable market inputs where possible, document assumptions, and include independent price verification.

Technology supports real-time margin forecasting, automated confirmations, exposure aggregation, and anomaly detection. Machine-learning methods may improve forecasts of basis or volatility, but they introduce data-snooping, instability, explainability, and governance risks. AI should complement—not replace—economic reasoning, independent validation, and human accountability.

11. Best-practice framework

A disciplined derivatives process should:

1. Define the economic exposure, including quantity, currency, timing, and risk measure.
2. Select the instrument with the closest underlying, maturity, liquidity, and settlement characteristics.
3. Determine whether the objective is price certainty, variance reduction, cash-flow protection, or tactical exposure.
4. Estimate hedge ratios using economically justified data and test their stability.
5. Model daily margin and liquidity requirements under adverse scenarios.
6. Monitor basis, roll costs, delivery provisions, and counterparty exposure.
7. Rebalance only under documented rules; excessive trading can increase costs and operational risk.
8. Report both accounting results and economic effectiveness.
9. Review the hedge after changes in forecast volume, market structure, regulation, or funding conditions.

Chapter Summary

Forwards and futures transfer price risk through commitments to transact in the future. Forward prices follow no-arbitrage relationships based on spot price, financing, income, storage costs, convenience yield, and— for currencies—interest-rate differentials. The value of an existing forward differs from its initial delivery price and must be discounted consistently.

Futures are standardized, exchange-traded, centrally cleared, and marked to market daily. Margining changes cash-flow timing and reduces bilateral credit exposure. Futures and forwards are approximately equivalent under deterministic rates, but stochastic rates can create a pricing difference. Basis measures the spot–futures relationship and should converge at maturity, although basis risk remains central to real-world hedging. Delivery options, cheapest-to-deliver assets, roll costs, collateral liquidity, and cross-hedging all complicate simple textbook pricing.

Effective risk management combines quantitative hedge-ratio estimation with governance, stress testing, liquidity planning, model validation, and clear communication of residual risks.

Key Terms

Forward contract: OTC agreement to transact at a future date at a specified price.

Futures contract: Standardized exchange-traded contract settled through a clearinghouse.

Long position: Position that benefits from an increase in the underlying price.

Short position: Position that benefits from a decrease in the underlying price.

Spot price: Current market price for immediate delivery.

Forward price: No-arbitrage delivery price for a new forward contract.

Delivery price: Price specified in an existing forward or futures contract.

Marking to market: Daily settlement of futures gains and losses.

Initial margin: Collateral required to open a futures position.

Maintenance margin: Minimum collateral balance before a margin call.

Variation margin: Payment made to restore collateral after adverse price movement.

Basis: Difference between spot and futures prices under a stated convention.

Convergence: Movement of futures and deliverable spot prices together at maturity.

Contango: Upward-sloping futures price curve.

Backwardation: Downward-sloping futures price curve.

Convenience yield: Non-cash benefit of holding physical inventory.

Cash-and-carry arbitrage: Strategy involving purchase of the asset and short sale of a forward or futures contract.

Cross-hedge: Hedge using a related but non-identical futures instrument.

Minimum-variance hedge ratio: Hedge ratio that minimizes variance of hedged price changes.

Cheapest to deliver: Eligible delivery asset most economical for the short to deliver.

Review Questions

Short Answer Questions

1. What is the principal institutional difference between a forward and a futures contract?
2. Why does known income reduce the theoretical forward price?
3. Define basis and explain convergence.
4. What is the purpose of maintenance margin?
5. Why can futures and forward prices differ when interest rates are stochastic?

Long Answer Questions

1. Derive the cash-and-carry pricing formula for a non-income-producing asset and discuss its assumptions.
2. Compare the credit, liquidity, settlement, and customization characteristics of forwards and futures.
3. Explain how storage costs and convenience yield affect commodity forward prices.
4. Discuss the role of delivery options and cheapest-to-deliver assets in interest-rate futures.
5. Explain the minimum-variance hedge ratio and evaluate its limitations in professional risk management.

Analytical / Application Questions

1. A stock index is at 3,500, the continuously compounded risk-free rate is 6%, the dividend yield is 1.5%, and maturity is nine months. Calculate the theoretical forward price and explain which assumptions may cause the market quote to differ.
2. A manufacturer will buy 5 million pounds of copper in four months and uses futures contracts covering 25,000 pounds each. Explain whether a long or short hedge is appropriate, calculate the unadjusted number of contracts, and identify at least three sources of residual risk.
3. A firm experiences a margin call on a futures hedge even though the hedge is expected to reduce its ultimate economic loss. Explain why this can occur and design a liquidity-control procedure to manage it.

Multiple Choice Questions with Answers

Question 1

Under continuous compounding, what is the theoretical forward price for a non-income-producing asset?

A. S₀e^rT

B. S₀e^−rT

C. (S₀−I)e^rT

D. S₀e^(r−q)T

Correct Answer: A. S₀e^rT

Question 2

Which feature most directly distinguishes futures from forwards?

A. Futures always have a higher payoff

B. Futures are standardized and marked to market

C. Forwards cannot be used for hedging

D. Forwards are always cash settled

Correct Answer: B. Futures are standardized and marked to market

Question 3

A currency has a domestic interest rate of 5% and a foreign interest rate of 3%. With spot equal to 1.20 domestic currency units per foreign unit and one-year maturity, the theoretical forward rate is closest to:

A. 1.1762

B. 1.2000

C. 1.2242

D. 1.2600

Correct Answer: C. 1.2242

Question 4

With basis defined as spot price minus futures price, what normally happens at maturity in a frictionless market?

A. Basis becomes permanently positive

B. Basis becomes permanently negative

C. Basis converges toward zero

D. Basis becomes equal to the interest rate

Correct Answer: C. Basis converges toward zero

Question 5

A firm expects to purchase an asset in the future and fears a price increase. Which futures strategy is generally appropriate?

A. Short hedge

B. Long hedge

C. Sell the asset immediately

D. Borrow the asset and sell it

Correct Answer: B. Long hedge

Question 6

If one futures contract represents 40,000 units and the exposure is 2,000,000 units, the unadjusted one-to-one hedge requires:

A. 20 contracts

B. 40 contracts

C. 50 contracts

D. 80 contracts

Correct Answer: C. 50 contracts

Question 7

What does maintenance margin represent?

A. The contract’s delivery price

B. The minimum collateral balance before a margin call

C. The exchange’s transaction fee

D. The present value of storage costs

Correct Answer: B. The minimum collateral balance before a margin call

Question 8

The minimum-variance hedge ratio is based on:

A. The ratio of spot price to futures price only

B. Covariance of spot and futures changes divided by futures-change variance

C. The risk-free rate divided by the dividend yield

D. The futures multiplier divided by contract maturity

Correct Answer: B. Covariance of spot and futures changes divided by futures-change variance

Question 9

For a commodity, convenience yield generally:

A. Increases the cost of carrying inventory

B. Reduces the theoretical forward price relative to a model without convenience yield

C. Eliminates basis risk

D. Equals the storage cost in all markets

Correct Answer: B. Reduces the theoretical forward price relative to a model without convenience yield

Question 10

A futures position can require cash funding before the hedge’s final economic outcome because:

A. Futures have no margin system

B. Daily marking to market transfers interim gains and losses

C. The spot asset is always delivered immediately

D. Forward prices are fixed by regulators

Correct Answer: B. Daily marking to market transfers interim gains and losses