Forward and Futures: Cost-of-Carry Pricing and Hedging

Derivatives markets dwarf the underlying cash markets they reference. Forward and futures contracts represent the simplest and most fundamental derivative instruments, serving as the building blocks for understanding more complex structures like options and swaps. These contracts allow market participants to lock in future prices today, transferring price risk from those who wish to avoid it to those willing to bear it.

A forward contract is a private agreement between two parties to buy or sell an asset at a specified future date for a price agreed upon today. No money changes hands at contract initiation. The agreed price, called the forward price, is set such that the contract has zero initial value to both parties. At expiration, the buyer (long position) pays the forward price and receives the asset, while the seller (short position) delivers the asset and receives payment.

Futures contracts serve the same economic purpose but trade on organized exchanges with standardized terms. The exchange interposes itself between buyer and seller, eliminating counterparty credit risk through daily settlement of gains and losses. This daily marking-to-market is the key distinction from forwards and introduces subtle pricing differences we'll explore later.

Understanding how to price these contracts requires no sophisticated mathematics, only the principle that markets should not offer free money. If you can construct two portfolios that produce identical future cash flows, they must have the same price today. This no-arbitrage principle, combined with the cost-of-carry model, gives us precise formulas for forward prices.

Before diving into pricing theory, we need to understand exactly what happens when you enter a forward contract and what payoffs result at expiration. The mechanics of these contracts form the foundation upon which all pricing and hedging analysis rests. By carefully examining how money flows at different points in time, we develop the intuition necessary for understanding why forward prices take the values they do.

When you enter a forward contract to buy an asset at price $F$ at time $T$, you are said to be long the forward. At expiration, you pay $F$ and receive an asset worth $S_T$ (the spot price at maturity). The payoff calculation is straightforward: you compare what you paid to what you received. Since you paid $F$ and received something worth $S_T$, your net gain or loss is simply the difference:

where:

• $S_T$: spot price of the asset at expiration
• $F$: agreed-upon forward price

This formula captures the essence of your position. If the asset price rises above the forward price, you profit because you locked in a purchase price lower than the prevailing market value. If it falls below, you lose because you committed to buying at a price higher than where the market ultimately settled. The payoff is linear in the spot price, meaning every dollar increase in $S_T$ translates directly into a dollar of profit, and every dollar decrease translates directly into a dollar of loss.

The short position (the party who agreed to sell) has the opposite payoff, which follows logically from the zero-sum nature of forward contracts:

where:

• $F$: agreed-upon forward price
• $S_T$: spot price of the asset at expiration

The short position profits when prices fall below the forward price because they locked in a sale price above market value. They lose when prices rise because they must deliver an asset worth more than the payment they receive. Whatever the long position gains, the short position loses, and vice versa.

Notice the linear payoff profiles evident in the chart. Unlike options, which provide asymmetric protection, forwards have symmetric profit and loss potential. The long benefits dollar-for-dollar from price increases and loses dollar-for-dollar from price decreases. There is no cap on potential gains and no floor on potential losses. This symmetry reflects the unconditional nature of the forward commitment: both parties are obligated to complete the transaction regardless of where the market moves.

A critical feature of forward contracts is that no money changes hands at initiation. The forward price $F$ is determined precisely so that the contract has zero value to both parties at the outset. This distinguishes forwards from options, where the buyer pays a premium upfront for the right (but not obligation) to transact.

The zero-initial-value property might seem arbitrary at first glance, but it follows directly from how markets establish forward prices through the forces of supply and demand. If a forward contract had positive value to the long at initiation, everyone would want to be long, and no one would willingly take the short side. This imbalance would drive the forward price up as longs compete for contracts, continuing until the advantage disappeared. Similarly, if the contract had positive value to the short, everyone would want to sell forwards, driving the price down. The forward price adjusts until neither party has an advantage, reaching an equilibrium where both sides view the contract as a fair exchange of risk.

This equilibrium condition provides the starting point for our pricing analysis. We don't need to model supply and demand explicitly; instead, we can use arbitrage arguments to determine what the forward price must be to ensure the contract has zero initial value. This approach leads us to the cost-of-carry model.

The forward price is determined by a no-arbitrage argument based on the cost of carrying the underlying asset until the delivery date. The core insight is deceptively simple: there are two ways to obtain an asset at a future date. You can either buy it forward, committing today to purchase at a predetermined price when you need it, or you can buy it today in the spot market and store it until you need it. Since both approaches deliver the same outcome, namely possession of the asset at the future date, they must cost the same in present value terms. If they didn't, you could profit risklessly by buying the cheaper approach and selling the more expensive one.

Consider an asset with current spot price $S_0$ and a forward contract expiring at time $T$ (measured in years). If the continuously compounded risk-free interest rate is $r$, the forward price must satisfy:

where:

• $F_0$: forward price
• $S_0$: current spot price
• $r$: continuously compounded risk-free interest rate
• $T$: time to expiration in years

This elegant formula states that the forward price equals the spot price grown at the risk-free rate for the time until delivery. The intuition is straightforward: if you choose to buy the asset today rather than forward, you must finance that purchase. The cost of that financing, accumulated over the contract period, is exactly what makes the two approaches equivalent.

To understand why this formula holds, and why it must hold in a well-functioning market, consider what happens if the forward price deviates from this value. Any deviation creates an arbitrage opportunity, which market participants will exploit until prices realign. Let us examine both cases.

Suppose the forward price is too high: $F_0 > S_0 e^{rT}$. You can lock in a riskless profit through the following cash-and-carry strategy. The name reflects the mechanics: you use cash to buy the asset and carry it until delivery.

1. Borrow $S_0$ at the risk-free rate $r$
2. Buy the asset in the spot market for $S_0$
3. Enter a short forward contract to sell at $F_0$ at time $T$

At expiration, we unwind all positions:

• Deliver the asset under the forward contract, receiving $F_0$
• Repay the loan: $S_0 e^{rT}$
• Profit: $F_0 - S_0 e^{rT} > 0$

This profit is guaranteed regardless of where the spot price ends up at expiration. You hold the physical asset, so there is no market risk; delivery is simply a matter of handing over what you already own. The profit requires zero initial investment because the borrowed funds exactly cover the spot purchase. This is the definition of arbitrage: riskless profit with no capital at risk.

Such opportunities cannot persist in competitive markets. Arbitrageurs would flood the market with this trade, selling forwards (pushing forward prices down) and buying spot (pushing spot prices up), until the price discrepancy disappears. The market quickly converges to $F_0 = S_0 e^{rT}$.

Conversely, if $F_0 < S_0 e^{rT}$, the reverse cash-and-carry strategy yields arbitrage profits. The logic is symmetric but the positions are reversed.

1. Short-sell the asset, receiving $S_0$
2. Invest $S_0$ at the risk-free rate $r$
3. Enter a long forward contract to buy at $F_0$ at time $T$

At expiration:

• Receive $S_0 e^{rT}$ from the investment
• Pay $F_0$ under the forward, receive the asset
• Return the asset to close the short position
• Profit: $S_0 e^{rT} - F_0 > 0$

Again, this is riskless profit with no initial investment. The forward contract guarantees you can acquire the asset at $F_0$ to close the short, eliminating any market risk. Arbitrageurs executing this strategy would buy forwards (pushing forward prices up) and sell spot (pushing spot prices down), eliminating the discrepancy.

The forward price is squeezed between these two arbitrage bounds, leaving only $F_0 = S_0 e^{rT}$ as the equilibrium. Any other price invites profitable trading that moves the market toward this value.

Many assets generate income while you hold them. Stocks pay dividends, bonds pay coupons, and foreign currencies earn interest in their home country. This income reduces the cost of carry because the holder benefits from owning the asset. As a forward buyer, you do not receive the asset until expiration. Because you forgo this income, you pay a lower forward price to compensate.

Discrete dividend case:

If the underlying pays a known cash dividend $D$ at time $t_D < T$, the forward price becomes:

where:

• $F_0$: forward price
• $S_0$: current spot price
• $D$: cash dividend amount
• $t_D$: time until the dividend payment (in years)
• $r$: risk-free interest rate
• $T$: time to expiration
• $e^{-r t_D}$: discount factor

The term $D e^{-r t_D}$ is the present value of the dividend. You subtract it from the spot price because you don't receive this dividend; only the current holder does. Think of it this way: if you buy the asset today, you get both the asset at expiration and the dividend along the way. If you buy forward, you only get the asset at expiration. The forward price must be lower to account for this missing benefit, and the reduction equals exactly the present value of what you forgo.

Continuous dividend yield:

For assets like equity indices where dividends are paid frequently by many constituents, it's convenient to model dividends as a continuous yield $q$. Rather than tracking hundreds of discrete payments, we approximate the aggregate dividend stream as a constant percentage of the index value paid continuously over time:

where:

• $F_0$: forward price
• $S_0$: current spot price
• $r$: risk-free interest rate
• $q$: continuous dividend yield (annualized)
• $T$: time to expiration

The dividend yield $q$ reduces the forward price because the spot holder earns income that you forgo. The formula has an elegant interpretation: the "net" cost of carry is $r - q$, representing financing costs minus income received. When dividends are high relative to interest rates, this net cost can be small or even negative. The following function implements the general cost-of-carry model:

The forward trades at a premium to spot because the interest cost of carrying the position ($r \times T = 2.5\%$) exceeds the dividend income ($q \times T = 0.75\%$). This net cost of carry is reflected in the forward price. In market terminology, we say the forward is trading at a premium or that the curve is in contango.

Physical commodities introduce additional complexity beyond what we encounter with financial assets. Unlike stocks or currencies, commodities incur real storage costs: warehousing, insurance, and potential spoilage. A barrel of oil doesn't sit for free; it requires tanks, security, and protection from the elements. Gold must be vaulted and insured against theft. Agricultural commodities may deteriorate over time or require climate-controlled storage. These costs increase the forward price because you must cover them:

where:

• $F_0$: forward price
• $S_0$: current spot price
• $r$: risk-free interest rate
• $u$: continuous storage cost rate (annualized)
• $T$: time to expiration

However, commodities also exhibit a phenomenon rarely seen in financial assets: convenience yield. Holding physical inventory can be valuable during supply disruptions or production bottlenecks. A manufacturer might pay a premium for immediate access to raw materials rather than waiting for a forward delivery. A refinery that runs out of crude oil must shut down production, incurring massive costs. The option to avoid such disruptions has real economic value.

Including convenience yield $y$, the pricing formula becomes:

where:

• $F_0$: forward price
• $S_0$: current spot price
• $r$: risk-free interest rate
• $u$: continuous storage cost rate
• $y$: continuous convenience yield
• $T$: time to expiration

When convenience yield is high, typically during supply disruptions or low inventory periods, forward prices can fall below spot prices. This condition is called backwardation, reflecting that the market "backs" into lower future prices because immediate possession is so valuable. When forward prices exceed spot prices (the normal case for non-yielding assets), the market is in contango. Understanding these term structure shapes is essential for commodity traders and hedgers, as they affect rolling strategies and storage economics.

The arbitrage arguments above reveal something powerful: we can replicate a forward contract using spot positions and borrowing or lending. This synthetic forward has identical payoffs to the actual forward contract. The ability to construct synthetic positions is not merely an academic curiosity; it has profound practical implications. It enables arbitrage when prices diverge, provides alternative execution when one market is more liquid than another, and forms the conceptual basis for pricing more complex derivatives.

To create a synthetic long forward on a non-dividend-paying asset, we combine a spot purchase with borrowing to fund it:

1. Buy the asset at $S_0$
2. Borrow $S_0$ at rate $r$ to fund the purchase

At time $T$, we evaluate our position:

• You own the asset worth $S_T$
• You owe $S_0 e^{rT}$
• Net value: $S_T - S_0 e^{rT}$

This exactly matches the payoff of a long forward with forward price $F_0 = S_0 e^{rT}$:

where:

• $S_T$: spot price at expiration
• $F_0$: forward price
• $S_0$: current spot price
• $r$: risk-free interest rate
• $T$: time to expiration

The synthetic position produces the same outcome in every possible state of the world. Whether prices rise dramatically, fall sharply, or stay flat, the synthetic and actual forward deliver identical results. This equivalence is what allows us to price forwards: since two portfolios with identical payoffs must have the same value (otherwise arbitrage profits exist), the forward price must be set to make the actual forward's value equal to the synthetic's value, which is zero at initiation.

Similarly, to create a synthetic short forward, we reverse the positions:

1. Short-sell the asset, receiving $S_0$
2. Invest $S_0$ at rate $r$

At time $T$:

• Investment grows to $S_0 e^{rT}$
• You owe the asset worth $S_T$
• Net value: $S_0 e^{rT} - S_T$

This matches the payoff of a short forward with price $F_0 = S_0 e^{rT}$. The synthetic short forward profits when the asset falls in value and loses when it rises, exactly mirroring the actual short forward position.

Synthetic forwards are useful when:

• The forward market is illiquid: You can create exposure using the more liquid spot market. Some assets trade actively in spot markets but have thin forward markets, making synthetics the preferred route.
• Forward prices deviate from fair value: Arbitrage brings prices back in line. By constructing synthetics, arbitrageurs profit from mispricings while simultaneously correcting them.
• Funding considerations differ: Your borrowing rate may differ from the rate embedded in market forward prices. Different participants face different funding costs, and synthetics allow each to use their own rates.

The ability to replicate forwards also underpins much of modern derivatives pricing. If we can express a derivative in terms of forwards and bonds, we can price it using the same no-arbitrage logic. Options, swaps, and exotic structures all build on this foundation. Understanding synthetic replication is therefore essential for anyone working in derivatives.

While forwards are over-the-counter contracts between two parties, futures are standardized contracts traded on organized exchanges. The CME Group, ICE, and Eurex are major futures exchanges trading contracts on equity indices, interest rates, currencies, and commodities. The move from bilateral to exchange-traded contracts fundamentally changes the risk profile and operational mechanics, though the economic exposure remains similar.

Several features distinguish futures from forwards, each addressing specific limitations of the bilateral forward market:

• Standardization: Contract sizes, delivery dates, and specifications are fixed by the exchange. This standardization creates fungibility: all March S&P 500 futures are identical, enabling liquid trading.
• Central clearing: The exchange clearinghouse becomes counterparty to all trades, eliminating bilateral credit risk. Buyers and sellers need not evaluate each other's creditworthiness; they only need confidence in the exchange.
• Daily settlement (marking-to-market): Gains and losses are settled daily through margin accounts. This prevents losses from accumulating to dangerous levels and ensures credit exposure remains manageable.
• Margin requirements: Traders must post collateral (initial margin) and maintain minimum balances (maintenance margin). This collateral backs the clearinghouse's guarantee.

The daily settlement process works as follows. At the end of each trading day, the exchange calculates a settlement price for each contract based on trading activity. Your margin account is credited or debited the day's price change multiplied by the contract size. If your account falls below the maintenance margin, you receive a margin call requiring immediate deposit of additional funds.

This process transforms a single future payment into a series of daily payments. Rather than waiting until expiration to realize your gains or losses, you experience them incrementally each day. We can simulate this process in Python to track the daily cash flows and cumulative P&L:

The daily settlement has an important implication: your P&L is realized incrementally each day, not as a lump sum at expiration. This means cash flows occur throughout the contract's life, affecting the time value of money. Daily gains can be withdrawn and invested; daily losses must be funded immediately. This timing difference creates subtle distinctions between futures and forward prices.

For constant interest rates, futures and forward prices are identical. The daily settlement merely changes when you receive money, not how much you ultimately receive. With constant rates, the present value of receiving cash daily equals the present value of receiving it all at expiration.

However, when interest rates are stochastic and correlated with the underlying asset, differences emerge. The daily settlement creates a subtle advantage or disadvantage depending on the correlation structure.

Consider a long futures position on an asset positively correlated with interest rates. When the asset price rises, rates tend to rise too. The daily gains are credited to your margin account and can be reinvested at higher rates. Conversely, when prices fall, rates fall, and you fund your losses at lower rates. This asymmetry benefits the futures holder: gains arrive when reinvestment rates are high, and losses occur when borrowing rates are low.

The opposite applies when asset prices are negatively correlated with rates. In this case, gains arrive when reinvestment rates are low, and losses occur when funding costs are high, disadvantaging the futures holder. The relationship is captured by the convexity adjustment:

where:

• $F^{\text{futures}}$: futures price
• $F^{\text{forward}}$: forward price
• $\rho$: correlation between the asset price and interest rates
• $\sigma_S$: volatility of the asset price
• $\sigma_r$: volatility of interest rates
• $T$: time to maturity

For most equity and commodity futures with short maturities (under one year), this adjustment is negligible, often just a few basis points. For interest rate futures like Eurodollar contracts, where the underlying is by definition correlated with rates, the adjustment matters and must be accounted for in pricing and hedging.

The primary economic function of futures markets is risk transfer. Producers, consumers, and investors use futures to hedge unwanted price exposure, converting uncertain future prices into known amounts. Speculators on the other side of these trades earn risk premiums for bearing the uncertainty that hedgers wish to avoid. This exchange benefits both parties: hedgers achieve certainty needed for planning and investment, while speculators are compensated for providing insurance.

A perfect hedge eliminates all price risk. If you own an asset and sell a futures contract on the same asset with the same delivery date, your position is fully hedged. The mathematics are straightforward:

where:

• $S_T$: spot price at expiration
• $F_0$: initial futures price

No matter what happens to the spot price, your total value equals the initial futures price. The spot gain or loss is exactly offset by the futures loss or gain. If the spot price rises by $10, you gain $10 on your asset but lose $10 on your short futures. If the spot price falls by $10, you lose $10 on your asset but gain $10 on your short futures. The net effect is always zero variation around the locked-in forward price.

In practice, perfect hedges are rare. The futures contract available may not match your exposure exactly, leading to basis risk. Understanding this risk is crucial for effective hedging.

Basis risk arises from three sources, each introducing uncertainty into what should be a certain hedge:

• Asset mismatch: The asset you're hedging differs from the futures underlying (e.g., hedging jet fuel with crude oil futures). Different assets may have different supply and demand dynamics.
• Maturity mismatch: Your exposure date doesn't match the futures expiration. You may need to roll positions or accept settlement at a different time than your underlying exposure.
• Grade or location mismatch: For commodities, the deliverable grade or location differs from your actual commodity. West Texas Intermediate oil delivered at Cushing, Oklahoma differs from Brent crude in Rotterdam.

When basis exists and changes unpredictably, the hedged position's profit becomes:

where:

• $h$: hedge ratio (number of futures per unit of spot exposure)
• $S_T, S_0$: spot price at expiration and initiation
• $F_T, F_0$: futures price at expiration and initiation

If the basis changes unexpectedly, hedging effectiveness suffers. You have transformed outright price risk into basis risk, which is typically smaller but not zero.

When the asset and futures don't move in perfect lockstep, we need to determine the optimal number of futures contracts to minimize risk. The minimum variance hedge ratio balances the tradeoff between unhedged exposure and hedge imperfection. Too few contracts leave residual exposure to the underlying asset; too many contracts introduce exposure to futures movements that don't correspond to the spot position.

Let $\Delta S$ be the change in spot value and $\Delta F$ be the change in futures price. The hedged portfolio change is:

where:

• $\Delta P$: change in the value of the hedged portfolio
• $\Delta S$: change in spot price
• $\Delta F$: change in futures price
• $h$: hedge ratio

Our objective is to choose $h$ to minimize the variance of $\Delta P$, thereby minimizing the uncertainty in our hedged position. To find the optimal hedge ratio, we first express the variance of the portfolio change using standard variance properties:

This variance expression is quadratic in $h$, opening upward (since the coefficient on $h^2$ is positive). It therefore has a unique minimum that we can find using calculus. To minimize variance, we take the derivative with respect to $h$ and set it to zero:

Solving for $h$:

where:

• $h^*$: minimum variance hedge ratio
• $\rho$: correlation between spot and futures price changes
• $\sigma_S$: spot price volatility
• $\sigma_F$: futures price volatility

This elegant result has a natural interpretation. The optimal hedge ratio is simply the regression coefficient from regressing spot changes on futures changes. It tells us how much the spot position moves, on average, for each unit move in the futures. This result is known as the optimal hedge ratio. We can calculate this ratio given the correlation and volatilities:

A hedge ratio greater than 1 means we need more futures notional than spot exposure because the futures is less volatile than the spot asset. Each dollar of spot movement requires more than a dollar of futures position to offset it. A hedge ratio less than 1 means the opposite: the futures is more volatile than the spot, so less notional is needed. Understanding why the ratio takes a particular value helps in assessing whether a hedging strategy makes economic sense.

To calculate the actual number of futures contracts needed, we must translate the hedge ratio into contract quantities:

where:

• $N$: number of contracts
• $h^*$: optimal hedge ratio

For equity index futures:

where:

• $N$: number of contracts
• $h^*$: optimal hedge ratio (or beta)
• $V$: portfolio value
• $F$: futures price
• $m$: contract multiplier

Let's work through a complete example of hedging equity portfolio risk with index futures. This practical application brings together the theoretical concepts we've developed and demonstrates how professional portfolio managers implement hedging strategies.

You hold $10 million in a diversified US equity portfolio. The portfolio has a beta of 1.15 relative to the S&P 500. With market uncertainty rising, you want to reduce market exposure over the next three months using E-mini S&P 500 futures.

The E-mini S&P 500 futures contract has a multiplier of $50 per index point. Current futures price is 5,000. Your goal is to determine how many contracts to sell to neutralize your portfolio's market sensitivity.

For equity portfolios, beta serves a similar role to the minimum variance hedge ratio. If the portfolio has beta $\beta$, a 1% market move translates to approximately a $\beta\%$ portfolio move. The hedge ratio is therefore $\beta$. This makes intuitive sense: a portfolio that moves more than the market (high beta) requires more futures contracts to hedge than a portfolio that moves less than the market (low beta). We can calculate the required number of contracts as follows:

You should short 46 E-mini S&P 500 futures contracts to hedge your portfolio's market exposure. The slight difference between the exact calculation and the rounded number creates a small residual exposure, but this is unavoidable since fractional contracts cannot be traded.

We can verify the hedge by simulating portfolio values across a range of market returns. This analysis demonstrates how the hedge performs in different market scenarios, from sharp declines to strong rallies:

The hedged portfolio remains relatively stable regardless of market moves. The slight upward slope in the hedged line reflects the portfolio's alpha: stock-specific returns that aren't eliminated by the index hedge. This is actually desirable. You retain alpha while eliminating beta (market) risk. In essence, the hedge isolates your stock-picking skill from the market's overall direction.

Sometimes full hedging isn't the goal. You might want to reduce beta from 1.15 to 0.5 rather than eliminating it entirely. Perhaps market conditions warrant caution but not complete defensive positioning. The number of contracts becomes:

where:

• $N$: number of contracts
• $\beta_{\text{current}}$: current portfolio beta
• $\beta_{\text{target}}$: desired target beta
• $V$: portfolio value
• $F$: futures price
• $m$: contract multiplier

This formula generalizes our full hedge calculation. When the target beta is zero, we recover the full hedge formula. When the target beta equals current beta, zero contracts are needed. The formula can also produce negative contract counts when you want to increase beta, indicating that long futures positions are required rather than short positions.

To go from beta 1.15 to a negative beta of -0.5, you would short significantly more contracts, effectively reversing market exposure. A negative beta means the portfolio profits when the market falls and loses when it rises. This might seem extreme, but some macro hedge funds deliberately take negative beta positions when they believe markets will decline.

Commodity futures serve producers and consumers who face price uncertainty in their core business operations. Unlike financial speculators, these commercial hedgers use futures not for profit but for risk management. Their goal is to reduce uncertainty and enable better business planning. As a farmer, you need to know what price to expect for next year's harvest to decide how much to plant. As an airline, you need fuel cost predictability to price tickets months in advance.

Suppose you expect to sell 100,000 barrels of crude oil in three months. Current spot price is $75/barrel, and the three-month futures trades at $76.50/barrel. You might worry that prices might fall before the sale, wiping out profit margins or even causing losses.

By shorting 100 crude oil futures contracts (each covering 1,000 barrels), you lock in approximately $76.50 per barrel. Whatever happens to spot prices over the next three months, your realized revenue per barrel will be close to the initial futures price. The code below demonstrates the revenue outcomes across various price scenarios:

Regardless of where spot prices end up, you receive approximately $7.65 million. If prices crash to $65, the futures gain compensates for the lower spot sale. If prices surge to $90, the futures loss offsets the spot windfall. You trade price uncertainty for price certainty. This tradeoff is valuable for business planning: you can budget, invest, and make operational decisions knowing revenues with confidence.

Suppose you need to purchase jet fuel in three months. Rather than face fuel price uncertainty, you can go long heating oil futures (a close substitute for jet fuel) to lock in an effective purchase price. Fuel is one of your largest operating costs, and price volatility creates significant earnings uncertainty.

The logic mirrors the producer hedge but with opposite positions. Going long futures profits when prices rise, offsetting higher spot purchase costs. If prices fall, the futures losses are offset by lower spot purchases. Either way, the effective cost is stabilized.

The airline example highlights cross-hedging: using futures on a related but different commodity. Jet fuel futures exist but may have lower liquidity than heating oil or crude oil futures. The tradeoff is between basis risk (jet fuel vs. heating oil price divergence) and liquidity risk (wider bid-ask spreads, harder to enter and exit positions).

Cross-hedging requires careful analysis of the relationship between the hedged asset and the futures contract. The correlation determines how much protection the hedge provides. If jet fuel and heating oil prices move in lockstep, the cross-hedge works well. If they sometimes diverge due to refinery issues, seasonal demand differences, or specification changes, residual risk remains. This simulation shows how imperfect correlation affects the effectiveness of the hedge:

The cross-hedge reduces cost variance substantially but doesn't eliminate it. The 85% correlation leaves residual basis risk from jet fuel-specific factors that heating oil doesn't capture. This residual risk is the price paid for using a more liquid hedging instrument. You must weigh the benefits of easier trading against the costs of imperfect protection.

The theoretical framework presented here assumes frictionless markets: perfect liquidity, no transaction costs, unrestricted borrowing and short-selling at the risk-free rate, and continuous trading. Real markets deviate from these assumptions in important ways. Understanding these limitations helps practitioners apply the theory appropriately and avoid pitfalls.

Transaction costs and market impact erode arbitrage profits. The cash-and-carry arbitrage requires buying spot, trading futures, and potentially financing or short-selling. Each leg incurs bid-ask spreads, commissions, and market impact. For the arbitrage to be profitable, the price discrepancy must exceed total costs. This creates no-arbitrage bands rather than a single price, and observed forward prices can deviate from theoretical values within these bands.

Short-selling constraints affect the reverse cash-and-carry arbitrage. For equities, short sellers must locate shares to borrow, pay lending fees, and post collateral. Some stocks are hard-to-borrow or unavailable for shorting entirely. When short-selling is costly or impossible, forward prices can trade above theoretical values without triggering corrective arbitrage.

Funding costs vary across participants. The arbitrage arguments assume everyone borrows and lends at the risk-free rate. In practice, hedge funds face different funding costs than pension funds, which differ from bank proprietary desks. This heterogeneity means the "fair" forward price differs across participants, and observed prices reflect marginal trading by the participant with the best funding access.

Margin and collateral requirements tie up capital. Futures positions require initial margin (typically 5-15% of notional value) plus variation margin as positions move. This capital has opportunity cost and may be constrained during market stress when margins increase just as trading opportunities arise. The daily settlement also means cash flow timing differs from forwards, which can matter for liquidity-constrained participants.

Model assumptions break down at extremes. The cost-of-carry model assumes known dividends, constant interest rates, and deterministic storage costs. In practice, dividends can be cut unexpectedly, rates move stochastically, and commodity storage costs depend on inventory levels and market conditions. During market dislocations, basis relationships can deviate dramatically from historical norms, causing hedges to fail precisely when they're needed most.

Despite these limitations, forward and futures markets generally price efficiently. Arbitrage may not be perfect or instantaneous, but it keeps prices close to theoretical values most of the time. The framework provides the foundation for more sophisticated analysis that incorporates transaction costs, funding constraints, and model uncertainty.

This chapter established the theoretical and practical foundations for pricing and hedging with forward and futures contracts. The key insights center on no-arbitrage pricing and risk transfer.

The cost-of-carry model provides the fundamental pricing relationship. The forward price equals the spot price compounded at the cost of carry: $F_0 = S_0 e^{(r - q + u - y)T}$, where $r$ is the financing rate, $q$ is income yield, $u$ is storage cost, and $y$ is convenience yield. Any deviation from this formula creates arbitrage opportunities through cash-and-carry or reverse cash-and-carry trades.

Forward contracts can be synthetically replicated. A long forward is equivalent to buying the asset and borrowing to fund the purchase; a short forward is equivalent to short-selling and investing the proceeds. This replication principle underlies all derivative pricing and enables arbitrage when market prices deviate from theoretical values.

Futures differ from forwards through daily settlement. Exchange-traded futures eliminate counterparty risk through central clearing and daily marking-to-market. For most practical purposes, futures and forward prices are identical, but stochastic interest rates introduce a convexity adjustment when asset prices and rates are correlated.

Hedging requires matching exposure characteristics. Perfect hedges use contracts identical to the underlying exposure. When mismatches exist, the minimum variance hedge ratio $h^* = \rho \sigma_S / \sigma_F$ minimizes residual risk. Basis risk from asset mismatch, timing mismatch, or specification differences reduces hedging effectiveness.

For equity portfolios, beta determines the hedge ratio. The number of index futures contracts to hedge or target a specific beta is $N = (\beta_{\text{current}} - \beta_{\text{target}}) \times V / (F \times m)$. This allows portfolio managers to adjust market exposure without trading the underlying stocks.

Commodity hedging transfers price risk from producers and consumers to speculators. Producers short futures to lock in sale prices; consumers go long to lock in purchase prices. Cross-hedging using related but not identical commodities is common when direct hedges are unavailable or illiquid.

Ready to test your understanding? Take this quick quiz to reinforce what you've learned about forward and futures pricing and hedging.