Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Forwards and Futures - Instrument Basics › Hedging with Futures

Michael BrenndoerferNovember 10, 202511 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 14Forwards and Futures - Instrument Basics › Hedging with Futures

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The primary purpose of futures markets is risk transfer. We use futures to reduce or eliminate price risk associated with our business activities.

A Hedging ExampleLink Copied

Consider an airline that will need to purchase 1 million gallons of jet fuel in three months. The airline faces the risk that fuel prices will rise, increasing its operating costs. To hedge this exposure, the airline can go long heating oil futures (a common proxy for jet fuel). The hedge works by creating a financial position whose gains offset losses on physical exposure, and vice versa. The calculation below demonstrates the number of contracts required and simulates the effective fuel cost under rising and falling price scenarios:

In[13]:

Code

# Airline jet fuel hedge
fuel_needed = 1000000  # gallons
heating_oil_contract_size = 42000  # gallons per contract
current_jet_fuel_price = 2.50  # $/gallon
current_heating_oil_futures = 2.45  # $/gallon

# Calculate number of contracts needed
contracts_needed = fuel_needed / heating_oil_contract_size
contracts_to_trade = round(contracts_needed)

# Simulate two scenarios
jet_fuel_in_3_months_high = 3.00  # prices rise
jet_fuel_in_3_months_low = 2.20  # prices fall
heating_oil_in_3_months_high = 2.95
heating_oil_in_3_months_low = 2.15


# Scenario analysis
def analyze_hedge(
    fuel_price_future, ho_price_future, ho_price_now, contracts, contract_size
):
    fuel_cost = fuel_needed * fuel_price_future
    futures_pnl = (ho_price_future - ho_price_now) * contracts * contract_size
    net_cost = fuel_cost - futures_pnl
    effective_price = net_cost / fuel_needed
    return fuel_cost, futures_pnl, net_cost, effective_price


scenario_high = analyze_hedge(
    jet_fuel_in_3_months_high,
    heating_oil_in_3_months_high,
    current_heating_oil_futures,
    contracts_to_trade,
    heating_oil_contract_size,
)
scenario_low = analyze_hedge(
    jet_fuel_in_3_months_low,
    heating_oil_in_3_months_low,
    current_heating_oil_futures,
    contracts_to_trade,
    heating_oil_contract_size,
)

Out[14]:

Console

Airline Jet Fuel Hedging Analysis
=======================================================
Fuel requirement: 1,000,000 gallons
Heating oil contract size: 42,000 gallons
Contracts needed: 23.81 → Trading 24 contracts
Current heating oil futures: $2.45/gallon

Scenario 1: Prices Rise
  Jet fuel spot price:    $3.00/gallon
  Fuel cost (unhedged):   $3,000,000
  Futures gain:           $504,000
  Net fuel cost:          $2,496,000
  Effective price:        $2.50/gallon

Scenario 2: Prices Fall
  Jet fuel spot price:    $2.20/gallon
  Fuel cost (unhedged):   $2,200,000
  Futures loss:           $-302,400
  Net fuel cost:          $2,502,400
  Effective price:        $2.50/gallon

The hedge stabilizes the airline's effective fuel cost around $2.50 per gallon regardless of whether prices rise or fall. When prices rise, the futures gain offsets the higher physical fuel cost. When prices fall, the futures loss offsets the savings on physical fuel. The mechanism is symmetrical: the hedge removes both downside risk and upside opportunity, locking in a predictable cost that enables better financial planning.

The small differences from the target price arise from basis risk: the imperfect correlation between jet fuel and heating oil prices. Because no jet fuel futures contract exists with sufficient liquidity, the airline must use a proxy contract that doesn't perfectly track its actual exposure. This hedging imperfection is unavoidable when the hedged asset differs from the futures underlying.

Out[16]:

Visualization

Line chart showing total fuel cost for hedged vs unhedged positions. — Comparison of total fuel costs for unhedged and hedged positions. The unhedged cost (red) increases linearly with prices, while the hedged cost (blue) remains stable near the target expenditure level.

Line chart showing effective price per gallon for hedged vs unhedged positions. — Effective price per gallon across varying spot scenarios. The hedge locks in the price near $2.50 per gallon regardless of price movements, significantly reducing cost volatility.

Key ParametersLink Copied

The key parameters for the hedging model capture the essential elements of hedge construction:

Position to hedge: The total volume of fuel needed (1,000,000 gallons). This defines the physical exposure that the financial hedge must offset. Accurately measuring the exposure is the first step in any hedging program.
Contract size: The volume of one heating oil contract (42,000 gallons). The contract size determines how many contracts are needed to match the exposure. In this case, the ratio yields approximately 24 contracts.
Contracts needed: The calculated number of contracts to offset the risk. Because partial contracts cannot be traded, rounding introduces a small amount of residual exposure that remains unhedged.
Effective price: The final cost per unit after accounting for both physical purchase and futures P&L. This measure captures the true economic outcome of the hedged position and should be relatively stable across price scenarios if the hedge is effective.

Types of HedgesLink Copied

Long hedge (buy futures): Used when you will need to purchase the underlying asset in the future. Examples include manufacturers hedging raw material costs, importers hedging currency exposure, and investors hedging future equity purchases.
Short hedge (sell futures): Used when you will need to sell the underlying asset or have an existing long position. Examples include farmers hedging crop sales, oil producers hedging production, and portfolio managers hedging equity exposure.

Hedge Ratio and Basis RiskLink Copied

A perfect hedge requires matching the futures position size to the underlying exposure. The hedge ratio calculates the number of contracts needed to match the dollar volatility of the position. The fundamental challenge in hedge construction is determining the right number of contracts: too few leaves residual exposure to price changes, while too many creates excess exposure in the opposite direction.

The hedge ratio formula scales the raw number of contracts (total position value divided by contract size) by the asset's sensitivity to futures price changes:

h = \frac{\text{Position to hedge}}{\text{Contract size}} \times \beta

where:

$h$ : hedge ratio (number of contracts)
$\text{Position to hedge}$ : total value of the asset exposure
$\text{Contract size}$ : notional value of one futures contract
$\beta$ : sensitivity of the hedged asset to the futures price

The first term in this formula, the position to hedge divided by contract size, gives the number of contracts that would be needed if the hedged asset moved dollar-for-dollar with the futures contract. This "naive" hedge ratio assumes perfect correlation and identical volatility between the hedged asset and the futures contract.

However, real-world hedging situations rarely involve perfect correlation. The factor $\beta$ adjusts for the sensitivity of the hedged asset to the futures price. When the hedged asset is more volatile than the futures contract, beta exceeds 1.0 and more contracts are needed. When the hedged asset is less sensitive to price changes than the futures, beta is less than 1.0 and fewer contracts suffice.

For index futures hedging a stock portfolio, the beta coefficient can be estimated using standard regression techniques:

\beta = \frac{\text{Cov}(R_{\text{portfolio}}, R_{\text{index}})}{\text{Var}(R_{\text{index}})}

where:

$\beta$ : sensitivity coefficient (beta)
$\text{Cov}(\cdot)$ : covariance between returns, measuring how the portfolio and index move together
$\text{Var}(\cdot)$ : variance of returns, measuring the variability of the index
$R_{\text{portfolio}}$ : return on the portfolio being hedged
$R_{\text{index}}$ : return on the futures index

The formula defines beta as the ratio of covariance to variance. It normalizes the co-movement of the portfolio and index by the index's variance to determine the optimal hedge ratio. Intuitively, the covariance captures how much the portfolio and index tend to move together, while dividing by the index variance scales this co-movement relative to how much the index itself moves. A beta of 1.0 implies the portfolio moves in lockstep with the index, while a beta of 0.5 implies it is half as sensitive. A portfolio with beta of 1.3 would require 30% more futures contracts than the naive ratio would suggest.

Basis risk arises because the price of the asset being hedged doesn't move perfectly with the futures price. Even with the optimal hedge ratio, residual risk remains because the relationship between the hedged asset and the futures contract varies over time. The basis is defined as the difference between the spot price and the futures price:

B = S - F

where:

$B$ : the basis
$S$ : spot price of the asset
$F$ : futures price of the asset

The basis reflects the cost of carry (such as interest and storage) and convenience yield. In an efficient market, the basis converges to zero at expiration as arbitrage forces the futures price to equal the spot price at delivery. Before expiration, however, the basis fluctuates based on changes in carrying costs, supply and demand imbalances, and market expectations.

Changes in the basis create hedging imperfections. Even if we perfectly match the quantity of the physical position with futures contracts, basis movements introduce uncertainty about the final hedge outcome. Basis risk is minimized by choosing futures contracts that closely match the hedged asset and by using nearby expiration dates, which tend to track spot prices more closely than distant contracts.

Out[18]:

Visualization

Line chart showing futures price starting above spot price and converging to meet it. — Convergence of spot and futures prices toward maturity. In a contango market, the futures price (red dashed) trades at a premium but narrows until it reaches equality with the spot price at expiration.

Line chart showing basis starting at -2.5 and rising to 0. — The basis narrowing toward zero over 90 days. The basis (spot minus futures) starts at a negative value and converges to zero at expiration as the cost-of-carry premium dissipates.