Core Path · L2.6.02

Hedging with forwards and futures

12 minHedgingDerivatives

Know these first

By the end you will be able to

  1. 1Size a hedge with the minimum-variance hedge ratio and work out the number of futures contracts.
  2. 2Explain basis and basis risk, and calculate the price a futures hedge locks in.
  3. 3Decide whether a hedger needs a long or a short hedge.
  4. 4Measure how much risk a hedge removes, and explain why correlation sets the limit.

Why it matters

Fuel is one of an airline's largest costs, and there are no futures on jet fuel that trade as actively as those on heating oil. So airlines hedge one product with another. How many contracts should they buy? Not one for one, it turns out, and the answer depends on a correlation.

Core explanation

Lock in a price, then mind the gap

A hedge is a position that gains when your exposure loses. It is the “transfer” response of L2.6.01: Four responses to risk: the price risk passes to whoever takes the other side.

Long hedge: buy futures

You will buy something later and fear the price rising. An airline buying fuel, a chocolate maker buying cocoa.

Short hedge: sell futures

You own or will sell something and fear the price falling. A farmer before harvest, an oil producer.

Long hedge or short hedge?

4 cards

  1. A wheat farmer in Kansas will harvest in September.

  2. An airline in Singapore buys jet fuel every month.

  3. A fund manager holds a portfolio of German shares and fears a fall next month.

  4. A pension fund will receive cash in three months and wants to invest it in shares at today's level.

Forwards: a tailored lock. A forward (L1.7.02: Forwards) can be written for exactly the asset, amount and date you need. Then the hedge is perfect: whatever the price does, you pay or receive the forward price. You give up any gain from a good move, and you take on the risk that the bank on the other side fails to pay.

Futures: standard, so rarely a perfect fit. Futures (L1.7.03: Futures and margining) trade on an exchange in standard sizes, for a few delivery months, on a few assets. They are cheap to trade and cleared, but your exposure seldom matches them. The asset may differ (a cross hedge), the date may differ, or both. And daily margin turns paper losses into cash calls.

Basis. The gap that a futures hedge leaves is measured by the basis: the spot price of what you are hedging minus the futures price, b = S − F. For the same asset, at the futures' delivery date, the basis shrinks to zero (convergence). Before that date, or for a different asset, nobody knows what it will be. If you buy futures at F₁ and close them when the spot price is S₂ and the futures price is F₂, you pay S₂ for the asset and gain F₂ − F₁ on the futures. Your effective price is

S2−(F2−F1)=F1+(S2−F2)⏟b2S_2 - (F_2 - F_1) = F_1 + \underbrace{(S_2 - F_2)}_{b_2}

You have swapped price risk for basis risk, which is usually much smaller. Try it:

ExploreWhat the hedge locks in

The futures were bought at F₁ = $2.40. Move the fuel price a long way: the effective price barely changes. Move the basis: it changes cent for cent.

Futures price F₂$2.58
Pay for fuel$2.65
Futures gain$+0.18
Effective price$2.47

= F₁ + b₂ = 2.40 + 0.07

How many contracts? If spot and futures moved exactly together, one unit of futures per unit of exposure would do. They don't. Two things matter, both from M2.5: First risk measures: the volatility σ of each price change (L2.5.01: Measuring volatility) and their correlation ρ (L2.5.03: Correlation and diversification). If the futures move more than the spot, you need fewer of them. If they move together only loosely, piling on more futures adds risk of its own. The best balance is the minimum-variance hedge ratio, h*.

The formal version

Write ΔS\Delta S and ΔF\Delta F for the changes in the spot and futures prices over the life of the hedge, σS\sigma_S and σF\sigma_F for their standard deviations and ρ\rho for their correlation (see the Math Atom on covariance and correlation). The hedge ratio hh is the number of futures units per unit of exposure.

How much risk is left? Put h∗=ρσS/σFh^* = \rho\sigma_S/\sigma_F back into the variance:

σV2=σS2+ρ2σS2−2ρ2σS2\sigma_V^2 = \sigma_S^2 + \rho^2\sigma_S^2 - 2\rho^2\sigma_S^2
σV2=σS2 (1−ρ2)σV=σS1−ρ2\sigma_V^2 = \sigma_S^2\,(1 - \rho^2) \qquad \sigma_V = \sigma_S\sqrt{1-\rho^2}

The hedge removes a share ρ2\rho^2 of the variance: its hedge effectiveness. With ρ = 0.9 it removes 81% of the variance, but the standard deviation only falls to 44% of what it was. High correlation is not optional.

A familiar slope. ρ σS/σF\rho\,\sigma_S/\sigma_F is also the slope of the best-fit line through a scatter of ΔS\Delta S against ΔF\Delta F: the same idea as an equity beta in L2.5.04: Sensitivities.

The formulas, for reference

h∗=ρ σSσFN∗=h∗ QAQFσV=σS1−ρ2h^* = \rho\,\frac{\sigma_S}{\sigma_F} \qquad N^* = h^*\,\frac{Q_A}{Q_F} \qquad \sigma_V = \sigma_S\sqrt{1-\rho^2}
b=S−Feffective price=F1+b2b = S - F \qquad \text{effective price} = F_1 + b_2

QA is the size of the exposure and QF the size of one futures contract, in the same units. Use price changes over the same horizon as the hedge.

Try it

Start at the naive hedge, h = 1, then press “Set h = h*”. Next, drag the correlation down to 0.5 and watch how little any hedge can do. Finally, make the futures twice as volatile as the spot: h* halves.

Try itHedge ratio explorer

Futures and forwards model →
Monthly changes, $ a gallon (illustrative). Solid line: slope h. Dashed: slope h*.
ΔF, futuresΔS, spot
Risk left after hedging, against the hedge ratio.
0.00.51.01.5hedge ratio hunhedged
h*0.857
Unhedged risk$0.200
Risk at your h$0.092
Lowest possible$0.087
Risk removed53.9%
Contracts47.6

Risk is the standard deviation of the monthly change, in dollars a gallon. Contracts: for 2m gallons, 42,000 gallons a contract.

Worked example

An airline hedges jet fuel with heating oil futures

A US airline will buy 2 million gallons of jet fuel in one month. It hedges with NY Harbor ULSD (heating oil) futures, each covering 42,000 gallons. Its analyst has the last 12 monthly price changes, in dollars a gallon. They are illustrative, made by a seeded generator, not market history.

Sizing the hedge

Step 1 of 7
  1. 1

    The data

    MonthΔS, jet fuelΔF, heating oil futures
    1+0.097+0.204
    2−0.029−0.116
    3+0.156+0.055
    4−0.206−0.163
    5−0.173−0.237
    6−0.059+0.080
    7+0.311+0.309
    8+0.197+0.179
    9+0.449+0.520
    10−0.129−0.011
    11−0.052+0.055
    12+0.067+0.058

Twelve months is a short history, so the estimates are rough; real hedgers use longer data and check that the relationship is stable. The method is the same.

Regulatory lens

A hedge can still make reported profit jump about if the derivative is marked to market and the item it hedges is not. Hedge accounting under IFRS 9 lets a firm match the two, if it documents the hedge, shows an economic relationship and sets a sensible hedge ratio. IFRS 9 dropped the old IAS 39 test that a hedge had to offset between 80% and 125% of the changes. M4.7: Accounting for risk covers it. In the EU, EMIR excludes a non-financial firm's hedging trades when working out whether it must clear its OTC derivatives.

Sparks

SparkTailing the hedge
Futures settle every day, so the gains and losses on a futures hedge arrive as cash along the way and can earn or pay interest. Hull describes a small fix, “tailing the hedge”: size the hedge with values instead of quantities, N* = h* × (value of the exposure) ÷ (futures price × contract size). It matters most for large, long hedges.
SparkRolling a hedge forward
Liquid futures are short-dated. A hedger with a long exposure often uses a near-month contract and rolls it into the next one before delivery, over and over. Each roll adds basis risk, and the margin calls in between need cash. Metallgesellschaft's oil hedges in 1993 are the classic case (L1.7.12: Derivatives disasters and their lessons).

Key takeaways

  • Buy futures (a long hedge) if you lose when the price rises; sell futures (a short hedge) if you lose when it falls.
  • A forward can lock a price exactly; futures are standard, so they leave basis risk.
  • Basis b = S − F. A futures hedge locks in about F₁ + b₂, so the unknown basis at close-out is the risk left.
  • The minimum-variance hedge ratio is h* = ρ σ_S / σ_F, and the number of contracts is N* = h* × Q_A / Q_F.
  • At h* the hedge removes a share ρ² of the variance; the standard deviation falls to σ_S √(1 − ρ²).

Reaction

Check what stuck

Question 1 · objective 1

The standard deviation of monthly changes in a spot price is 0.030, that of the futures price 0.036, and their correlation 0.9. What is the minimum-variance hedge ratio, to two decimal places?

Question 2 · objective 1

With that hedge ratio, how many futures contracts hedge an exposure of 500,000 units, if each contract covers 1,000 units?

contracts

Question 3 · objective 2

An oil producer sells futures at $80.00 a barrel. It closes the hedge early, when the spot price of its oil is $76.50 and the futures price is $77.20. What price per barrel has it locked in?

$

Question 4 · objective 3

A chocolate maker must buy cocoa next spring. Which hedge fits?

Question 5 · objective 4

At the minimum-variance hedge ratio, the correlation between spot and futures changes is 0.9. What share of the variance does the hedge remove?

Glossary

Hedge
A position taken to offset the risk of another position, so that losses on one are made up by gains on the other.
Long hedge
Buying futures to protect against a rise in the price of something you will need to buy.
Short hedge
Selling futures to protect against a fall in the price of something you own or will sell.
Cross hedge
Hedging one asset with futures on a different but related asset, such as jet fuel with heating oil futures.
Basis
The spot price of the asset being hedged minus the futures price of the contract used: b = S − F.
Basis risk
The risk that the basis at the date a hedge is closed is not what was expected, so the hedge is imperfect.
Convergence
The futures price moving towards the spot price as the delivery month approaches.
Minimum-variance hedge ratio
The number of futures units per unit of exposure that makes the hedged position's variance as small as possible: h* = ρ σ_S / σ_F.
Hedge effectiveness
The share of the variance of the exposure that the hedge removes. At h* it equals ρ².

Deep Dives & Origins

Deep Dives

  • John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Hedging Strategies Using Futures”. Intermediate.

  • Louis H. Ederington, “The Hedging Performance of the New Futures Markets”, Journal of Finance 34(1), 1979. Advanced.

Origins

  • John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
  • Leland L. Johnson, “The Theory of Hedging and Speculation in Commodity Futures”, Review of Economic Studies 27(3), 1960.
  • Louis H. Ederington, “The Hedging Performance of the New Futures Markets”, Journal of Finance 34(1), 1979.
  • IASB, IFRS 9 Financial Instruments (2014), Chapter 6, Hedge accounting.
  • Regulation (EU) No 648/2012 (EMIR), Article 10, Non-financial counterparties.

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Four responses to risk

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Hedging with options