Commodity Markets and Futures › Hedging with Commodity Futures

Hedging is the primary economic function of futures markets. By taking offsetting positions in futures, commercial participants can transfer price risk to others willing to bear it.

Consider a farmer expecting to harvest 50,000 bushels of corn in six months. Current corn futures for delivery in six months trade at $4.50 per bushel. The farmer can lock in this price by selling 10 futures contracts (each contract covers 5,000 bushels).

If corn prices fall to $4.00 at harvest

• The farmer receives $4.00 per bushel in the physical market
• The futures position profits $0.50 per bushel ($4.50 - $4.00)
• Net effective price: $4.50 per bushel

If corn prices rise to $5.00 at harvest

• The farmer receives $5.00 per bushel in the physical market
• The futures position loses $0.50 per bushel ($4.50 - $5.00)
• Net effective price: $4.50 per bushel

In both scenarios, the hedge locks in the $4.50 price. The farmer sacrifices upside potential in exchange for downside protection.

Perfect hedges are rare in practice. The commodity being hedged may differ from the commodity underlying the futures contract in grade, quality, location, or timing of delivery. Even when the commodity matches perfectly, spot and futures prices do not move in lockstep. These imperfections mean that hedgers must carefully consider how large a futures position to take relative to their spot exposure.

To determine the optimal hedge ratio $h$, we seek to minimize the variance of the hedged portfolio's value changes. This is a classic optimization problem: we want to find the position size that most effectively reduces risk. The approach begins by writing out the variance of the hedged position and then finding the value of $h$ that minimizes this variance.

Consider a position that is long the spot commodity and short $h$ futures contracts. The change in the portfolio's value equals the change in spot price minus $h$ times the change in futures price. The variance of this hedged position can be expanded using standard properties of variance:

where:

• $\text{Var}(\Delta S - h \Delta F)$: variance of the hedged position value changes
• $\sigma_S$: standard deviation of spot price changes
• $\sigma_F$: standard deviation of futures price changes
• $h$: hedge ratio
• $\rho$: correlation between spot and futures price changes

This expression decomposes risk into three parts, each with a clear economic interpretation:

• $\sigma_S^2$: the variance of the unhedged spot position, representing the baseline risk we seek to reduce
• $h^2 \sigma_F^2$: the variance added by the futures position, which increases with the square of the hedge ratio
• $- 2h \rho \sigma_S \sigma_F$: the covariance term that reduces total risk, provided $\rho > 0$, by capturing the offsetting movements between spot and futures

The key insight is that the third term provides the risk reduction we seek. When spot prices rise, futures prices tend to rise as well (assuming positive correlation), so the short futures position generates losses that offset the gains from the long spot position. The optimization challenge is to balance the risk reduction from the covariance term against the additional variance introduced by the futures position itself.

To find the minimum variance, we take the derivative with respect to $h$, set it to zero, and solve for $h$. This standard calculus technique identifies the point where the variance function reaches its minimum:

where:

• $h^*$: optimal hedge ratio
• $\rho$: correlation between spot and futures price changes
• $\sigma_S$: standard deviation of spot price changes
• $\sigma_F$: standard deviation of futures price changes

This formula shows that the optimal hedge depends on two factors: the strength of the relationship between spot and futures prices, as captured by the correlation $\rho$, and the relative volatility of the two instruments, expressed as the ratio $\sigma_S/\sigma_F$. If futures are more volatile than the spot price, a smaller futures position is needed to offset the risk, because each futures contract "punches above its weight" in terms of risk contribution. If the correlation is imperfect, the hedge ratio scales down accordingly, since the offsetting effect is weaker.

Basis risk means that even a hedged position retains some price exposure. The hedge converts outright price risk into basis risk, which is typically smaller and more predictable, but not zero. A wheat farmer hedging with Chicago wheat futures still faces the risk that local basis conditions change between the time the hedge is placed and the time the physical grain is sold.