Hedging & Basis Riskintermediate

Cross Hedging

Cross hedging uses a futures contract on a different but correlated asset when no liquid future exists for the exact exposure. The classic case is an airline that hedges jet fuel with WTI or Brent crude futures, or a Pacific salmon farmer that hedges with Norwegian salmon futures. The procedure introduces additional basis risk because the hedged and hedging assets do not move in lockstep. The optimal hedge ratio is the minimum-variance ratio $h^* = \rho\,\sigma_S / \sigma_F$ , not the naive 1:1.

Why it matters

If no umbrella exists in your size, take a slightly bigger or smaller one. The shelter is imperfect but better than walking unprotected in the rain. Cross hedging works the same way. A jet fuel buyer cannot find jet fuel futures, so it picks the most correlated liquid instrument, scales the contract count by the relative volatilities, and accepts the residual mismatch as the price of using an imperfect substitute.

Formulas

Minimum-variance hedge ratio for cross hedging

h^* = \rho \times \frac{\sigma_S}{\sigma_F}

\rho

is the correlation between changes in spot and futures prices.

\sigma_S

and

\sigma_F

are their standard deviations. Source: Hull (2022) §3.5.

Optimal number of cross-hedge contracts

N^* = h^* \times \frac{Q_A}{Q_F}

Q_A

is the size of the cash-market position and

Q_F

is the size of one futures contract in the same units.

Hedge effectiveness (R-squared)

R^2 = \rho^2

Fraction of spot-price variance removed by the cross hedge. A correlation of 0.9 implies

R^2 = 0.81

, so 81% of price variance is hedged.

Worked examples

Scenario

An Australian airline expects to buy 1,000,000 barrels of jet fuel in three months. Studies of monthly changes show jet fuel and Brent crude futures have $\rho = 0.92$ , $\sigma_S = 0.032$ and $\sigma_F = 0.028$ . Each Brent futures contract covers 1,000 barrels.

Solution

Hedge ratio $h^* = 0.92 \times (0.032 / 0.028) = 1.05$ . Optimal contracts $N^* = 1.05 \times (1{,}000{,}000 / 1{,}000) = 1{,}051$ . The airline buys 1,051 Brent crude futures. Hedge effectiveness $R^2 = 0.92^2 = 0.85$ , so 85% of jet fuel price variance is removed. The remaining 15% is cross-hedge basis risk.

Scenario

A Vietnamese rice mill needs to hedge a Thai jasmine rice purchase, but no liquid jasmine rice future exists. The mill chooses CBOT rough rice futures, observing $\rho = 0.70$ , $\sigma_S = 0.045$ , $\sigma_F = 0.040$ .

Solution

Hedge ratio $h^* = 0.70 \times (0.045 / 0.040) = 0.79$ . The mill uses 79 contracts per 100,000-unit notional position. Effectiveness $R^2 = 0.49$ , meaning only half the price variance is hedged. Because the rice grades and quality bands differ markedly, cross-hedge quality is poor and the mill may prefer over-the-counter swaps with a regional dealer instead.

Common mistakes

✗Cross hedging is as effective as a direct hedge. The variance removed is $R^2 = \rho^2$ of the spot variance. Unless $\rho$ is close to one, a meaningful residual remains. For airlines using crude futures to hedge jet fuel, typical correlations of 0.80 to 0.90 deliver $R^2$ between 64% and 81%.
✗A naive 1:1 contract count is fine. Differences in volatility between the two assets mean the optimal ratio is $h^* = \rho\,\sigma_S / \sigma_F$ , often well above or below one. Picking 1:1 either over-hedges or under-hedges the position.
✗Higher correlation always means a higher hedge ratio. The hedge ratio depends on both correlation and the volatility ratio. Two perfectly correlated assets with very different volatilities still need a non-unit hedge ratio so that the dollar value of the futures position matches the dollar exposure of the spot position.

Revision bullets

•Hedge with a correlated but different asset
•Used when no liquid future for exact exposure exists
•Optimal $h^* = \rho\,\sigma_S/\sigma_F$
•Effectiveness $R^2 = \rho^2$
•Higher correlation improves hedge quality
•Always introduces extra basis risk versus a direct hedge

Quick check

Cross hedging is necessary when:

If $\rho = 0.80$ , $\sigma_S = 0.04$ and $\sigma_F = 0.05$ , the optimal cross-hedge ratio is closest to:

Connected topics

Basis Risk Min Var. HR

In learning paths

Hedging & Risk Path

Sources

Hull (2022), §3.5
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Section 3.5 develops the minimum-variance hedge ratio in the cross-hedge setting and illustrates with an airline jet-fuel hedging example.
Ederington (1979)
Ederington, Louis H. The Hedging Performance of the New Futures Markets. Journal of Finance, Vol. 34, No. 1, March 1979, pp. 157-170.
Original derivation of hedging effectiveness as $R^2$ in a regression of spot changes on futures changes, the workhorse statistic for cross-hedge quality.
Adams & Gerner (2012)
Adams, Zeno and Mathias Gerner. Cross Hedging Jet-Fuel Price Exposure. Energy Economics, Vol. 34, No. 5, 2012, pp. 1301-1309.
Empirical comparison of WTI, Brent, gasoil, and heating-oil futures as cross hedges for jet fuel, showing how the optimal instrument depends on horizon.
CME Group, Hedging with Equity Index Futures
CME Group. Hedging and Risk Management for Equity Index Futures. CME Education Centre, accessed 2026.
Discusses the use of broad index futures as cross hedges for narrower equity portfolios, a common professional application of cross hedging.

How to cite this page

Dr. Phil's Quant Lab. (2026). Cross Hedging. Derivatives Atlas. https://phucnguyenvan.com/concept/cross-hedging

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