Hedging & Basis Riskadvanced

Minimum Variance Hedge Ratio

The minimum-variance hedge ratio $h^*$ is the proportion of the spot exposure to be hedged with futures that minimises the variance of the hedged position. It equals $h^* = \rho\,\sigma_S / \sigma_F$ , where $\rho$ is the correlation between the change in the spot price and the change in the futures price, and $\sigma_S$ , $\sigma_F$ are their standard deviations. The result is the slope of the regression $\Delta S = a + h\,\Delta F + \varepsilon$ , and the regression $R^2 = \rho^2$ measures hedge effectiveness.

Why it matters

A naive hedger sells one futures contract per unit of spot exposure. That is only optimal when the spot and the futures change by the same amount on average. If futures are twice as volatile as the spot, half as many contracts will offset the average shock. If correlation is below one, the additional uncertainty from the futures leg means the variance-minimising ratio is further reduced. The formula bundles both adjustments into a single number that can be estimated by a simple OLS regression.

Formulas

Minimum-variance hedge ratio

h^* = \rho_{SF} \times \frac{\sigma_{\Delta S}}{\sigma_{\Delta F}}

Equivalently

h^* = \mathrm{Cov}(\Delta S, \Delta F) / \mathrm{Var}(\Delta F)

, the OLS slope from regressing spot changes on futures changes. Source: Hull (2022) §3.5.

Optimal number of futures contracts

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

Q_A

is the size of the position being hedged and

Q_F

is the size of one futures contract, both in the same units. Round to the nearest whole contract.

Hedge effectiveness

R^2 = \rho^2

Fraction of variance in the hedged-position changes removed by hedging at

h^*

. Reported in Ederington (1979) as the standard measure of hedge quality.

Worked examples

Scenario

An airline plans to buy 2,000,000 gallons of jet fuel in one month. Heating oil futures cover 42,000 gallons each. From historical monthly data, $\sigma_{\Delta S} = 0.0263$ , $\sigma_{\Delta F} = 0.0313$ , and $\rho = 0.928$ .

Solution

Hedge ratio $h^* = 0.928 \times (0.0263 / 0.0313) = 0.7777$ . Optimal contracts $N^* = 0.7777 \times (2{,}000{,}000 / 42{,}000) = 37.03$ , so the airline buys 37 heating oil futures. Hedge effectiveness $R^2 = 0.928^2 = 0.861$ , meaning 86% of jet fuel price variance is removed. This worked example follows Hull (2022) Example 3.3.

Scenario

An Australian grain trader hedges 50,000 tonnes of barley using ASX Eastern Wheat futures (50 tonnes per contract). Regression of monthly barley price changes on wheat futures changes gives $h^* = 0.85$ .

Solution

Optimal contracts $N^* = 0.85 \times (50{,}000 / 50) = 850$ . The trader shorts 850 ASX wheat futures. With $h^*$ implying a correlation around 0.85 to 0.90 depending on volatilities, hedging effectiveness lies in the 70 to 80 percent range. The remaining variance is barley-specific basis risk.

Common mistakes

✗The hedge ratio is always one. $h^* = 1$ only when $\rho = 1$ and $\sigma_S = \sigma_F$ . In every other case, the optimal ratio differs from unity. The naive 1:1 hedge typically over-hedges or under-hedges and leaves more variance in the portfolio.
✗A high correlation is sufficient. Hedge quality also depends on the relative volatilities. Two assets with $\rho = 0.99$ but very different volatility scales still need a non-trivial $h^*$ to balance dollar exposures.
✗The ratio is constant over time. Volatilities and correlations vary with market conditions. Hull (2022) recommends **re-estimating $h^*$ ** when the regression window shifts substantially or when the market regime changes.

Revision bullets

•** $h^* = \rho\,\sigma_S / \sigma_F$ ** minimises hedged variance
•Equals the OLS slope of $\Delta S$ on $\Delta F$
•Optimal contracts $N^* = h^* \times Q_A / Q_F$
• $h^* = 1$ only when $\rho = 1$ and equal volatilities
•Hedge effectiveness $R^2 = \rho^2$
•Re-estimate when market regimes shift

Quick check

If $\rho = 1.0$ and $\sigma_S = \sigma_F$ , the minimum-variance hedge ratio is:

An exporter has spot exposure with $\sigma_S = 0.020$ . The hedge instrument has $\sigma_F = 0.040$ and correlation $\rho = 0.90$ with the spot. The minimum-variance hedge ratio is closest to:

Connected topics

Delta Hedge Cross Hedge Tailing

In learning paths

Hedging & Risk Path

Sources

Hull (2022), §3.5, Example 3.3
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Derives $h^* = \rho\,\sigma_S / \sigma_F$, links it to the OLS regression slope, and works through the airline jet-fuel example used in our case study.
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.
Foundational paper that estimated minimum-variance hedge ratios and reported hedge effectiveness as the regression $R^2$, the standard benchmark since.
Johnson (1960)
Johnson, Leland L. The Theory of Hedging and Speculation in Commodity Futures. Review of Economic Studies, Vol. 27, No. 3, 1960, pp. 139-151.
Earliest derivation of the minimum-variance hedge ratio in a mean-variance utility framework, predating Ederington's empirical estimation.
CME Group, Equity Index Hedging
CME Group. Hedging and Risk Management for Equity Index Futures. CME Education Centre, accessed 2026.
Applies the minimum-variance framework to equity portfolios and discusses how to translate the ratio into contract counts.

How to cite this page

Dr. Phil's Quant Lab. (2026). Minimum Variance Hedge Ratio. Derivatives Atlas. https://phucnguyenvan.com/concept/minimum-variance-hedge-ratio

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