The Minimum-Variance Hedge Ratio

Every worked example so far quietly assumed one futures contract on your asset offsets one unit of your exposure, one-for-one. That's exactly right when the futures contract is written on your exact asset. But basis risk taught us that's often not available — the airline hedges jet fuel with heating-oil futures, the portfolio manager hedges a stock basket with index futures. When spot and futures don't move in perfect lockstep, "one-for-one" is no longer obviously the right ratio. How many futures contracts actually minimize the risk of the hedged position? That's a genuine optimization problem, and it has a clean closed-form answer.

Setting up the optimization

Let \Delta S be the change in the value of one unit of the asset being hedged over the life of the hedge, and \Delta F the change in the futures price over the same period. A hedger who holds the exposure and takes a short futures position sized at h units of futures per unit of exposure ends up with a change in the value of the hedged position of

\Delta V = \Delta S - h\,\Delta F.

We want the h — the hedge ratio — that makes \Delta V as predictable as possible, i.e. that minimizes its variance. Writing \sigma_S, \sigma_F for the standard deviations of \Delta S and \Delta F, and \rho for their correlation, the variance of the hedged position is

\operatorname{Var}(\Delta V) = \sigma_S^2 + h^2 \sigma_F^2 - 2h\rho\,\sigma_S \sigma_F.

This is a plain upward-opening parabola in h — differentiate and set to zero:

\frac{d}{dh}\operatorname{Var}(\Delta V) = 2h\sigma_F^2 - 2\rho\,\sigma_S\sigma_F = 0 \quad\Longrightarrow\quad h^{*} = \rho\,\frac{\sigma_S}{\sigma_F}.

The variance-minimizing hedge ratio is h^{*} = \rho \, \dfrac{\sigma_S}{\sigma_F}, where \rho is the correlation between spot and futures price changes and \sigma_S,\sigma_F their standard deviations.
The optimal number of futures contracts is N^{*} = h^{*}\,\dfrac{Q_A}{Q_F}, where Q_A is the size of the exposure being hedged and Q_F is the size of one futures contract, both in the same units.
N^{*} is rounded to the nearest whole contract — futures can't be split.

Read h^{*} as a familiar object: it's exactly the slope of the ordinary least-squares regression line of \Delta S on \Delta F. Hedging optimally is the same computation as fitting the best straight-line predictor of the spot change from the futures change — which makes sense, since a good hedge is one where the futures leg "explains away" as much of the spot's wobbliness as it can.

Worked example: cross-hedging jet fuel with heating oil

An airline needs to hedge Q_A = 2{,}000{,}000 gallons of jet fuel using NYMEX heating-oil futures, each contract covering Q_F = 42{,}000 gallons. Studying two years of monthly price changes, the risk desk estimates:

\sigma_S = 0.0263, \qquad \sigma_F = 0.0313, \qquad \rho = 0.928.

Step 1 — the hedge ratio.

h^{*} = \rho\,\frac{\sigma_S}{\sigma_F} = 0.928 \times \frac{0.0263}{0.0313} \approx 0.780.

So each gallon of jet-fuel exposure should be hedged with about 0.78 gallons' worth of heating-oil futures — not the naive 1.00 a one-for-one hedge would use, because heating oil is the more volatile of the two (\sigma_F > \sigma_S), so a smaller futures position produces a matching-sized offset.

Step 2 — the number of contracts.

N^{*} = h^{*}\,\frac{Q_A}{Q_F} = 0.780 \times \frac{2{,}000{,}000}{42{,}000} \approx 37.1 \;\longrightarrow\; 37 \text{ contracts, long.}

The airline buys 37 heating-oil futures contracts. It isn't a perfect hedge — with \rho = 0.928, about 1 - \rho^2 \approx 14\% of the spot price's variance survives the hedge unexplained — but it's the best that a heating-oil future can do, and it's dramatically better than either not hedging at all or blindly matching contracts one-for-one to jet-fuel gallons.

Seeing it: the parabola being minimized

Dividing through by \sigma_S^2 gives a clean, unit-free version of the variance formula:

\frac{\operatorname{Var}(\Delta V)}{\sigma_S^2} = 1 + h^2\Big(\frac{\sigma_F}{\sigma_S}\Big)^{\!2} - 2h\,\rho\,\frac{\sigma_F}{\sigma_S},

plotted below against h. Its minimum sits exactly at h^{*} = \rho\,\sigma_S/\sigma_F. Drag \rho toward 1 and watch the minimum drop toward zero — a near-perfect hedge is possible when spot and futures are almost perfectly correlated. Drag it toward 0 and the curve flattens into an almost pointless hedge: no choice of h helps much when the futures price barely tracks the spot price at all.

The very same formula, with a shortcut, is exactly how the next lesson hedges an equity portfolio with stock-index futures. There, the "correlation times volatility ratio" collapses to a single familiar number: the portfolio's beta. That's not a coincidence — beta is defined as \operatorname{Cov}(\Delta S, \Delta F)/\operatorname{Var}(\Delta F) = \rho\,\sigma_S/\sigma_F when F is the market index, i.e. beta is the minimum-variance hedge ratio for that special case. Every stock-index hedge you'll ever compute is a corollary of the formula on this page.

Two traps that catch people who've memorized the formula without thinking about it:

Confusing \rho with h^{*}. A very high correlation does not by itself mean "hedge one-for-one." If heating-oil futures are twice as volatile as jet fuel (\sigma_F = 2\sigma_S) even with \rho = 1, the optimal ratio is h^{*} = 0.5, not 1 — using a full one-for-one position with a much twitchier instrument would over-hedge and actually add variance back in.
Rounding N^{*} to a whole number of contracts is unavoidable (you can't trade 37.1 contracts) but it isn't free — the leftover fractional exposure is itself a small, un-hedged residual, on top of whatever basis risk the correlation being less than 1 already leaves behind. For a small book relative to the contract size, that rounding error can be a meaningful fraction of the total exposure — a reason large hedgers prefer contracts with a small notional per unit.