Futures — Part 2: hedging properly (hedge ratio and basis risk)

Educational content. Hedging reduces one risk but introduces others (basis, leverage). Not a recommendation.

⟵ Part 1: how futures work · Back to the pillar

Hedging transforms risk, it doesn't erase it

In Part 1 we saw how a future works. Here, the use that matters: hedging. When you hedge an exposure with an opposite futures position, you lock in a future selling price equal to the initial futures price + the ending basis. In other words: you don't zero the risk, you transform it from full price risk (unmanageable) into basis risk (small and manageable).

Basis and convergence

The basis is the difference between spot and futures price: basis = S − F. It's not random: it's the cost of carry (Part 1). The key principle for a hedger is convergence: as expiry approaches the basis narrows to zero — at expiry S = F. Meanwhile, though, the basis fluctuates, and that's the residual risk. Rigorously, hedging is a bet on the basis. This is basis risk: why no hedge is perfect.

The minimum-variance hedge ratio: how many contracts

A "one-to-one" hedge assumes your asset and the future move identically (correlation = 1): almost never true. The minimum-variance hedge ratio corrects for this:

h = ρ · (σ_S / σ_F) — (position opposite to the exposure)

where ρ is the correlation between changes in your asset (S) and the future (F). It is, in effect, the beta of your exposure to the future. Example: asset less volatile than the future (σ_S/σ_F = 0.8) and correlation 0.9 → h = 0.72: you hedge 72% of the notional, not 100%. When ρ < 1 a residual risk remains (the unhedged variance) equal to σ_S² · (1 − ρ²): the hedge has minimized, not erased, risk.

Cross-hedging: when the exact future doesn't exist

Often there's no future on your specific asset: you hedge with a correlated but different-class underlying — e.g. a corporate bond portfolio hedged with government bond futures. It works as far as the correlation is high and stable; that's exactly where basis risk lives.

Controlling beta and duration without touching the portfolio

The same logic enables synthetic risk control. For equities, the hedge ratio is adjusted for the portfolio's beta to the index future: you compute the exact number of contracts to move risk from the current profile to the desired one — without selling a single stock. For fixed income you think in duration: to cut duration to a target, you compute how many bond futures shorten rate sensitivity.

The operational choice: stack vs strip hedge

• Stack hedge — put the whole position on the most deferred contract and hold it. Simple; exposes you only to basis risk at termination. But the deferred contract is often less liquid.
• Strip hedge — start with the near contract and then roll (close the near, open the next), repeating to the horizon. Tighter, but exposes you to rollover risk: the final price depends on the calendar spread at the roll.

There's no right answer: you choose between the certainty of a known basis risk (stack) and the risk of unknown future rates on the roll cost (strip). It's a judgment on your scenario.

In short

Hedging = transforming risk into basis risk; the basis converges to zero at expiry; the minimum-variance hedge ratio (ρ·σ_S/σ_F) tells you how much to hedge; cross-hedging is for when the exact future doesn't exist; beta and duration are controlled synthetically; stack vs strip is the operational choice. A hedge doesn't eliminate risk: it reduces and manages it. It all starts at the derivatives pillar.

Sources

J. Hull, Options, Futures and Other Derivatives (minimum-variance hedge ratio, basis) · CME Group — education · Borsa Italiana — glossary.

Educational content, not a recommendation. Past performance does not guarantee future results.