Cost of carry and the no-arbitrage band
Carrying spot to delivery costs money: forgone funding (r), storage and insurance (u), minus income (q — dividends, lease) and minus convenience yield (y — the hidden benefit of physically owning a consumable). The carry rate is C = r + u − q − y and fair value is F₀ = S₀·e^(C·T). If futures trade rich, cash-and-carry arbitrage sells them and buys spot; if cheap, the reverse trade kicks in — pinning futures inside a no-arbitrage band whose width is transaction costs.
Hedging with futures: hedge ratio and basis risk
When the exposure is the underlying itself, a one-to-one hedge works. Otherwise the minimum-variance ratio h* = ρ·(σ_S/σ_F) minimizes the variance of the hedged position, and N* = h*·V_S/(F × multiplier) converts it to contracts. Hedge effectiveness HE = 1 − Var(hedged)/Var(unhedged) is a historical, in-sample diagnostic — not a promise. What remains is basis risk: spot and futures never move in perfect lockstep.
Worked example: fuel hedge for a transport operator
- Exposure 100,000 liters; correlation with gasoil futures ρ = 0.9, σ_diesel = 12%, σ_F = 10%.
- h* = 0.9 × 1.2 = 1.08 → hedge 108% of the exposure.
- Futures at $0.80/liter-equivalent, contract 42,000 liters → N* ≈ 2.6 → buy 3 contracts.
- Prices rise 10%: spot loss ≈ $10,000; futures gain ≈ $10,080 — nearly complete, with a small residual.
نتیجه: Hedges are never complete and should not be: the residual basis risk is what turns a raw exposure into a manageable one.
Futures hedging accepts daily settlement: in adverse paths the margin call consumes cash before the offsetting spot gain arrives. Margin budgeting is part of the cost of hedging, not a technicality.