Convergence and basis decay
A futures price converges to spot at delivery — the basis decays to zero by expiry. Only the endpoint is enforced; the path is noisy.
Try this
Contango. r > q, so F starts 4.08 above spot. The positive basis decays down to zero by delivery.
Futures Fₜ 104.08Spot Sₜ 100.00Basis Fₜ − Sₜ +4.08Time to delivery (T − t) 1y
Sweep time t toward delivery0y
Spot S₀100.00
Financing rate r6%
Carry / dividend yield q2%
Horizon to delivery T1y
Fₜ = Sₜ · e^((r − q)·(T − t)) ⇒ basisₜ = Sₜ · (e^((r − q)·(T − t)) − 1) → 0 as t → TThe factor (T − t) shrinks to zero at delivery, so the exponential collapses to 1 and the basis vanishes — no matter where the spot wandered. No-arbitrage only pins down the endpoint Fₜ = Sₜ at delivery; in between, the basis is whatever carry and noise make it.
Discuss. Turn on the bumpy path and raise σ until the spot swings wildly. The basis still hits exactly zero at delivery. Why is the endpoint guaranteed by no-arbitrage but the path is not? What trade would you put on if the basis stayed wide one day before delivery?