Convergence of Futures to Spot
As a futures contract approaches its last trading day, the futures price converges to the spot price , so at expiry . The mechanism is arbitrage. Any positive gap would let a trader sell futures and buy spot, deliver immediately, and pocket the difference. The same logic in reverse closes any negative gap. Basis, defined as , collapses to zero at the last settlement, which is the foundation of hedge accuracy.
Why it matters
The futures premium reflects the cost of carry that remains between today and delivery. As time passes, less financing and storage stand between the two prices, so the premium melts away. By the morning of the last trading day, only seconds of carry are left, and the futures and spot quotes must agree to within transaction costs. If they did not, anyone could sell the dearer and buy the cheaper for an immediate riskless profit, which is exactly what arbitrageurs do. Convergence is not a coincidence, it is an arbitrage identity timed to delivery.
Formulas
Worked examples
June wheat futures trade at $310 per tonne with spot at $308 two days before expiry. Storage and financing cost over two days is negligible.
The two-dollar gap is too large for the remaining carry. A trader sells futures at $310 and buys spot at $308 for a quick cash-and-carry. By the last settlement the gap collapses, and they deliver the wheat at $310 to lock in the difference of roughly $2 per tonne less carry. Convergence drives within hours.
An expiring December ASX SPI 200 futures contract has spot index 7,250 and futures 7,254 one hour before final settlement.
The remaining net carry is essentially zero over one hour, so the four-point gap is just noise plus transaction costs. The official opening price mechanism on settlement morning forces to within tick precision, and any residual difference is absorbed into the final mark-to-market through the margin account.
Common mistakes
- ✗Convergence is smooth and predictable. The basis can be volatile during the life of the contract because spot and futures respond to news at different speeds. Only the end point is enforced, not the daily path.
- ✗Convergence guarantees a perfect hedge. It guarantees the gap closes at expiry, not before. A hedger who closes early faces basis risk, defined as uncertainty in at the close-out date.
- ✗Cash-settled contracts do not converge. They do. The final settlement price is calculated from spot market values (for SPI 200, the opening prices of constituents on the third Thursday of the expiry month), so the futures quote is forced to equal that reference at the end.
Revision bullets
- • as , enforced by arbitrage
- •Final basis for a clean contract
- •Path can be bumpy even though the endpoint is fixed
- •Cash-settled contracts converge via the final settlement price
- •Convergence underpins hedge effectiveness and basis risk
Quick check
At the final settlement of a deliverable futures contract:
Convergence ensures that:
Connected topics
In learning paths
Sources
- Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.Section 3.3 introduces basis and convergence in the context of hedging. Section 5.2 develops the arbitrage logic that forces $F_T = S_T$.
- Australian Securities Exchange. ASX SPI 200 Index Futures Contract Specifications and Settlement. ASX, 2024.Defines the special opening quotation that anchors final settlement of the SPI 200 contract to the underlying index value.
- CME Group. Final Settlement Prices and Delivery Mechanisms. CME Group Education, 2024.Plain-English description of how final settlement procedures force convergence in liquid US contracts.