Spot-Futures Parity

The arbitrage relationship that anchors the price — where the parity formula comes from, how to exploit a violation, and why real-world frictions set the limits.

Spot-futures parity is the no-arbitrage condition that ties the price of a futures contract to the current price of the underlying asset. It is not a forecast or a model assumption — it is a constraint enforced by replication. Whenever the futures price deviates far enough from parity, a riskless profit is available; arbitrageurs collect it, and the deviation closes. Every futures desk and the futures pricing calculator on this site uses this relationship as the foundation. Understanding it tells you not just what the fair futures price is, but exactly how far it can stray before money is left on the table.

The spot futures parity formula

For an asset with current spot price S, continuously compounded risk-free rate r, continuous dividend yield q, and time to delivery T in years, the fair futures price is:

F = S · e^{(r − q)T}

This is the cost of carry formula in its general form. The logic is a replication argument. You can create a synthetic long futures position by borrowing S dollars at rate r, buying the underlying, and holding it to delivery while receiving dividends at yield q. At expiry the position is worth exactly S · e^{(r − q)T}. If the listed futures price F differs from that, one side of the trade is mispriced relative to the other. For a non-dividend-paying asset the formula simplifies to F = S · e^rT, which is just the future value of the spot price at the risk-free rate.

Worked example: S = 5,000, r = 5.25% continuously compounded, q = 1.30%, T = 0.25 (90 days). Fair value is 5,000 · e^{(0.0525 − 0.013) × 0.25} = 5,000 · e^0.009875 ≈ 5,000 · 1.009924 ≈ 5,049.62. If the futures contract is listed at 5,060, it is rich to fair value by roughly 10.38 index points.

Cash-and-carry arbitrage when the future is rich

When F > S · e^{(r − q)T}, the futures contract is overpriced relative to its replicating portfolio. The correct trade is the cash-and-carry:

1. Borrow S dollars at the risk-free rate r for time T.
2. Buy the underlying at S and hold it, collecting dividends or yield q.
3. Sell (short) the futures contract at the inflated price F.

At expiry, deliver the underlying against the futures short, receive F, repay the loan at S · e^rT, and net the dividends received. The riskless profit per unit is:

Π = F − S · e^{(r − q)T}

Using the numbers above: Π = 5,060 − 5,049.62 = 10.38 index points. On one E-mini S&P 500 contract (multiplier $50), that is $519 locked in with no directional exposure. Scale to institutional size and the incentive to close the gap is obvious.

Reverse cash-and-carry when the future is cheap

When F < S · e^{(r − q)T}, the futures contract is underpriced. The mirror trade is the reverse cash-and-carry:

1. Short-sell the underlying at S, receiving the proceeds.
2. Invest the proceeds at rate r for time T.
3. Buy the futures contract at the depressed price F.
4. At expiry, take delivery via the long futures, return the borrowed shares, and pay out any dividends owed.

The profit is S · e^{(r − q)T} − F. The mechanics are symmetric; the difficulty is asymmetric, which is why the no-arbitrage band is not centered on fair value.

The no-arbitrage band and real-world frictions

In practice, parity is not a single number but a band. Frictions prevent arbitrageurs from capturing every deviation:

• Transaction costs. Bid-ask spreads, commissions, and market impact on both legs add a round-trip cost that must be exceeded before the trade pays. For index futures this might be 1–3 index points; for less liquid contracts it widens considerably.
• Borrow cost and short constraints. The reverse cash-and-carry requires shorting the underlying. Securities lending fees raise the effective rate above r, and for hard-to-borrow names the short may be unavailable entirely. This pushes the lower bound of the band down.
• Dividend uncertainty. The formula uses an expected continuous yield q, but actual dividends are discrete and can be cut. For single-stock futures, estimating q incorrectly turns an apparent arbitrage into a dividend-risk bet.
• Financing rate uncertainty. Repo rates are not perfectly locked in for the full horizon, particularly for longer-dated contracts, introducing basis risk on the funding leg.
• Mark-to-market settlement. Exchange-traded futures are marked daily. Variation margin flows create a subtle convexity difference versus the theoretical forward price, though for most practical purposes the effect is small.

The band is typically tightest for liquid equity index futures (S&P 500, Nasdaq-100) where the underlying basket is cheap to replicate and borrow is abundant. It widens for commodity futures, where storage costs, convenience yield, and delivery logistics dominate, making the cost-of-carry model a rougher approximation.

Why index futures track parity so tightly

Equity index futures — especially front-month contracts on the major U.S. indices — are among the most efficiently priced instruments in any market. The constituent stocks are all highly liquid, dividends are well-forecast, and the financing market is deep. Program trading desks monitor the basis in real time and execute the cash-and-carry mechanically when it opens. The result is that the E-mini S&P 500 future rarely strays more than a few tenths of an index point from fair value during normal trading hours. Deviations large enough to profit after transaction costs are measured in seconds, not minutes.

This tight relationship has a practical consequence: when you see the futures price, you can back out where the spot market is trading even before the cash open. Index arbitrage works in both directions, so futures lead spot price discovery, particularly at market open and around major macro releases.

Relation to put-call parity

Spot-futures parity and put-call parity are the same idea applied to different instruments. Put-call parity says a synthetic long forward (long call, short put at the same strike and expiry) must equal the discounted forward price; otherwise there is an arbitrage between the options and the underlying. Spot-futures parity says the listed futures price must equal the cost of replicating forward exposure via the cash market. Both are no-arbitrage constraints that link derivative prices to the replicating portfolio — the futures parity anchors the forward price level, and put-call parity then anchors the options surface around that level. Violate either and there is a direction-neutral profit available; enforce both and prices become self-consistent.