Futuresintermediate

Arbitrage in Futures

Q: If the traded futures price is below the cost-of-carry value, an arbitrageur should:

Buy futures and short the spot. Underpriced futures mean F < S_0 e^(r-q)T, so a reverse cash-and-carry trade buys the cheap futures while shorting the spot and investing the proceeds at r. At expiry, deliver the asset received from the futures into the short to close the loop.

Q: S_0 = 50, r = 5\% continuously, T = 0.5, no income. Market futures price is 53. Per-unit arbitrage profit is approximately:

1.27. Fair value F_0 = 50 × e^0.025 = 50 × 1.02532 ≈ 51.27. Market futures at 53 are overpriced by 53 - 51.27 = 1.27 per unit. The cash-and-carry locks in this amount net of trading costs.

Futures arbitrage exploits any deviation between the traded futures price and the cost-of-carry fair value $F_0 = S_0 e^{(r - q) T}$ . When the futures price is too high, a cash-and-carry trade buys spot, finances it at $r$ , and sells futures to lock in a riskless profit. When the futures price is too low, the reverse cash-and-carry trade short-sells spot, invests proceeds at $r$ , and buys futures. Both strategies require no net capital at inception, so any positive expected profit attracts traders until the gap closes.

Try it yourself

Cost of carry explorer

A future is just the spot price carried forward to maturity. Interest and storage push it up; a convenience or dividend yield pulls it down. Compounding the net carry more often nudges the price up toward the continuous limit.

Compounding frequency — F at 0.5y

Contango. F sits 2.53 above spot — positive net carry means it costs more to hold the asset than the yield it throws off.

Spot S100.00

Fair futures price F (continuous)102.53

Carry components (price impact over 0.5y)

Interest carry (raises F)+2.53

Storage cost (raises F)+0.00

Convenience / dividend (lowers F)+0.00

Net carry+2.53

Spot S100.00

Risk-free rate r5%

Time to maturity T0.5y

Storage cost u0%

Convenience / dividend y0%

F = S · e^((r + u − y)·T)As the compounding frequency rises, the discrete price climbs toward the continuous limit. Raise r or T to see the four curves fan apart. Storage (u) and convenience/dividend (y) are modelled as yields added to r for teaching simplicity, not as separate cash flows.

Why it matters

Arbitrage is the enforcement mechanism behind futures pricing. If June crude futures trade noticeably above the cost-of-carry level, the market is offering free money to anyone willing to buy oil today, store it, and deliver it in June. The buying pressure in spot and the selling pressure in futures push prices back to fair value almost instantly in liquid markets. In illiquid or constrained markets, arbitrage bounds widen because shorting the spot is costly or transaction costs are high. Real arbitrage is bounded by transaction costs, repo specials, and credit limits, not by absolute zero.

Before you read on — recall

If the traded futures price is below the cost-of-carry value, an arbitrageur should:

Formulas

No-arbitrage condition

F_0 = S_0 e^{(r - q) T}

Cash-and-carry when overpriced

F_0 > S_0 e^{(r - q) T} \Rightarrow \text{sell futures, buy spot, finance at } r

Reverse cash-and-carry when underpriced

F_0 < S_0 e^{(r - q) T} \Rightarrow \text{buy futures, short spot, invest proceeds at } r

Profit equals the mispricing minus financing and transaction frictions. Hull (2022) §5.2.

Worked examples

Scenario

$S_0 = 100$ , $r = 5\%$ continuously compounded, $T = 1$ year. Market futures price is $F = 108$ .

Solution

Fair value $F_0 = 100 \times e^{0.05} = 105.13$ . Futures are overpriced by 2.87. Cash-and-carry: borrow 100 at 5%, buy one unit of spot, sell one futures at 108. At delivery deliver the asset, receive 108, repay 100 × $e^{0.05}$ = 105.13. Risk-free profit $= 108 - 105.13 = 2.87$ per unit, before transaction costs.

Scenario

A June ASX SPI 200 futures contract trades at 7,180 while spot is 7,200 with two months to expiry. Continuously compounded $r = 4.5\%$ and dividend yield $q = 4.0\%$ .

Solution

Fair value $F_0 = 7{,}200 \times e^{(0.045 - 0.040) \times 2/12} = 7{,}200 \times e^{0.000833} \approx 7{,}206$ . The futures price 7,180 is below fair value by about 26 points. Reverse cash-and-carry: buy futures cheap, short the index basket, invest proceeds. Profit roughly 26 × 25 = A$650 per contract, less basket borrowing and short-sale costs that often eat most of it in equity markets.

Common mistakes

✗Arbitrage opportunities are common and risk-free for retail traders. They are neither. Bid-ask spreads, financing costs, and short-sale frictions consume nearly all observed mispricings in liquid markets. Studies of S&P 500 futures show pricing errors fall almost entirely inside the no-arbitrage band once full costs are accounted for.
✗Arbitrage requires capital. The textbook trade is self-financing. Borrowing funds purchase the spot, and the short side pays for itself. Capital is needed only for margin and to absorb temporary mark-to-market losses before convergence.
✗Cash-and-carry only applies to physical commodities. The logic is identical for equity index futures, bond futures, and FX futures, with the dividend or foreign rate playing the role of $q$ . Hull (2022) Chapter 5 develops each case.

Revision bullets

•Cash-and-carry sells overpriced futures and buys spot
•Reverse cash-and-carry buys underpriced futures and shorts spot
•Both strategies enforce $F_0 = S_0 e^{(r-q)T}$
•Real-world arbitrage band set by transaction costs and repo
•Short-sale frictions are the binding constraint in equity arbitrage

Quick check

If the traded futures price is below the cost-of-carry value, an arbitrageur should:

$S_0 = 50$ , $r = 5\%$ continuously, $T = 0.5$ , no income. Market futures price is 53. Per-unit arbitrage profit is approximately:

Connected topics

Cost of Carry Convergence PCP Parity Arb.

More in Futures

Futures 101 Long/Short Payoff Cont. Pricing Discrete Pricing Eff. Price

In learning paths

Futures Path

Sources

Hull (2022), §5.2 and §5.6
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Standard textbook derivation of cash-and-carry and reverse cash-and-carry arbitrage that pins the no-arbitrage futures price.
Cornell & French (1983)
Cornell, Bradford, and Kenneth R. French. The Pricing of Stock Index Futures. Journal of Futures Markets, 3(1), 1983, pp. 1 to 14.
Original empirical analysis of S&P 500 futures mispricings against cost-of-carry fair value and the role of transaction costs.
MacKinlay & Ramaswamy (1988)
MacKinlay, A. Craig, and Krishna Ramaswamy. Index-Futures Arbitrage and the Behavior of Stock Index Futures Prices. Review of Financial Studies, 1(2), 1988, pp. 137 to 158.
Quantifies the no-arbitrage band in equity index futures and shows mispricings rarely exceed transaction costs.
CFTC Cash and Carry Glossary
U.S. Commodity Futures Trading Commission. CFTC Glossary: Cash and Carry. CFTC, accessed 2026.
Regulator-side definition of cash-and-carry arbitrage and the role of short selling in enforcing futures pricing.

How to cite this page

Dr. Phil's Quant Lab. (2026). Arbitrage in Futures. Derivatives Atlas. https://phucnguyenvan.com/concept/arbitrage-futures