The theoretical price of a futures contract takes only a few lines to calculate. The market price, meanwhile, keeps moving around it. Is every gap an arbitrage opportunity? Usually not. In this article we look at when the gap can actually be traded.
Fair value: the cost of carry
Buying a basket of shares today and holding it to expiry is no different from buying the same basket at expiry through a futures contract, apart from the cost of cash and the dividends the basket will pay. That equivalence gives the theoretical value of the futures price:
Here S is the spot price, r the annual interest rate, τ the time to expiry in years (days / 365), and D the value at expiry of the dividends expected before expiry. The difference between the futures price and the spot price is called the basis.
Example
Assume an index with a spot value of 10,000 points, an annual interest rate of 40%, 30 days to expiry and no dividends before expiry:
The theoretical basis is about 329 points. In a high-rate environment the basis becomes a meaningful share of the spot price, so small differences in the rate assumption move the theoretical price noticeably.
Not one price but a band
The formula assumes a single interest rate. In reality, borrowing and lending rates differ, and transaction costs come on top. The no-arbitrage price is therefore not a single point but a band:
C is the total cost of the two-legged trade. If the futures price is above the upper bound, borrowing to buy the basket and selling the future (cash and carry) leaves a profit. Below the lower bound, the reverse trade is needed: sell the basket, buy the future and put the cash on deposit.
Example, continued
Let the borrowing rate be 42%, the deposit rate 38%, and the total transaction cost 20 points in each direction:
| Calculation | Value | |
|---|---|---|
| Upper bound | 10,000 × (1 + 0.42 × 30 / 365) + 20 | 10,365.21 |
| Fair value | 10,000 × (1 + 0.40 × 30 / 365) | 10,328.77 |
| Lower bound | 10,000 × (1 + 0.38 × 30 / 365) − 20 | 10,292.33 |
The band is about 73 points wide. As long as the futures price stays inside it, there is no risk-free arbitrage, however striking the deviation from fair value may look.
The real costs that set the band
- Commissions and taxes: Commissions paid on both legs, on entry and exit, and the taxes that come with them.
- Spread and market impact: The bid–ask spread of every share in the index basket, and large orders pushing the price. This item grows quickly as the basket gets bigger.
- Margin and funding: The opportunity cost of margin posted for the futures position, and any additional margin needed during the trade.
- Dividend uncertainty: If the amount and timing of dividends to be declared before expiry are not certain,
Dis an estimate and its error flows straight into profit. - Tracking error: A basket built from fewer shares than the full index does not track it exactly.
- Short-selling constraints: Lower-bound arbitrage requires selling the basket. For a participant who does not already hold it, that means the cost and limits of borrowing shares, which is why the lower bound works less well in practice than the upper.
- Final settlement: For cash-settled contracts, the gap between how the settlement price is set and the prices at which the basket is closed out.
Reading it backwards: the implied rate
Solving the formula for the rate gives the implied interest rate hidden in the futures price:
Comparing the implied rate with your own funding cost is the most practical way to assess the opportunity. A fund, for instance, can earn the implied rate on its cash by buying the basket and selling the future instead of placing the cash on deposit; if the implied rate is above the deposit rate, that means a higher return without taking a directional position.
Why speed matters
Prices that break out of the band do not stay there for long. Capturing the opportunity requires watching the cash and futures markets at the same time, executing both legs almost simultaneously, and managing the open-position risk that arises when one leg fills and the other does not. That is why cash–futures arbitrage is as much an infrastructure problem as a mathematical one.
Conclusion
A deviation from fair value is not a signal on its own. The tradable opportunity is the gap that remains once borrowing and lending rates, real transaction costs and execution risk are taken into account. Measuring that gap correctly and acting on it quickly is what determines the value of the strategy.
This article is for general information only and is not investment advice. Numerical examples are illustrative and may not reflect actual market conditions, taxes or cost structures.