Insights
How to Price an Index Future
Alphanume Team · May 28, 2026
Dividends, rates, and fair value for equity indices.
Index futures pricing comes down to one equation and two numbers most traders underweight: the risk-free rate and the dividend yield. The fair value of a futures contract is simply the spot index level compounded forward at the net cost of carrying the position — borrowing money to buy the basket, minus the dividends you collect while holding it. Get those two inputs right and you can calculate fair value in your head, explain why the future sometimes trades above spot and sometimes below, and understand the "fair value" figure that scrolls across financial news screens every pre-market session. The futures pricing calculator on this site automates the arithmetic; this post explains the mechanics underneath it.
The pricing formula for index futures
For an equity index that pays a continuous dividend yield, the theoretical futures price is:
F = S · e(r − q)T
where:
- S is the current index level (spot)
- r is the continuously compounded risk-free rate
- q is the continuously compounded dividend yield of the index
- T is the time to expiry in years
- e(r − q)T is the net carry factor
The exponent (r − q) is the cost of carry: what it costs to finance a long position in the basket, minus what you earn from dividends along the way. When financing costs exceed dividends, carry is positive and the future prices above spot. When dividends exceed financing costs, carry is negative and the future prices below spot.
A worked S&P-style example
Suppose the index is at 5,400, the annualised risk-free rate is r = 5.25%, the index dividend yield is q = 1.40%, and the futures contract expires in 73 days (T = 73/365 ≈ 0.2000 years).
Net carry rate: r − q = 5.25% − 1.40% = 3.85% = 0.0385 per year.
Carry factor: e0.0385 × 0.2000 = e0.0077 ≈ 1.00772.
Fair value: F = 5,400 × 1.00772 ≈ 5,441.7.
The carry in index points is 5,441.7 − 5,400 = 41.7 points. At the standard E-mini S&P 500 multiplier of $50 per point, that 41.7-point carry represents $2,085 per contract in forward financing costs net of dividends. The total notional of one contract at fair value is 5,441.7 × $50 = $272,085.
Crucially, both the carry and the absolute notional scale linearly with spot. If the index were at 5,800, the carry at the same r − q and T would be 5,800 × (1.00772 − 1) ≈ 44.8 points — more index points of carry, same percentage.
Premium, discount, and what drives the sign
The sign of the carry — and therefore whether the future trades at a premium or discount to spot — is entirely determined by the relationship between r and q:
- r > q (normal environment): Financing costs dominate. The future trades at a premium to spot. You pay more for the future because if you bought the basket outright, the interest on borrowed capital would cost more than the dividends you receive. This is the typical setup for US equity index futures in a positive-rate environment.
- r < q (high-yield or low-rate environment): The dividend stream outweighs the cost of financing. The future trades at a discount to spot. This occurred in many European equity markets during the negative-rate era, and it occurs for high-dividend emerging market indices.
- r = q: Carry is zero. The future is priced at spot. Rare in practice, but a useful limiting case.
Understanding dividends matters most here. The index dividend yield used in the formula is not just the trailing yield — it reflects when dividends are expected to go ex-date before expiry. Large constituent ex-dates cause visible step-downs in futures fair value, which is why the futures can appear to cheapen abruptly around major ex-dividend dates.
Pre-open fair value and what the financial media quotes
The "fair value" number quoted each morning before the US cash market opens is simply the formula above, evaluated with the current overnight futures price compared to the previous cash close. Broadcasters typically express it as the difference between the futures price and fair value: if the futures are trading 8 points above fair value, the implied cash open is 8 points above the prior close adjusted for carry.
More precisely, if Fmarket is where the futures are trading and Ffair is the theoretical price, the implied cash index open is:
Simplied = Sprior close + (Fmarket − Ffair)
A futures premium above fair value signals buyers; a discount signals sellers. What actually moves the future overnight — earnings, macro data, geopolitical events — gets translated back into index-point terms through this relationship.
Basis, convergence, and index arbitrage
The basis is the difference between the futures price and the spot index: basis = F − S. At initiation, basis equals the carry (41.7 points in our example). At expiry, basis must equal zero — the futures contract settles to the cash index by definition, so F and S converge.
Basis erodes predictably as time passes. If nothing changes in r, q, or S, the futures price declines relative to spot (when carry is positive) as T shrinks toward zero. A long futures position therefore loses value relative to spot simply from the passage of time, at a rate proportional to the carry rate.
What keeps the futures from straying far from fair value is index arbitrage: if the futures trade materially above fair value, a program trading desk buys the basket of stocks and simultaneously sells the futures, locking in a riskless spread. If the futures trade below fair value, the desk buys futures and sells the basket short. These trades are automated and execute within milliseconds, which is why large, liquid index futures — the E-mini S&P 500 in particular — rarely deviate from fair value by more than a fraction of a point under normal market conditions.
| Condition | Futures vs. spot | Arbitrage response |
|---|---|---|
| F > Fair value | Future overpriced | Buy basket, sell futures |
| F < Fair value | Future underpriced | Buy futures, sell basket short |
| F = Fair value | No edge | No trade |
The speed and capital behind index arb desks means that deviations are self-correcting. Persistent mispricing signals something else — a constraint on short-selling, a dividend surprise, or a sudden change in funding rates — not a free lunch.
Practical limits of the model
The continuous dividend yield assumption is a simplification. Real indices pay lumpy, discrete dividends clustered around quarterly ex-dates. For short-dated contracts this lumpiness matters: a large constituent going ex-dividend three days before expiry creates a step-change in carry that the smooth formula misses. Practitioners maintain an explicit dividend schedule — expected dividend points per period — and substitute that for the q·T term when precision is required.
Funding rates also fluctuate. The overnight rate used to finance positions is not constant across the life of a contract. For contracts beyond the front month, the carry calculation should use the forward interest rate for that period, not the current overnight rate. The formula F = S · e(r − q)T remains correct; it just requires the right r for each expiry.
These refinements matter most at the edges — very short-dated contracts near ex-dividend dates, or longer-dated back-month contracts in volatile rate environments. For most practical purposes — understanding whether the futures are rich or cheap to spot, sizing a hedge, or reading a pre-open fair value screen — the basic formula is both sufficient and exact.