A forward contract and a futures contract both let you lock in a price for buying an asset at a future date. They sound identical. In a world of constant interest rates, they produce the same price. But the moment interest rates start to fluctuate, a surprising and important gap opens between them — the Forward-Futures Spread. Understanding this gap is essential for anyone building algorithmic strategies on exchange-traded derivatives like NIFTY futures.
A private agreement between two parties: buy asset at price K on date T. No cash changes hands until maturity. Counterparty risk exists throughout the life of the contract. Typically OTC (over-the-counter).
An exchange-traded contract. Every day, gains and losses are settled in cash immediately via the margin account. No counterparty risk — the exchange guarantees performance. Traded on NSE, CME, etc.
The key difference: a forward settles once at maturity; a futures contract settles every single day through the margin account.
The forward price K is set so that the contract has zero initial value — no money changes hands at inception. If it were set any other way, an immediate arbitrage would exist. The no-arbitrage condition pins the forward price precisely.
The arbitrage proof: If the forward price were higher than S(t)/B(t,T), you could borrow money at the risk-free rate, buy the stock today, and simultaneously sell the forward — locking in a guaranteed riskless profit greater than the borrowing cost. Markets eliminate this instantly. The no-arbitrage condition forces For_S(t,T) = S(t)/B(t,T) exactly.
The futures price is determined differently. Because gains and losses are settled every day, the futures price must be a Martingale under the risk-neutral measure ℙ̃. This means the futures price today is simply the expected value of the asset at maturity, under ℙ̃.
The daily settlement means you collect (or pay) the price change each day. Here is a four-day example for a NIFTY futures contract:
| Day | Futures price | Daily change | Margin payment | Interest earned |
|---|---|---|---|---|
| 0 | ₹18,000 | — | — | — |
| 1 | ₹18,240 | +₹240 | Receive ₹240 | ₹240 × r overnight |
| 2 | ₹18,100 | −₹140 | Pay ₹140 | Lost ₹140 early |
| 3 | ₹18,450 | +₹350 | Receive ₹350 | ₹350 × r overnight |
| 4 (expiry) | ₹18,450 | — | — | Position closed |
The critical insight: when the NIFTY rallies (Day 1, Day 3), the futures holder receives cash immediately and can reinvest it at the prevailing overnight rate. When the market falls (Day 2), they pay immediately. This asymmetry — getting wins early when rates may be high and paying losses early when rates may be low — is exactly what creates the Forward-Futures Spread.
Now we can state the precise relationship between the two prices. The spread depends on the covariance between the discount factor D(T) and the asset price S(T). This covariance captures exactly the interest-rate timing advantage (or disadvantage) of daily settlement.
| Scenario | Logic | Covariance | Spread |
|---|---|---|---|
| Constant rates | D(T) is non-random. No correlation with S(T). | = 0 | For = Fut |
| Positive correlation: stock ↑ when rates ↓ | High S(T) occurs when D(T) is high (low rates). Futures gains come when reinvestment rates are low — disadvantage. | > 0 | For > Fut |
| Negative correlation: stock ↑ when rates ↑ | High S(T) occurs when D(T) is low (high rates). Futures gains come when reinvestment rates are high — advantage. | < 0 | For < Fut |
The spread is driven by what happens to reinvestment rates when the stock moves. If stocks and rates move in opposite directions (common in equity markets), the forward trades above the futures.
For equity indices like NIFTY, stocks and interest rates are often negatively correlated: when markets rally, central banks may raise rates (reducing bond prices), but the dominant effect is often that strong economic growth means both stocks and rates rise together — making the correlation positive and Forward > Futures. This is why NIFTY futures often trade at a slight discount to the forward price implied by carrying costs alone.
Shreve presents a single unified formula that values any asset producing a stream of cash flows — bond coupons, dividend payments, futures margin calls. It is the most general risk-neutral pricing formula, and it subsumes all the special cases seen throughout Chapter 5.
dC(u) = coupon payments at each date. The formula gives the present value of all future coupons discounted at the stochastic risk-free rate. This is bond pricing in a term structure model.
dC(u) = daily mark-to-market settlement = d(Fut_S(u,T)). Plugging this into the formula and noting that Fut_S is a Martingale recovers the futures price formula exactly.
This formula is the "Swiss Army Knife" because it is completely agnostic about the type of cash flow. Whether C(u) is continuous (like continuous dividends δ·S dt from Chapter 21), discrete (quarterly coupons), or random (futures margin calls), the same formula applies. Risk-neutral pricing is universal — it prices everything by discounting risk-neutral expectations.
Simulate interest rate paths and compute both the forward price and the futures price (risk-neutral expectation of S(T)). Vary the correlation between the stock and interest rates and watch the Forward-Futures spread change sign and magnitude.
Red = distribution of S(T) | Blue = distribution of discount factor D(T) | Gold line = futures price (E[S(T)]) | Green line = forward price
This chapter brought together the most powerful ideas in continuous-time finance. Here is the complete journey from Girsanov to Forwards and Futures:
Chapter 22 — forwards, futures and the spread in three ideas:
Ready for Chapter 6: Connections to Partial Differential Equations — where the probabilistic machinery of Chapter 5 (risk-neutral expectations) and the calculus machinery of Chapters 10–14 (the Black-Scholes PDE) are proved to be two sides of the same coin via the Feynman-Kac formula.