Forward And Futures Prices Under Changing Rates

Why Forward and Futures Prices Can Differ

A forward contract and a futures contract can both set a price today for a transaction at a future date. The key cash-flow distinction is that a forward generally settles once at maturity, while a futures contract is marked to market through regular settlement payments. If interest rates are deterministic, the timing difference does not create a difference in their prices under standard assumptions. If rates are stochastic, however, gains and losses settled before maturity may be reinvested or financed at rates that depend on the state of the world.

The resulting price difference is not a generic premium added to every futures contract. Its direction depends on how the terminal spot price moves with discount factors. To isolate that effect, assume the underlying is a tradable asset with no income, storage cost, or other carrying cost; markets are frictionless; there is no default; and margin gains and losses can be funded or invested at the risk-free short rate. The probabilities used below are risk-neutral probabilities consistent with this model, not ordinary real-world probabilities.

Definition: Let $S_T$ be the underlying asset's spot price at maturity $T$, and let $D(0,T)$ be the discount factor from $T$ to time 0. The forward delivery price $K_0$ is the delivery price that makes a newly entered forward contract have value zero at time 0. The futures price $G_0$ is the quoted price for a contract whose gains and losses are settled regularly. Under the idealized assumptions above, the futures quote is evaluated using the money-market risk-neutral measure, while the forward delivery price weights terminal outcomes by their discount factors.

Deterministic Rates: The Prices Are Equal

Suppose the effective rates over the contract term are known in advance. With one annual effective rate $i$ for $T$ years and no income or carrying costs, buying the asset now and financing it to maturity costs $S_0(1+i)^T$. The no-arbitrage delivery price is therefore $S_0(1+i)^T$. Daily settlement does not change the final price in this deterministic-rate setting: the timing of each settlement payment does not expose the holder to uncertainty in its reinvestment rate.

The same conclusion holds if the known effective rate changes over the term. For annual periods with known rates $i_1,\ldots,i_T$, the accumulation factor is the product of the period accumulation factors. Both prices use that same known accumulation.

Worked Example: A Known Rate Changes After Year One

An asset with no income or carrying costs is worth $100 today. A two-year contract is subject to a known effective annual rate of 4% in year one and 6% in year two. Find the no-arbitrage forward delivery price and futures price, assuming regular futures settlement and no funding frictions.

Method and assumptions. Since both annual rates are known in advance, accumulate the spot price by each annual factor. Under these deterministic rates, the forward and futures prices coincide.

The two-year accumulation factor is

$$ (1.04)(1.06)=1.1024. $$

Thus the forward delivery price is

$$ K_0=100(1.04)(1.06)=100(1.1024)=110.24. $$

The futures price is the same:

$$ G_0=110.24. $$

Check. An asset purchased for $100 and financed through the two known annual rates has a maturity cost of $110.24. A different contract price would permit a financing-and-delivery arbitrage under the stated assumptions. The common mistake avoided here is using one rate for both years or treating the two annual rates as additive.

Deriving the Stochastic-Rate Adjustment

Now suppose discount factors and the terminal spot price can vary across outcomes. A forward with delivery price $K_0$ pays $S_T-K_0$ at maturity to its long position. Its time-0 value is zero when the discounted expected payoff is zero. Write $E_Q$ for expectation under the stated risk-neutral measure. The zero-value condition is

E_Q\!\left[D(0,T)(S_T-K_0)\right]=0.

Expanding and solving for $K_0$ gives the forward delivery price. The futures price, under the standard daily-settlement model, is the risk-neutral expected terminal spot price. Substituting the futures price into the forward formula gives a direct comparison.

Set the forward's value to zero.
$E_Q[D(0,T)S_T]-K_0E_Q[D(0,T)]=0$, because the delivery price is constant across outcomes.

Solve for the forward delivery price.
$K_0=E_Q[D(0,T)S_T]/E_Q[D(0,T)]$. The denominator is the time-0 price of one unit paid at maturity.

Use the daily-settlement futures price.
Under the stated model, $G_0=E_Q[S_T]$. This expectation is not discounted as a single payment at maturity because futures gains and losses are settled along the way.

Separate the covariance effect.
Using $E_Q[DS_T]=E_Q[D]E_Q[S_T]+\operatorname{Cov}_Q(D,S_T)$, obtain $K_0=G_0+\operatorname{Cov}_Q(D(0,T),S_T)/E_Q[D(0,T)]$.

This derivation explains why deterministic rates make the prices equal: if $D(0,T)$ is constant, its covariance with $S_T$ is zero. With stochastic rates, the covariance can be positive, negative, or zero. A positive covariance makes the forward price higher than the futures price; a negative covariance makes it lower. Independence between the discount factor and terminal spot price also gives zero covariance, even if rates themselves are stochastic.

Theorem: Under the stated idealized assumptions,

K_0=\frac{E_Q[D(0,T)S_T]}{E_Q[D(0,T)]}, \qquad G_0=E_Q[S_T], \qquad K_0-G_0= \frac{\operatorname{Cov}_Q(D(0,T),S_T)}{E_Q[D(0,T)]}.

A discount factor is smaller when the corresponding accumulated interest rate is higher. Consequently, the sign of the covariance with $S_T$ is not determined by saying simply that interest rates are positively correlated with the asset price: the discount factor moves in the opposite direction from the rate. Always assess the covariance using the variable in the formula, $D(0,T)$.

Worked Stochastic-Rate Comparisons

Worked Example: High Spot Prices Occur with Higher Discount Factors

Consider a one-year underlying asset with no income. In a two-outcome risk-neutral model, each outcome has probability $0.50$. In outcome A, $D(0,1)=0.90$ and $S_1=\$90$; in outcome B, $D(0,1)=1.00$ and $S_1=\$120$. The futures contract has regular daily settlement. Find the time-0 forward and futures prices, and compare them.

Method and assumptions. Use the risk-neutral expectations stipulated by the model. The discount factor $0.90$ corresponds to a one-year effective rate of $1/0.90-1=0.111111$, or about 11.1111%; a discount factor of $1.00$ corresponds to a 0% effective rate. These are uncertain outcomes, so the deterministic-rate equality does not apply automatically.

First compute the expected discount factor:

$$ E_Q[D]=0.50(0.90)+0.50(1.00)=0.45+0.50=0.95. $$

The futures price is the expected terminal spot price:

G_0=E_Q[S_1]=0.50(90)+0.50(120)=45+60=\$105.

For the forward, calculate the discounted expected spot price:

$$ E_Q[DS_1]=0.50(0.90)(90)+0.50(1.00)(120) =40.50+60.00=100.50. $$

Therefore,

K_0=\frac{100.50}{0.95}=\$105.789473\ldots.

The forward price exceeds the futures price by approximately $\$0.78947$. Check the covariance: $E_Q[D]E_Q[S_1]=0.95(105)=99.75$, so $\operatorname{Cov}_Q(D,S_1)=100.50-99.75=0.75$. The adjustment is $0.75/0.95=0.789473\ldots$, agreeing with the direct calculation. Rounded to the nearest cent, $K_0=\$105.79$, $G_0=\$105.00$, and the forward is $\$0.79$ higher. The common mistake avoided is assuming that stochastic rates still imply equality.

Worked Example: High Spot Prices Occur with Lower Discount Factors

For another one-year asset with no income, each of two risk-neutral outcomes has probability $0.50$. In outcome A, $D(0,1)=0.90$ and $S_1=\$120$; in outcome B, $D(0,1)=1.00$ and $S_1=\$90$. Find the forward delivery price and futures price. Assume regular futures settlement and the same idealized funding conditions.

Method and assumptions. Again use the expected terminal spot for the futures price and the discounted expected terminal spot divided by the expected discount factor for the forward. The first outcome's one-year effective rate is $1/0.90-1=11.1111\%$; the second outcome's rate is 0%.

The expected discount factor and expected terminal spot are

E_Q[D]=0.50(0.90)+0.50(1.00)=0.95, \qquad G_0=E_Q[S_1]=0.50(120)+0.50(90)=105.

The discounted expected spot is

$$ E_Q[DS_1]=0.50(0.90)(120)+0.50(1.00)(90) =54+45=99. $$

Thus,

K_0=\frac{99}{0.95}=104.210526\ldots.

The forward price is below the futures price. The covariance confirms the direction: $E_Q[D]E_Q[S_1]=0.95(105)=99.75$, so $\operatorname{Cov}_Q(D,S_1)=99-99.75=-0.75$. Dividing by $0.95$ gives $K_0-G_0=-0.789473\ldots$. Rounded to the nearest cent, $K_0=\$104.21$ and $G_0=\$105.00$. The common mistake avoided is reversing the sign by reasoning directly from the rates instead of checking the discount-factor covariance.

Common Exam Traps and When the Model Changes

The equality of forward and futures prices is a useful result, but only under its assumptions. When rates are deterministic, all settlement cash flows can be accumulated or discounted at known factors. When rates are stochastic, settlement timing matters because a gain received early can earn an uncertain rate, while an early loss must be financed at an uncertain rate. The covariance formula summarizes this effect in the idealized model.

Wrong: “Both contracts deliver the same asset, so their prices must always match.” Delivery at the same maturity does not eliminate the effect of different settlement timing when rates are stochastic. Check whether the discount factor is random and whether its covariance with $S_T$ is zero.
Wrong calculation: $K_0=E_Q[S_T]$ for every forward. Correction: this is the futures expectation under the stated model. For the forward, use $K_0=E_Q[D(0,T)S_T]/E_Q[D(0,T)]$.
Wrong sign test: infer the price difference from the correlation between interest rates and the asset price. Correction: use $\operatorname{Cov}_Q(D(0,T),S_T)$; discount factors move inversely with accumulated rates.
Wrong probability: substitute a real-world average spot forecast into a no-arbitrage formula. Correction: the expectation formulas here use risk-neutral probabilities consistent with the pricing model.
Wrong scope: apply the simple cost-of-carry formula without checking income, storage, funding, or margin assumptions. Correction: this tutorial's formulas exclude those effects; if they are present, the cash flows and pricing model must include them.

In practice and on problems aligned with the SOA Exam FM (Financial Mathematics) syllabus, first classify the interest-rate assumption. If rates are fixed, accumulate using the stated rates and compare the prices directly. If rates are stochastic, identify the measure and settlement assumptions, calculate the expected discount factor and discounted spot, and use the covariance as a sign and arithmetic check.

Check Your Understanding

For each question, state any equality or difference between the forward and futures prices using the assumptions given. Round dollar prices to the nearest cent where needed.

An asset with no income is worth $250. The known effective annual rates are 3% for year one and 5% for year two. Find both two-year prices. (Answer: forward and futures prices are both $270.38.)
In a one-year, two-outcome risk-neutral model, each outcome has probability 0.50. The pairs $(D(0,1),S_1)$ are $(0.96,\$95)$ and $(1.00,\$115)$. Find the forward and futures prices. (Answer: $E[D]=0.98$, $E[DS_1]=\$103.10$, so the forward is $\$105.20$ and the futures price is $\$105.00$.)
The discount factor is equally likely to be 0.92 or 0.98, and the terminal spot is equally likely to be $80 or $120, independently of the discount factor. Find both prices. (Answer: $E[D]=0.95$, $E[S_1]=\$100$, and $E[DS_1]=\$95$, so both prices are $100.00.)
In three risk-neutral outcomes with probabilities $0.25, 0.50, 0.25$, the pairs $(D(0,1),S_1)$ are $(0.80,\$80)$, $(0.90,\$100)$, and $(1.00,\$120)$. Find both prices and state which is higher. (Answer: $E[D]=0.90$, $E[S_1]=\$100$, $E[DS_1]=\$91$; the forward is $101.11 and the futures price is $100.00.)
A model gives a one-year forward price of $102, a futures price of $100, and $E_Q[D(0,1)]=0.96$. Find $\operatorname{Cov}_Q(D(0,1),S_1)$ and interpret its sign. (Answer: $\operatorname{Cov}_Q(D,S_1)=0.96(102-100)=1.92$; it is positive, so the forward exceeds the futures price.)

What You'll Learn