Forward And Futures Pricing With Deterministic Rates

Why Compare Forward and Futures Prices?

A forward contract fixes an exchange at a future date, with the contract value typically settled at that date. A futures contract also fixes a future exchange price, but it is marked to market: gains and losses are settled as they arise. That difference in settlement timing can matter because cash received early can earn interest, while cash paid early must be funded.

When interest rates are deterministic, the rates applicable to future settlement cash flows are known in advance. Under standard frictionless pricing assumptions, this removes the uncertainty that can otherwise make daily settlement affect the fair futures price. The fair forward and futures prices are then equal for contracts on the same underlying asset with the same maturity and delivery terms. Their cash flows along a particular realized path need not be equal.

Definition: A deterministic interest-rate structure is one in which the accumulation and discount factors for each future date are known today. Rates may vary from period to period; they need not be constant. A fair forward delivery price is the delivery price that gives a newly entered forward contract value zero. A fair futures price is the quoted futures price at which a newly entered futures position has value zero, before any required collateral.

The equality relies on matching contract terms and on idealized pricing assumptions: no taxes, transaction costs, default, or restrictions on borrowing, lending, or investing settlement cash flows. Assume also that the underlying asset is available for delivery and that any income or carrying costs are either absent or known and treated consistently. The result is not a claim that the contracts have identical settlement cash flows.

Common Price for a No-Income Asset

Let $S_0$ be the spot price of an asset that pays no income before delivery at time $T$. Write $D(0,T)$ for the known discount factor from time $T$ to time 0, and $A(0,T)=1/D(0,T)$ for the corresponding accumulation factor. If the effective annual rates for successive periods are $i_1,\ldots,i_n$, then $A(0,T)=\prod_{k=1}^{n}(1+i_k)$.

The forward price follows from cash-and-carry: buy the asset now and borrow its cost until delivery. The amount of the borrowing due at $T$ is $S_0A(0,T)$. Alternatively, a short forward position delivers the asset at $T$ and receives the delivery price. To avoid an arbitrage at inception, the delivery price must match the financed purchase cost.

K_{\mathrm{forward}}=S_0A(0,T)=\frac{S_0}{D(0,T)}

With a flat effective annual rate $i$ and a term of $n$ years, $A(0,T)=(1+i)^n$, or equivalently $D(0,T)=v^n$, where $v=1/(1+i)$. With a changing deterministic curve, use the accumulation factors for the actual periods rather than applying one annual rate to the whole term.

Worked Example: A Forward and Futures Price on a Changing Rate Curve

An asset with spot price $250 pays no income. Delivery is in three years. The known effective annual rates are 4% for the first year, 5% for the second year, and 6% for the third year. Find the fair forward delivery price and the fair futures price.

Method and assumptions. Use cash-and-carry for the forward price, then apply the deterministic-rate equality to the futures price. Assume the asset is deliverable, there are no transaction costs or other carrying costs, and the stated rates apply to each successive year.

Step 1: Accumulate the spot cost. The three-year accumulation factor is $1.04\times1.05\times1.06=1.15752$.

Step 2: Calculate the forward price. $250\times1.15752=\$289.38$.

Step 3: Calculate the futures price. Since the rates are deterministic and the contract terms match, the fair futures price is also $\$289.38$.

Check. Discounting the delivery amount gives $\$289.38/1.15752=\$250$, the current spot price. A common mistake is to use only the last rate, 6%, for all three years; that would ignore the stated first- and second-year rates.

Why Daily Settlement Does Not Change the Fair Price

A useful justification uses risk-neutral pricing. In this framework, the expected present value of a traded contract’s cash flows is zero at a fair price. Let $Q$ denote the risk-neutral probability measure, and let $F_j$ be the futures price just after settlement at date $t_j$. A long futures position receives the variation margin $F_j-F_{j-1}$ at that date. Its time-0 value is the sum of the discounted expected settlement amounts.

For a no-income asset, when rates are deterministic, the futures price at a settlement date is the conditional risk-neutral expectation of the delivery-date spot price. This conditional expectation property makes the futures-price sequence a martingale under $Q$: each next price has conditional expectation equal to the current price. Consequently each settlement increment has conditional expectation zero. Since the discount factors are known, discounting and taking expectations does not introduce an additional rate uncertainty.

F_j=E_Q[S_T\mid\mathcal{F}_{t_j}], \qquad E_Q[F_j-F_{j-1}\mid\mathcal{F}_{t_{j-1}}]=0

At expiration, $F_n=S_T$. Summing the discounted expected futures settlement increments gives zero, consistent with a newly entered futures position having zero value. The forward’s value is also zero when its delivery price equals the risk-neutral expected spot price at $T$, expressed with the same deterministic discounting. More directly, deterministic rates make the money-market and $T$-forward probability measures equivalent for this pricing comparison, so both contracts use the same expected delivery value. Thus their fair initial prices agree.

K_{\mathrm{forward}}=F_0

The key condition in this argument is that interest rates are known. If rates are stochastic, the amount earned or paid on early futures settlements is uncertain and may be related to changes in the underlying price. The futures and forward prices can then differ; the deterministic-rate equality should not be applied automatically.

Worked Example: Known Income Before Delivery

A share costs $100 and will pay a known cash dividend of $4 at the end of year 1. A contract delivers the share at the end of year 2. The effective annual rate is 5% in both years. Find the fair forward and futures prices.

Method and assumptions. Subtract the present value of the known income from the spot cost, then accumulate the net cost to delivery. Assume the dividend amount and payment date are certain, and that the same share and delivery date apply to both contracts.

Step 1: Discount the dividend. Its time-0 present value is $\$4/1.05=\$3.8095238$.

Step 2: Find the net spot cost. $\$100-\$3.8095238=\$96.1904762$.

Step 3: Accumulate for two years. The accumulation factor is $1.05^2=1.1025$. Therefore the forward price is $\$96.1904762\times1.1025=\$106.05$.

Step 4: Apply the equality. The fair futures price is also $\$106.05$, since the rates are deterministic and the income is known and treated consistently.

Check. Discounting the delivery price and adding the dividend’s present value gives $\$106.05/1.1025+\$4/1.05=\$96.1904762+\$3.8095238=\$100$. A common mistake is to ignore the dividend and quote $\$100\times1.1025=\$110.25$.

Equal Prices Do Not Mean Equal Realized Cash Flows

Daily settlement changes when cash is received or paid. The following example isolates that timing effect. It uses risk-neutral probabilities to check fair prices, not to claim that the two contracts produce identical outcomes in each state.

Worked Example: Settlement Timing Along Two Possible Paths

A non-income asset costs $100 today. The effective annual rate is 10%. A two-year model has two equally likely risk-neutral paths. At year 1 the spot price is either $120 or $100; with no income, the corresponding year-2 prices are $132 and $110. Find the fair forward and futures prices, then compare their cash flows on each path.

Method and assumptions. First calculate the risk-neutral expected delivery price. Then track the futures settlement at year 1 and accumulate that cash flow to year 2 at the known 10% rate. Assume settlement occurs at year 1 and at expiration, with no other intervening price changes.

Step 1: Find the expected year-2 spot price. $(\$132+\$110)/2=\$121$.

Step 2: Check consistency with the spot price. The present value is $\$121/1.1^2=\$121/1.21=\$100$. Thus the fair forward delivery price is $\$121$. Under deterministic rates, the fair initial futures price is also $\$121$.

Step 3: Calculate year-1 futures prices and settlements. On the path with year-1 spot $120, the year-1 futures price is $\$120\times1.1=\$132$, so the long receives $\$132-\$121=\$11$. On the path with year-1 spot $100, the year-1 futures price is $\$100\times1.1=\$110$, so the long pays $\$110-\$121=-\$11$.

Step 4: Accumulate each settlement to year 2. The year-1 cash flows become $\$11\times1.1=\$12.10$ and $-\$11\times1.1=-\$12.10$. The year-2 forward payoffs are, respectively, $\$132-\$121=\$11$ and $\$110-\$121=-\$11$.

Check. Each contract’s risk-neutral expected value is zero at its fair initial price: the forward payoff averages $(\$11-\$11)/2=0$, and the accumulated futures settlement averages $(\$12.10-\$12.10)/2=0$. The cash flows differ on each path, but not in expected value. The common mistake is to require the accumulated futures settlement to equal the forward payoff state by state; fair-price equality does not assert that.

Common Exam Traps

When comparing these contracts, separate the initial fair quote from the value or realized cash flows of a position. The equality is a pricing result under stated assumptions, not a statement that settlement mechanics are interchangeable in every calculation.

Using a single rate when the curve varies. Accumulate each period at its own known rate, or use the given term discount factor.
Forgetting known income. For a no-arbitrage delivery price, known income received before delivery reduces the net cost of holding the asset.
Equating realized cash flows. A futures gain or loss may arrive before expiration and earn or incur interest. It need not match the forward payoff on a particular path.
Applying the equality with stochastic rates. If future funding rates are uncertain, the settlement timing can affect pricing. Deterministic-rate reasoning alone is insufficient.
Confusing price with contract value. The fair price is the quote for a new contract. The value of an existing contract depends on its agreed delivery price and current market conditions.

Key takeaway: With deterministic rates, matching terms, and consistent treatment of income and costs, a forward and futures contract have the same fair initial price. Daily marking to market changes the timing of cash flows, so it does not make the realized gains and losses identical.

Check Your Understanding

For each problem, assume frictionless markets and deterministic rates unless the question says otherwise.

A no-income asset costs $150. Delivery is in two years, and the effective annual rate is 4% in each year. Find the fair forward and futures prices. (Answer: $162.24 for each.)
A no-income asset costs $80. The effective annual rate is 3% for the first year and 7% for the second. Find the fair delivery price for year 2. (Answer: $88.168 for both the forward and futures price.)
A share costs $200 and pays known income of $6 at the end of year 1. The effective annual rate is 5% in each of the two years to delivery. Find the common fair price. (Answer: $214.20.)
A no-income asset costs $300. The effective annual rates over the next three years are 2%, 4%, and 5%. Find the common fair price for delivery at the end of year 3. (Answer: $334.152.)
An asset costs $500 and pays known income of $10 at the end of year 1 and $15 at the end of year 2. Delivery is at the end of year 2. The effective rates are 4% in year 1 and 6% in year 2. Find the common fair price. (Answer: $525.60.)

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