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Lesson 1 of 5 · 11 min

Pricing a futures contract at inception

At inception a futures contract is worth zero and its price is set by the same carry arithmetic as a forward: the spot price, adjusted for income and costs, grown at the risk-free rate to maturity.

Video lesson · 5 min. Watch it first, then use the text below to go deeper.

In short

A futures contract is a standardised, exchange-traded forward commitment. Like a forward, its value at inception is zero: $V_0(T) = 0$ .
With no costs or benefits of holding the underlying, the futures price is $f_0(T) = S_0(1+r)^T$ , identical to the forward price $F_0(T)$ .
For indexes (equity, bond, commodity, credit) and FX, continuous compounding is preferred: $f_0(T) = S_0e^{rT}$ .
With income I and costs C in present value terms: $f_0(T) = [S_0 - PV_0(I) + PV_0(C)](1+r)^T$ . Costs raise the futures price; benefits lower it.
A futures price well below the level implied by cash costs and benefits hints at a convenience yield.

Same starting point as a forward

A futures contract is a forward commitment that trades on an exchange, with standard sizes and maturities and a clearinghouse standing between buyer and seller. At inception, no money changes hands for the contract itself (margin is a deposit, not a price), so neither party holds an asset or a liability: $V_0(T) = 0$ .

The price that achieves this is fixed by the same no-arbitrage argument used for forwards. Buying the underlying today and holding it to T costs $S_0$ plus the interest forgone on that cash. With no other costs or benefits, the futures price is $f_0(T) = S_0(1+r)^T$ .

So at inception, a futures price $f_0(T)$ and a forward price $F_0(T)$ on the same underlying with the same maturity are the same number. The differences between the two contracts appear only later, through the way gains and losses are settled.

Silver at USD 30.00 per ounce with r = 3% and no storage cost. The no-arbitrage futures price starts at the spot price and rises with maturity at the risk-free rate: the longer the contract, the more interest the buyer saves by not paying for the metal today.

Discrete or continuous compounding

For a single asset, such as a share or an ounce of a metal, use discrete compounding: $f_0(T) = S_0(1+r)^T$ with T in years.

When the underlying is a portfolio, such as an equity, bond, commodity or credit index, or when two currencies and two interest rates are involved, as with FX futures, continuous compounding is the norm: $f_0(T) = S_0e^{rT}$ , where r is a continuously compounded rate.

The two give slightly different numbers for the same quoted rate. For an index at 5,000 with r = 3% and three months to maturity, $5{,}000e^{0.03 \times 0.25}$ = 5,037.64, while $5{,}000 \times 1.03^{0.25}$ = 5,037.09. On the exam, use the convention the question gives you.

Adding costs and benefits of carry

Holding the underlying can generate income or benefits (I), such as dividends or coupons, and can create costs (C), such as storage and insurance. Expressed as present values at t = 0, they adjust the spot price before it is grown to maturity: $f_0(T) = [S_0 - PV_0(I) + PV_0(C)](1+r)^T$ .

Benefits lower the futures price: the futures buyer does not receive the dividend, so it should pay less. Costs raise it: the futures buyer avoids paying for storage, so the seller, who carries the asset, must be compensated through a higher price. When costs exceed benefits the futures price sits above the no-carry price; when benefits exceed costs it sits below.

If a commodity futures price is well below even the level that cash costs and benefits explain, holders of the physical commodity may be enjoying a convenience yield: a non-cash benefit of having the commodity on hand, for example to keep a plant running during a shortage.

Carry situation	Futures price versus no-carry price
No costs or benefits	$S_0(1+r)^T$
Costs > benefits	Higher
Benefits > costs	Lower
Far below even with benefits	Suggests a convenience yield

Aluminium at USD 2,400 per tonne, r = 4%, three months to maturity. Without carry costs the futures price is 2,423.65; a USD 15 storage cost payable at maturity adds exactly 15 more, to 2,438.65. Costs push the futures price further above spot.

Formulas

Value at inception: $V_0(T) = 0$
Holds for futures and forwards alike when the price is set at its no-arbitrage level.
Futures price, no costs or benefits: $f_0(T) = S_0(1+r)^T$
Discrete compounding, used for single assets. Same number as the forward price F₀(T).
Futures price, continuous compounding: $f_0(T) = S_0e^{rT}$
Preferred for index underlyings and FX; r is continuously compounded.
Futures price with income and costs: $f_0(T) = [S_0 - PV_0(I) + PV_0(C)](1+r)^T$
PV₀(I) and PV₀(C) are present values at t = 0 of income and costs paid during the contract's life.

Worked examples

Example 1 Silver futures with no carry

Silver trades at USD 30.00 per ounce and can be stored at no cost. The risk-free rate is 3%. What is the no-arbitrage six-month futures price, and what is a 5,000-ounce contract worth in price terms?

Example 2 Storage cost and dividend income

(1) Aluminium trades at USD 2,400 per tonne, r = 4%, and storage of USD 15 per tonne is payable in three months. What is the three-month futures price? (2) A share trades at EUR 80.00 and pays a EUR 1.20 dividend in six months. With r = 2.5%, what is the one-year futures price?

Example 3 Reading a low futures price

Copper futures trade well below the price implied by spot, the risk-free rate and storage costs, and no arbitrageur steps in. What is the most likely explanation?

On the exam

Expect a calculation of $f_0(T)$ with a dividend or storage cost, sometimes with continuous compounding for an index, and statements that futures and forwards have the same price and zero value at inception.

Adding the future value of income instead of subtracting its present value (or forgetting to discount it at all).
Thinking the initial margin is the price of a futures contract. The contract's value at inception is still zero.
Using discrete compounding when the question gives a continuously compounded rate for an index or FX contract.

Check yourself

Q1A share trades at CHF 120 and will pay a CHF 3 dividend at the futures maturity in one year. The risk-free rate is 2%. The one-year futures price is closest to:
Q2At inception, the value of a futures contract priced at its no-arbitrage level is:
Q3An equity index is at 4,000 and the continuously compounded risk-free rate is 2.5%. The six-month index futures price is closest to:
Q4Compared with an otherwise identical commodity that is free to store, a storage cost payable during the contract's life will most likely make the futures price:
Q5Gas oil futures trade far below the no-arbitrage price based on spot, the risk-free rate and storage costs. This most likely indicates:

0/5Check yourself

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