Show how the absence of arbitrage fixes a forward price
The idea
The price of a forward contract is set by the cost of replicating it, not by a forecast of where the underlying will trade. Buying the asset today with borrowed money and holding it to the delivery date produces exactly what the forward delivers, so the forward price must equal the spot price compounded at the financing rate, adjusted for anything the asset pays or costs while held. Income received while holding lowers the forward price; storage costs raise it. If the quoted price departs from that, the arbitrage is mechanical - buy the cheap side, sell the dear one, and the two positions cancel at delivery leaving the difference.
Formulas
Forward price of an asset with no cash flows
F₀ = S₀ × (1 + r)ᵗ
- S₀
- the spot price today
- r
- the risk-free rate
- t
- the years to delivery
- F₀
- the price agreed today for delivery then
Not a forecast of the spot price. It is what stops an arbitrage: borrow, buy the asset, hold it, deliver it. Any other forward price lets somebody do that for free money.
A practice question
The price of a forward contract is determined principally by:
- the cost of replicating the position today - correct
- a forecast of the future spot priceWhy not: The most common misconception here. The forward price comes from carrying costs, and arbitrage enforces it whatever anyone expects.
- the historical average priceWhy not: History has no role; the price is what makes a replicating trade break even from today.
A forward price is set by what it would cost to get the same result by trading today: borrow money, buy the asset now and hold it until delivery. For example, if the asset costs $100 and the borrowing rate is 5%, buying now and holding for a year costs $105 at delivery, so a forward price of $105 makes both routes equal. No forecast of the future price is needed. If the forward were priced differently, traders could make a risk-free profit, which pushes it back to this level.
A harder one
Spot is $100 and the one-year financing rate is 5%, with no income or storage. The one-year forward price is:
- $95Why not: Subtracts the financing cost. The buyer defers payment for a year, so the price rises rather than falls.
- $100Why not: The spot price, which would be the forward only at a zero financing rate.
- $105 - correct
The spot price is today's price. To get the forward price, imagine buying the asset now with borrowed money. After a year you would owe the loan plus interest. Forward price = Spot x (1 + r)^T = 100 x 1.05 = $105. The forward must be priced at this same figure, or traders could make a risk-free profit. It cannot be lower than spot, because the buyer is delaying payment for a year, and that delay is worth the interest.
The hardest tier
Describing the forward price as the market's forecast of the future spot price is:
- correct, since arbitrage forces it to equal the expected spot priceWhy not: Arbitrage works on the cost of replicating the position, which is known today, so no forecast is needed.
- partly correct, since expectations set it and carry costs adjust itWhy not: Expectations play no part: the forward is fixed by financing, storage and income, whatever the market expects.
- mistaken, since it is set by the cost of carry, not by expectations - correct
People often think the forward price is a prediction of the future price, but it is not. It comes from the cost of copying the position today: borrow, buy the asset and hold it. Forward price = Spot x (1 + r)^T, adjusted for income and storage. Even if every investor expects the price to double, arbitrage keeps the forward at this level. For example, a $100 asset with 5% financing has a one-year forward of $105 whether the market expects $90 or $130 in a year.
In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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