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.
Borrowing to buy the asset now delivers exactly what the forward delivers, so the two must cost the same.
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
Buying now with borrowed money costs 105 to repay in a year, so the forward must be priced at the same figure.
The hardest tier
Spot is $50, the financing rate is 4% and the asset pays $1 before delivery in one year. The forward price is closest to:
- $53.00Why not: Adds the $1 income instead of subtracting it, on top of the $2 financing.
- $51.00 — correct
- $49.00Why not: Subtracts the income and the financing. Financing raises the forward price; only the income lowers it.
Compounding 50 at 4% gives 52, and the dollar of income the forward buyer forgoes reduces it to about 51.
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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