Forwards & Futures

Lock a price today with no forecast: the fair forward is the cost of carry.

The idea

A forward contract fixes today the price of a trade that happens later. The two sides agree now that at time $T$ one will hand over one unit of an asset and the other will pay a delivery price $F$, set at signing and unchangeable. No money moves when the contract is written; it is a promise, not a purchase. And it binds both sides: each must trade at $F$ however the asset's price has moved by then.

Finding the fair $F$ requires no forecast. Anyone who wants the asset at time $T$ has a second route to it: borrow the spot price $S$ today, buy the asset immediately, and hold it. That route delivers the same unit on the same day and costs $S(1+r)^{T}$ then — the loan, grown at the risk-free rate. Two ways of obtaining the same thing at the same moment must cost the same, or an arbitrage exists, so

$F = S(1+r)^{T}.$

This is the cost-of-carry price: the spot price plus the cost of financing the wait. The asset's price at time $T$ appears nowhere in it, because the carrier owns the unit throughout and is bound to deliver it, so its value on delivery day changes nothing. With interest quoted continuously the same argument gives $F = Se^{rT}$; only the growth factor changes.

Ways to work on it

• Walkthrough. Derive F = S(1+r)^T from the carry arbitrage, run it in both directions, and read off the expiry payoff.
• Practice. Compute fair delivery prices, arbitrage profits from off-market quotes, and payoffs at expiry.
• Hardest. Price a forward on an asset paying a known income, or back out the implied financing rate.

Not sure where to start? Take the ten-question placement test.