A forward contract obligates the long side to buy at the delivery price K, no matter what the spot price does. Turning that sentence into a formula for exactly how much money changes hands is the whole content of this lesson — and, plotted, it's the simplest payoff diagram in all of derivatives: a single straight line.

Deriving the payoff, by cases (there's only one case)

Take the long side first. At maturity T the long is obligated to buy the asset for K, and that asset is worth S_T on the open market. If they wanted to, the long could immediately sell what they just bought at the market price and pocket the difference:

Compare that with a call option's payoff, (S_T - K)^+: the call needed a case split, because the holder could always walk away when S_T < K. The forward's long has no such escape — the formula is S_T - K in every case, full stop, with no \max(\cdot, 0) clamping it at zero. When S_T < K that payoff is simply negative: the long loses money, for real, exactly as if they'd bought high and sold low.

The short side is obligated to sell at K an asset worth S_T on the market — the mirror image, and (since it's a zero-sum bilateral contract) the exact negative of the long's payoff:

This is what "symmetric risk" means for a forward: the long gains a dollar for every dollar S_T rises above K, and loses a dollar for every dollar it falls below — in both directions, without limit. A long option position, by contrast, has a floor: its worst case is losing the premium it paid, full stop. That missing floor is the single biggest structural difference between a forward/futures position and an option position, and it's why margin (Lesson 4) exists at all — an exchange needs collateral against a loss that, in principle, has no bound.

Worked example: pricing the jet-fuel hedge exactly

Return to the airline from Lesson 1: long a six-month forward on N = 2{,}000{,}000 gallons of jet fuel at K = \$2.68. The per-gallon payoff formula is S_T - K; multiply by the notional N to get the total dollar payoff:

With S_T = \$3.10: 2{,}000{,}000 \times (3.10 - 2.68) = 2{,}000{,}000 \times 0.42 = \$840{,}000 — exactly the saving computed by hand in Lesson 1, now read straight off a single linear formula. With S_T = \$2.20: 2{,}000{,}000 \times (2.20 - 2.68) = 2{,}000{,}000 \times (-0.48) = -\$960{,}000 — a loss, because the formula doesn't clamp at zero the way an option's would.

Straight lines, not hockey sticks

Plot payoff against the terminal price S_T and the contrast with an option is immediate. The long forward (rising line) and short forward (falling line) are straight all the way through — they cross zero exactly at S_T = K and keep going in both directions. The faint dashed curve is a long call's payoff (S_T-K)^+ at the same strike, included only for contrast: it's flat (zero) below K and only turns upward beyond it. Drag K and watch all three pivot around the same point on the axis.