Options & Payoffs

A financial derivative is a contract whose value derives from something else — an underlying asset price S (a stock, an index, a currency). The contract specifies a payoff: a known function of the underlying that is paid out at a fixed future date, the maturity T. Nothing else about the contract matters for its terminal value — only the number S_T that the underlying happens to print at time T.

The three workhorse contracts are the call, the put, and the forward. Each is pinned down by two numbers: the strike K (a reference price written into the contract) and the maturity T. The whole of this lesson is reading off, by cases, what each one pays.

The call payoff, by cases

A European call gives its holder the right, not the obligation, to buy one unit of the underlying at the strike K at maturity T. A rational holder exercises that right only when it is worth something. Split on whether the option finishes in the money (S_T > K) or out of the money (S_T \le K).

Step 1 — value the right when S_T > K. The holder buys at K and immediately sells at the market price S_T, banking the difference:

\text{payoff} = S_T - K \quad (> 0).

Step 2 — value the right when S_T \le K. Buying at K something worth only S_T \le K would lose money, so the holder simply walks away and the option expires worthless:

\text{payoff} = 0.

Step 3 — fuse the two cases. The contract pays the larger of S_T - K and 0, which is exactly the positive part (\cdot)^+ = \max(\cdot,\,0):

C_T = \max(S_T - K,\; 0) = (S_T - K)^+.

That single expression encodes both branches — a function of S_T alone, kinked at the strike. Plotted against S_T it is the famous hockey stick: flat on the floor up to K, then rising at slope 1.

The put payoff, by cases

A European put is the mirror image: the right to sell one unit at the strike K at maturity. Now the holder profits when the price has fallen below the strike.

Step 1 — when S_T < K: sell at the high strike K something worth only S_T on the market, pocketing the gap:

\text{payoff} = K - S_T \quad (> 0).

Step 2 — when S_T \ge K: selling below the market price is a loss, so the holder declines and the put expires worthless:

\text{payoff} = 0.

Step 3 — fuse. Again the positive part collapses both branches:

P_T = \max(K - S_T,\; 0) = (K - S_T)^+.

Same hockey stick, reflected: it slopes down at slope -1 from a peak of K at S_T = 0, hits the floor at S_T = K, and stays there.

The forward: an obligation, not a right

A forward contract is the simplest of the three. It is an obligation — there is no "walk away" — to buy the underlying at K at maturity. The holder must take delivery at K and can sell at S_T whatever happens:

F_T = S_T - K,

with no positive part anywhere. Because there is no choice, the payoff is the straight line S_T - K through the point (K, 0) — it goes negative when S_T < K. The forward is genuinely the difference of a call and a put at the same strike, a fact we will exploit in put–call parity: (S_T - K)^+ - (K - S_T)^+ = S_T - K.

For strike K and maturity T, the terminal payoff is a function of the underlying price S_T alone:

European call: C_T = (S_T - K)^+ = \max(S_T - K,\,0) — the right to buy.
European put: P_T = (K - S_T)^+ = \max(K - S_T,\,0) — the right to sell.
Forward: F_T = S_T - K — the obligation to buy (may be negative).

Every contract has two sides. The long holds the right or obligation as stated; the short (the writer) takes the opposite payoff, -(S_T - K)^+ for a written call. The long can never lose more than the premium paid; the short of a call has unbounded downside as S_T \to \infty — which is why writing naked calls keeps risk managers awake.

Before maturity a call's price splits into two parts. Its intrinsic value is what it would pay if exercised now, (S_t - K)^+; the rest is time value, the premium for the chance that the price moves favourably before T. Time value is largest for an at-the-money option and decays to zero as t \to T, when only intrinsic value survives and the price collapses onto the hockey stick. Pricing that time value — the area between the smooth pre-maturity curve and the kinked payoff — is precisely what the Black–Scholes formula will compute.

Calls, puts, and forwards are the vanilla derivatives — their payoff depends only on the final price S_T. Exotic options break that simplicity by depending on the whole path (S_t)_{0 \le t \le T}:

A barrier option springs into or out of existence if the price ever touches a level H — e.g. a knock-out call pays (S_T - K)^+ only if S_t < H for all t.
An Asian option pays off on the average price, \big(\bar{S} - K\big)^+ with \bar S = \tfrac1T\int_0^T S_t\,dt, smoothing out end-date manipulation.

Path-dependence makes exotics far harder to price, but the no-arbitrage machinery we build next handles them all — the payoff is just a more elaborate function of the path.

Drag the strike, watch the kink move

Below are the two hockey sticks plotted against the terminal price S_T: the call (S_T - K)^+ rising to the right of the strike, and the put (K - S_T)^+ falling away to the left. Slide K and watch both kinks slide together — the strike is the hinge of every option payoff.