Option Payoffs¶

The previous section defined what options are and introduced the key contract terms. We now turn to the central question: what is an option actually worth at maturity? Imagine watching a single call option through expiration day. As long as the stock sits below the strike, the option is worth nothing — a flat line. The instant the stock crosses the strike, value begins accumulating dollar-for-dollar with the stock. The resulting graph is a kinked, piecewise-linear curve — the famous "hockey stick" — and that shape, not the algebra, is the geometric object at the heart of every later pricing argument.

The answer depends entirely on the relationship between the stock price \(S_T\) and the strike price \(K\) at expiration. The payoff of an option is its value at maturity, before accounting for the premium paid to enter the position. Understanding payoff functions is the first step toward pricing options at any earlier time.

Call Option Payoff¶

The holder of a European call option has the right to buy the underlying asset at the strike price \(K\) at maturity \(T\). If the stock price \(S_T\) exceeds \(K\), the holder exercises and gains \(S_T - K\). If \(S_T \leq K\), exercise is worthless and the holder lets the option expire. The payoff is therefore

which can be written in piecewise form as

The notation \((x)^+ = \max(x, 0)\) will appear throughout this text. It captures the holder's right to walk away: the payoff is never negative.

Put Option Payoff¶

The holder of a European put option has the right to sell the underlying asset at the strike price \(K\) at maturity \(T\). If \(S_T < K\), selling at \(K\) is more favorable than selling at the market price, and the holder gains \(K - S_T\). If \(S_T \geq K\), the holder does not exercise. The payoff is

or equivalently

Note the symmetry: a call profits from upward moves and a put profits from downward moves, but in both cases the payoff is bounded below by zero.

Payoff Diagrams¶

The following diagrams show the characteristic "hockey stick" shapes of call and put payoffs for a strike of \(K = 100\).

Long call payoff: flat at zero for \(S_T \leq K\), then rising linearly.

xychart-beta
    title "Long Call Payoff (K = 100)"
    x-axis "Stock Price at Maturity" [60, 70, 80, 90, 100, 110, 120, 130, 140]
    y-axis "Payoff" 0 --> 40
    line [0, 0, 0, 0, 0, 10, 20, 30, 40]

Long put payoff: linearly decreasing for \(S_T < K\), then flat at zero.

xychart-beta
    title "Long Put Payoff (K = 100)"
    x-axis "Stock Price at Maturity" [60, 70, 80, 90, 100, 110, 120, 130, 140]
    y-axis "Payoff" 0 --> 40
    line [40, 30, 20, 10, 0, 0, 0, 0, 0]

Payoff Profiles¶

The following table shows the shape of call and put payoffs across a range of stock prices, for a strike of \(K = 100\):

The call payoff is a "hockey stick" shape: flat at zero for \(S_T \leq K\), then rising linearly with slope 1 for \(S_T > K\). The put payoff is the mirror image: linearly decreasing for \(S_T < K\), then flat at zero. The kink at \(S_T = K\) is the defining geometric feature of option payoffs and the source of the nonlinearity that makes pricing nontrivial.

Long and Short Positions¶

Every option trade has two sides. The long party is the buyer (holder) of the option; the short party is the seller (writer). Their payoffs are mirror images:

The writer's payoff is the negative of the holder's payoff. The option market is zero-sum at expiration: every dollar gained by the holder is lost by the writer, and vice versa. This is why the writer demands compensation upfront — the premium — in exchange for accepting this obligation. The same long/short identity propagates into profit-and-loss profiles for combined strategies (see § The Four Elementary Positions).

Numeric Example¶

Consider a European call and a European put, both with strike \(K = 100\).

Suppose \(S_T = 118\) at maturity:

• Call payoff (long): \((118 - 100)^+ = \$18\). The holder buys at $100 and immediately has an asset worth $118.
• Put payoff (long): \((100 - 118)^+ = \$0\). The holder would not sell at $100 what is worth $118.
• Call payoff (short): \(-\$18\). The writer must deliver the asset at $100, losing $18.

Now suppose instead \(S_T = 87\):

• Call payoff (long): \((87 - 100)^+ = \$0\). No exercise.
• Put payoff (long): \((100 - 87)^+ = \$13\). The holder sells at $100 an asset worth only $87.
• Put payoff (short): \(-\$13\). The writer must buy at $100 what is worth $87.

These payoffs represent value at expiration only. To determine whether a trade was profitable overall, one must subtract the premium paid (for the long) or add the premium received (for the short). This raises a natural question: if the payoff depends on the unknown future price \(S_T\), how should the premium be determined today? This is the central problem of option pricing, and we turn to it next.

Exercises¶

Exercise 1. A European call option has strike \(K = 75\). Compute the call payoff at maturity for each of the following stock prices: (a) \(S_T = 60\), (b) \(S_T = 75\), (c) \(S_T = 90\).

Exercise 2. A European put option has strike \(K = 50\). The holder paid a premium of $4. (a) Compute the payoff and profit when \(S_T = 38\). (b) Find the stock price \(S_T^*\) at which the holder breaks even (profit \(= 0\)).

Exercise 3. A trader writes (sells) a European call with strike \(K = 200\) and receives a premium of $15. (a) Write the writer's profit as a function of \(S_T\). (b) What is the maximum profit? (c) Is there a maximum loss? Explain.

Exercise 4. Show that for any stock price \(S_T \geq 0\) and strike \(K > 0\), the call and put payoffs satisfy

Exercise 5. A trader holds a long call and a short put, both European with the same strike \(K\) and maturity \(T\). Using the identity from Exercise 4, show that the combined payoff at maturity equals \(S_T - K\). Why does this mean the combined position behaves like a forward contract?