Option Payoff

Option payoff is the value an option delivers to its holder at expiry, as a function of the price of the underlying asset at that time. For a call it is the amount by which the underlying exceeds the strike price, or nothing. For a put it is the amount by which the strike exceeds the underlying, or nothing. Subtracting the premium paid gives the profit or loss on the position.

Payoffs are the simplest way to describe what an option, or a combination of options, does. They are drawn as payoff diagrams, with the final price of the underlying on the horizontal axis and the gain or loss on the vertical axis. Their bent, hockey-stick shape is what distinguishes options from shares and from a futures contract, whose payoffs are straight lines.

For the holder of a call option, the payoff at expiry is:

For the holder of a put option, it is:

• $S_T$ is the price of the underlying at expiry.
• $K$ is the strike price.
• $\max$ takes the larger of the two values.

The profit is the payoff less the option premium paid at the start. The seller, or writer, of the option has the mirror image: the writer keeps the premium and pays out whatever the holder receives. Ignoring costs, options are a zero-sum game between the two sides.

A trader buys a call with a strike of 100 for a premium of 4. If the share ends at 110, the payoff is $110 - 100 = 10$ and the profit is $10 - 4 = 6$. If it ends at 102, the payoff is 2 and the result is a loss of 2. If it ends at or below 100, the option expires worthless and the loss is the full premium of 4. The position breaks even at $100 + 4 = 104$.

A trader buys a put with a strike of 100 for a premium of 3. If the share ends at 90, the payoff is $100 - 90 = 10$ and the profit is 7. If it ends at or above 100, the loss is the premium of 3. The break-even point is $100 - 3 = 97$. For US equity options each contract covers 100 shares, so these figures are multiplied by 100.

• Long call. The loss is limited to the premium, and the gain is unlimited as the underlying rises.
• Short call. The gain is limited to the premium, and the loss is unlimited as the underlying rises, unless the writer owns the shares, as in a covered call.
• Long put. The loss is limited to the premium, and the gain grows as the underlying falls, up to the strike less the premium if the price reaches zero.
• Short put. The gain is limited to the premium, and the loss grows as the underlying falls.

The asymmetry is the point of the instrument. A buyer pays a known amount for a payoff that can be many times larger. A seller receives a known amount and accepts an uncertain, possibly large, liability.

Because payoffs at expiry simply add, complex positions can be analysed by summing the payoffs of their parts. Buying a call and a put at the same strike gives a straddle, which pays off if the underlying moves far in either direction. Buying one option and selling another at a different strike gives an option spread, which limits both gain and loss. Four options can be arranged into an iron condor or a butterfly spread, which profit if the underlying stays in a range.

The same arithmetic shows that options can replicate other instruments. A long call combined with a short put at the same strike and expiry has the payoff $S_T - K$, which is the payoff of a forward purchase at $K$. This relationship, known as put-call parity, ties the prices of calls, puts and the underlying together, and any deviation large enough to cover costs is removed by arbitrage.

A payoff diagram shows only the position at expiry. Before then, the option is worth more than its payoff line suggests because it still has time value, and its price responds to volatility and the passage of time as well as to the underlying. These effects are described by the option Greeks. With an American option, the holder may exercise early, so the writer's obligation can arrive before the expiry date.

The diagram also says nothing about likelihood. A position may show a large potential gain over a range of prices the underlying is very unlikely to reach, and a small gain over the range it will probably occupy. Assessing a trade requires weighting the payoff by the probability of each outcome, which is what pricing models and measures such as probability of profit attempt. Transaction costs, margin requirements for short positions and, for physically settled options, the cost of handling the delivered shares are further practical matters outside the diagram.