9.3 Option Payoffs, Moneyness, and Risk

Option contract rights

An option gives its buyer a right and its seller (the writer) an obligation. A call option grants the right to buy the underlying at the exercise (strike) price, X. A put option grants the right to sell at X. The buyer pays a premium for the right; the writer receives the premium and accepts the matching obligation, so the writer may have to sell (short call) or buy (short put) at X.

American-style options can be exercised any time up to expiration; European-style options only at expiration. Level I usually focuses on expiration payoffs, where exercise style matters less than the payoff formula itself.

Payoff versus profit

This distinction prevents many exam errors. Payoff is the value at exercise or expiration; profit nets the premium. For a call, payoff = max(0, S - X). For a put, payoff = max(0, X - S). The buyer subtracts the premium for profit; the writer adds the premium received.

Worked example: a call with X = 50 and the underlying at 58 has payoff max(0, 58 - 50) = 8; if the premium was 3, the buyer's profit is 5. If the underlying is 47 at expiration, payoff = 0 and the buyer loses only the 3 premium. The breakeven for a long call is X + premium = 53.

Moneyness

Moneyness compares the underlying price S with the exercise price X. A call is in-the-money (ITM) when S > X, at-the-money (ATM) when S = X, and out-of-the-money (OTM) when S < X. For a put, the inequalities reverse. An OTM option still has value because future price movement may turn it profitable.

Intrinsic value is the value from immediate exercise (the payoff at the current S); time value is option value beyond intrinsic value, reflecting the chance of favorable movement before expiration. An OTM option has zero intrinsic value but positive time value. Higher expected volatility raises option value because the buyer captures favorable moves while downside is capped at the premium. Longer time to expiration and (for calls) a higher risk-free rate also raise value; income such as dividends on the underlying lowers call value and raises put value.

Common option strategies

A protective put = long underlying + long put. It preserves upside above the current price while limiting downside below X, after the premium, like insurance on a stock. A covered call = long underlying + short call. It earns premium income but caps upside beyond X; it fits a neutral to mildly bullish view and offers only premium-sized cushion against a large decline. A naked long call gives leveraged upside with loss limited to premium; a naked long put gives bearish exposure or protection with limited loss; writing options collects premium but transfers the favorable optionality to the buyer.

Structured aid: option question workflow

1. Identify call or put.
2. Identify long (buyer) or short (writer).
3. Compare S with X to classify moneyness.
4. Compute payoff with max(0, S - X) or max(0, X - S).
5. Net the premium to get profit if asked.
6. Link the payoff shape to the investor's objective.

The six determinants of option value

Level I expects you to know the direction each input moves an option's value, even without computing a price. Summarize them and the sign of their effect:

Volatility is the only input that raises both call and put value, because greater dispersion increases the chance of a large favorable move while the holder's loss stays capped at the premium. A higher risk-free rate raises call value (the call defers paying the exercise price, so the deferred cash earns more interest) and lowers put value. Dividends or other income paid before expiration lower the underlying's expected price path, hurting calls and helping puts.

These signs explain the worked profit numbers above: a deep out-of-the-money option is almost all time value, which decays toward zero as expiration approaches if the underlying does not move.

Exam focus

When the stem asks for payoff, ignore the premium; when it asks for profit or breakeven, include it. The call breakeven is X plus the premium; the put breakeven is X minus the premium. Compare S and X from the option holder's perspective. The most common conceptual trap is assuming buyers can lose more than the premium; they cannot, because they can simply let an out-of-the-money option expire worthless.

Writers, by contrast, have collected a fixed premium but may face large adverse moves, so a short (naked) call is the classic example of asymmetric, potentially unlimited downside, while a short put has large but bounded downside because the underlying cannot fall below zero.