Every structure here is built from four atoms (long call, short call, long put, short put) plus, sometimes, the underlying. Every payoff diagram ever drawn is generated by two rules applied mechanically:
- Rule 1 (payoff). At expiry a long call pays
max(S_T − K, 0); a long put paysmax(K − S_T, 0); a short pays the negative; the underlying paysS_T. There is nothing else. - Rule 2 (profit). Profit is payoff minus net premium paid (or plus net received), where the structure's net premium is the algebraic sum across legs.
Do not memorize diagrams, memorize the rules and derive the diagram, because the moment a question uses a strike or a ratio you have not seen, memory fails and derivation does not. The professional habit is a small table: one row per price level, one column per leg, add the columns. Ninety seconds, and right.
How to find a breakeven and avoid common mistakes
A breakeven is not a shape, it is a root, the price at which profit crosses zero. You find it by writing the profit function in the region where the crossing happens and solving. Long call: above K, profit is S_T − K − c, so S_T* = K + c. Bull call spread: between the strikes, profit is S_T − K₁ − D, so S_T* = K₁ + D. A straddle has two linear pieces and therefore two roots, K ± (c + p). Get the region wrong and the arithmetic is flawless and meaningless.
Two conventions cause more errors than any concept. First, textbook profit ignores financing of the premium, fine for exams, not fine for a real hedge held a quarter at 6.5%, so every place financing matters here carries the premium forward explicitly. Second, the multiplier: a US equity option covers 100 shares; SPX options carry a $100 multiplier; an Indian F&O contract carries an exchange-set lot size that is revised from time to time. Every per-share number must be multiplied by the multiplier and the contract count before it is a currency amount, and a very large share of real-world option errors is exactly this multiplication done once too few or once too many times.
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The eight structures, with their arithmetic
Covered call: long underlying, short a call at K > S₀. Profit = (S_T − S₀) + c − max(S_T − K, 0). BE S₀ − c; max profit K − S₀ + c; max loss S₀ − c. You sold the right tail for cash. "Income" is the most expensive euphemism in retail finance: the honest description is you sold your upside above K for a known amount today, and whether that is good depends on whether c exceeds the value of what you sold, a volatility view, not a yield.
Protective put: long underlying, long a put at K < S₀. Profit = (S_T − S₀) − p + max(K − S_T, 0). BE S₀ + p; max loss K − S₀ − p; upside unlimited less p. Insurance, with a premium paid whether or not the event occurs. The right question is never "does it protect" (arithmetic says yes) but "what does it cost per year": the section on hedging an equity exposure makes that a number.
Collar: long underlying, long put K_p, short call K_c, net premium D = p − c. Profit = (S_T − S₀) + max(K_p − S_T, 0) − max(S_T − K_c, 0) − D. BE S₀ + D; max profit K_c − S₀ − D; max loss K_p − S₀ − D. The workhorse of concentrated-position management: it converts an open-ended holding into a band, and by choosing K_c you can usually make D any number, including zero.
Verticals. Bull call spread (debit D = c₁ − c₂): BE K₁ + D, max profit (K₂ − K₁) − D, max loss D. Bear put spread (debit D = p₂ − p₁): BE K₂ − D, max profit (K₂ − K₁) − D, max loss D. Bull put spread (credit C = p₂ − p₁): BE K₂ − C, max profit C, max loss (K₂ − K₁) − C. Verticals cap both ends: a directional view with a known maximum loss, paid for by selling the tail beyond the far strike.
Straddle: long call and put at the same K, cost c + p. BEs K ± (c + p); max loss c + p at S_T = K. Strangle, K_p < K_c, cheaper, wider: BEs K_p − (p + c) and K_c + (p + c). Both are long volatility, and both lose if the underlying moves less than the market already charged you to expect, the trap that IV crush dissects.
Long butterfly (calls): long K₁, short two K₂, long K₃, equally spaced, debit F = c₁ − 2c₂ + c₃. Max profit (K₂ − K₁) − F at K₂; max loss F; BEs K₁ + F, K₃ − F. A capped-loss bet that the underlying finishes near K₂, the cheapest way to say "the market is overpaying for a move."
Calendar spread: short a near-dated option, long a far-dated one at the same strike. No static expiry payoff (the legs expire at different times); it is a pure position in the term structure of volatility and in theta. Risk reversal, long an out-of-the-money call, short an out-of-the-money put (or the reverse); in FX this is the standard name for a collar and the standard way skew is quoted (see hedging an FX exposure).
Choosing a structure: the direction × volatility grid
Two views, not one, determine the structure. Everyone has a view on direction; the professional also has a view on whether options are cheap or dear relative to the move expected.
| Volatility higher than priced | Volatility ≈ as priced | Volatility lower than priced | |
|---|---|---|---|
| Bullish | Long call; call ratio | Bull call spread (debit) | Bull put spread (credit); covered call |
| Neutral on direction | Long straddle / strangle | , | Butterfly (never a naked short straddle) |
| Bearish | Long put | Bear put spread (debit) | Bear call spread (credit) |
| Hedging a long you own | Protective put | Collar | Covered call |
The left column buys optionality and pays theta daily; the right column sells it, collects theta and carries tail risk; the middle is the honest place to sit with a directional view and no volatility view, and much professional option use lives exactly there, in verticals, precisely because a vertical is far less sensitive to being wrong about volatility than an outright option is. The bottom row is where this program's readers mostly live: you own a business, you have a view on its price, and you are choosing what to give up.