Payoff at Expiration

Four shapes, four break-evens

At expiration an option is worth its intrinsic value and nothing else. A call is worth the amount by which the stock exceeds the strike, floored at zero; a put is worth the amount by which the strike exceeds the stock, floored at zero. From that one sentence the four basic payoffs follow. A long call loses its premium below the strike, breaks even at strike plus premium, and then rises dollar for dollar with the stock. A long put loses its premium above the strike, breaks even at strike minus premium, and rises dollar for dollar as the stock falls toward zero. A short call is the mirror image of the long call — it keeps the premium below the strike and loses without limit above it — and a short put is the mirror image of the long put, keeping the premium above the strike and losing as the stock falls, with a maximum loss reached only if the stock goes to zero. Working the numbers makes the symmetry concrete. On a $48.00 stock the $50 call at $2.40 needs $52.40 — the strike plus the premium — and at $58 it is worth $8.00 of intrinsic value, so the profit is $560 a contract against $240 paid. The $50 put at $4.10 needs $45.90 and at $42 it is worth $8.00, keeping $390 of the $800 it captured. The asymmetry between those two outcomes is the market’s opinion, not an accident: the put cost more, so the same eight-dollar move pays less. And the two short sides deserve their own line, because the “premium collected” framing hides the shape: a short $55 call for $0.85 keeps $85 if the stock stays below $55 and then loses $100 for every dollar the stock rises above it, which is why the credit is the least informative number on the ticket. Two consequences are worth carrying out of the beginner rung. The first is that the payoff is a picture of outcomes rather than probabilities: a long call’s unlimited upside and a short put’s near-total loss are both drawn on the same diagram that shows the premium, and the diagram says nothing about how likely either is. That is why the payoff and the odds have to be read together, and why the later lessons put implied volatility and probability beside it. The second is that every structure in the subject is a sum of these four curves. A vertical is a long call plus a short call; an iron condor is four; a covered call is a long share position plus a short call; the wheel is a short put then a short call. Learning the four shapes is therefore not a beginner exercise that is left behind, it is the algebra that all the rest of the subject is written in. Four positions on a $48 stock — Long $50 call at $2.40: Break-even $52.40 · loss capped at $240 · profit $560 at $58 · Long $50 put at $4.10: Break-even $45.90 · loss capped at $410 · profit $390 at $42 ← · Short $55 call at $0.85: Keeps $85 below $55 · loss open-ended above it · Short $50 put at $4.10: Keeps $410 above $50 · loss grows all the way down to zero ← Break-even is strike plus premium for a long call and strike minus premium for a long put. The writer’s break-even is the same number from the other side, because the two payoffs are reflections.

What the diagram hides

The payoff diagram is a picture of the outcome at one instant — the close on expiration day — and three things fall outside it. The first is time: before expiry the position has extrinsic value, so the same contract can be sold for more than the diagram shows, which is the entire reason an early exit can beat holding to the final bell. A long call that is only $1.00 in the money with two weeks left is worth more than $100 of intrinsic value, and the difference is what a trader is actually buying and selling before expiry. The second is the path. The diagram cannot distinguish a stock that drifted from $48 to $58 over two months from one that gapped there overnight, and the two experiences are completely different for the holder: a drift allows the position to be managed, a gap does not. That is why the risk of a long option is not really the premium — it is the premium, and the inability to act on the way. The third is the writer’s capital. A short put’s payoff looks tame because the loss only reaches its maximum in a total collapse, but the cash that has to be reserved to buy the shares, and the margin that can be called while the position is open, are not on the diagram at all. The honest way to read a payoff is therefore to add two annotations to it: what the position requires in cash or collateral, and where the invalidation is — the price at which the thesis, rather than the position, is wrong. • At expiration: intrinsic value only, so every payoff is straight lines. • Long call break-even = strike + premium; long put break-even = strike − premium. • Short premium positions are the mirror images: capped gain, and the loss is the other side’s. • The diagram shows one instant — it says nothing about time value, the path, or the capital required. • Before expiry, extrinsic value means a position can be worth more (or less) than its payoff shows. The most common beginner error is reading a short option’s payoff as safe because the gain is capped and “known”. A short put writes unlimited downside against a bounded credit, and a short call does the same with no floor at all. The shape of a writer’s loss is the shape of the buyer’s gain, and the buyer paid for it — so the question to ask is never “how big is the credit”, but “what would I have to own, and at what price, if this finishes against me”.

Payoff arithmetic: adding the shapes

Because every payoff is straight lines and flat lines, combining positions is arithmetic rather than geometry. Build a sum by working left to right across the price axis and tracking two things at each strike: the height of the line, and its slope. The slope is the position’s exposure to the next dollar of stock movement, so a naked long call is flat below its strike and rises one-for-one above it, while a naked short call is flat below and falls one-for-one above. Strikes are the only places the slope can change, and between them the payoff is straight — which is what makes the diagram readable at a glance. That gives the whole subject a mechanical procedure. Take a long $50 call and a short $55 call, one contract each. Below $50 the slope is zero, because nothing is in the money. Between the strikes the long call is climbing while the short one is still flat, so the slope is plus one. Above $55 the long is climbing and the short is falling, so the slopes cancel and the line goes flat. The result is the classic bull call spread — a flat loss below, a linear climb in between, a plateau above — and none of it required a picture, because the slope bookkeeping produced the shape. The same method explains the structures that look complicated. Add a long put to a long call at one strike and the flat regions cancel above and below, leaving a V whose slope is minus one below the strike and plus one above it: a long straddle. Add a long call at $50 and a short put at $45 and the shape becomes linear almost everywhere, which is the synthetic long stock of the parity lesson, and it is why that position carries the same leverage as shares bought on borrowed money. An iron condor is four kinks and four slope changes, and it can be assembled, checked and priced as four additions rather than memorised as a picture. Two habits follow. First, predict the number of kinks before drawing anything: a structure drawn across *n* distinct strikes has *n* changes of slope, and a diagram showing more or fewer means an arithmetic slip somewhere. Second, read the slopes as exposures and the plateaus as caps. The best and worst outcomes are simply the highest and lowest heights the piecewise line reaches at its flat ends, which turns the earlier instruction about finding the worst case into something you can compute rather than hunt for. • Between strikes the payoff is straight; only at strikes can the slope change. • Track the slope as exposure to the next dollar: zero flat, plus one stock-like, minus one short. • A long call plus a short call above it is a vertical: flat, then linear, then plateau. • A long call plus a long put at one strike is a straddle; a call plus a short put is synthetic stock. • Count the kinks before drawing: *n* strikes means *n* slope changes, and anything else is an error. This is the algebra that makes the later structures legible. A wheel is a short put whose outcome turns into a long share position, a collar has three kinks, and an iron condor has four — all of them read by adding shapes that were introduced as four lines in this lesson.

Sources

• Long and short call and put payoff diagrams, with break-evensMcMillan, “Options as a Strategic Investment”
• Intrinsic value at expiration and the geometry of the payoffCBOE Education — Options Payoff Diagrams
• Payoff symmetry and why the writer’s profile is the mirror imageNatenberg, “Option Volatility and Pricing”