Summary

Every option position has a characteristic at-expiration payoff, and the four basic profiles combine into every spread. Put-call parity is the no-arbitrage relationship that ties calls, puts, stock, and carry costs together — it makes synthetics possible and explains why calls and puts of the same strike have near-mirror greeks¹².

Payoff Profiles at Expiration

Position	Max loss	Max gain	Directional bias
Long call	Premium	Unlimited	Bullish
Short naked call	Unlimited	Premium	Bearish/neutral
Long put	Premium	Strike − premium	Bearish
Short naked put	Strike − premium	Premium	Bullish/neutral

Protected positions collapse into each other via parity: a protective put (stock + put) has the profit shape of a long call; a covered call has the shape of a short put².

Put-Call Parity

For European options at the same strike and expiration:

Call − Put = Stock − PV(Strike)      (no dividends)
Call + Strike − Interest + Dividend = Put + Stock   (with dividends)

Intuition: below the strike, a long call and a married put (stock + put) have identical payoffs — but the call ties up no capital, so it carries an interest advantage that pricing must offset. Arbitrageurs force prices to the point where the relationship holds¹. Parity is exact for European options; American early-exercise rights cause small, tradable divergences (deep-ITM puts trading at parity while calls keep time value, and call/put asymmetries near ex-dividend dates)¹².

The Four Synthetics

Adding stock changes only delta — gamma, vega, and theta signs follow the option leg¹.

Real position	Synthetic construction
Long call	Long put + long stock
Short call	Short put + short stock
Long put	Long call + short stock
Short put	Short call + long stock

Example: a 0.55-delta call plus short stock (−1.00) gives a synthetic put at −0.45 delta, tracking the real put. Synthetic long stock = long call + short same-strike put (a "combo"); its effective purchase price is the strike ± net debit/credit¹.

Conversions and Reversals

Structure	Legs	Exposure
Conversion	Long stock + short call + long put	~flat everything; short carry (negative rho)
Reversal	Short stock + long call + short put	~flat everything; long carry (positive rho)

These are the market maker's bread and butter: flatten all risk and harvest bid-ask edges at scale. They carry pin risk at expiration — an unwanted stock position if assignment is guessed wrong — and institutional traders use boxes (a conversion across two strikes, worth the PV of the strike distance) and jelly rolls (across expirations) to warehouse capital and roll inventory¹.

Why This Matters for Premium Sellers

• Parity guarantees the short put and covered call are the same risk under the hood — "get paid for selling insurance" claims must be made with that equivalence in mind.
• Any spread is a combination of these synthetic building blocks; understanding them is prerequisite to structuring defined-risk underwriting positions.

Links

• Option Mechanics
• Edge and Expectancy
• Source depth: Trading Option Greeks Ch 6, put-call-parity topic

Footnotes

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1. Trading Option Greeks, Ch 6 — parity equation, four synthetics, conversions/reversals, boxes, pin risk. ↩↩↩↩↩↩

2. Hull, options-basics topic — parity bounds, American early exercise, spread/combination payoffs. ↩↩↩

3. Trading Option Greeks, Ch 1 — at-expiration diagrams for the four basic positions. ↩