Synthetic stock and protective put payoff construction

Signed holdings and the riskless book

Holdings are signed so a long position counts as a plus and a short position counts as a minus when profit or loss is combined across legs. Under that signing, a book that is long the stock, long the put, and short the call has a combined profit or loss of zero that does not change with the underlying price.

That combination is the riskless three-security book. The combined profit or loss of the book equals the sum of the profit or loss on each individual holding.

Synthetic stock from a long call and short put

Buying a call and selling a put at the same exercise price is a synthetic option position whose profit-or-loss profile matches owning the stock. That package is synthetic stock.

In that synthetic-stock package, the long call receives gains above the shared exercise price and the short put absorbs losses below it.

The protective put as a synthetic call

A long stock plus a long put, called a protective put, is a synthetic option position whose profit-or-loss profile matches a long call. That package is a synthetic call.

A protective put holds the long put against long stock so downside is bounded. It is the live construction of the synthetic call.

Rearrange the identity to isolate a payoff

Rearranging the zero-sum three-security identity isolates either the stock payoff or the call payoff as a constructed package of the remaining holdings. The same-exercise-price long call and short put reconstruct the stock. The long stock and long put reconstruct the call.

A synthetic option position is a constructed book whose profit-or-loss profile is meant to match another security by pairing options or pairing stock with an option. Editorial: finish that substitution check before the package is treated as an entry, hedge, or abstention rule of its own.