Constructing synthetic calls and puts with stock

What the two packages are

A synthetic-option-position is a call or put payoff rebuilt by pairing the opposite option with a long or short stock leg of matching size. An option-spread is a multi-leg package whose net payoff is compared with a single listed option that shares the same strike and expiration.

From listed option to synthetic twin

A long call combined with a short stock position offsets upside from the call against upside risk in the stock, leaving a payoff shape equivalent to a long put. A long put combined with a long stock position reproduces the payoff shape of a long call.

The conversion-leg is the stock purchase or short sale that flips a listed option into the opposite synthetic payoff. Adding or removing stock is the conversion step that moves a position from a listed call to a synthetic put, or from a listed put to a synthetic call.

Parity as the pricing identity

Put-call-parity is the identity that links same-strike, same-expiry call and put values through the underlying, the strike, financing, and dividends. Parity prices the call from the stock, the strike, the put, financing, and dividends, and prices the put by rearranging that same identity.

Standard pricing construction assigns the same volatility to a call and a put that share a strike and expiration, so leftover time value is treated as equal once the other known price parts match.

When the matching payoff is exact

The matching-payoff identity is exact for european-style options that cannot be exercised before expiration. The same stock-plus-option construction is treated as a close approximation for american-style options, which can be exercised early, when meaningful time value remains.

A relative-cost check

A same-strike put offered below the value implied by the listed call can be paired with stock to assemble a synthetic call that costs less than buying the listed call. That comparison is a relative-cost check between the two packages.