2.2d.4. Synthetic Stocks

Synthetic Stock

A synthetic stock is a portfolio that replicates holding one share of the underlying by combining options with a risk-free position.

Using European options with the same strike $K$ and maturity $T$, put–call parity implies
$C-P=S-K{e}^{-rT}\Rightarrow S=C-P+K{e}^{-rT}.$
So a synthetic long stock is long a call, short a put, and lend $K{e}^{-rT}$ today so the deposit grows to $K$ at $T$. By the law of one price, this synthetic position should be priced the same as the actual stock (up to implementation frictions).

More generally, put–call parity lets you “solve” for other synthetics by rearranging the same identity, so you can replicate a call, a put, or a risk-free loan using the other instruments.